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March 8, 2010

A Perspective on Higher Category Theory

Posted by Tom Leinster

A few of us here at the Café decided that it would be good to have a short series of posts in which each of us (at least, each of us who wants to) says something about his overall take on higher category theory. So, this is big-picture stuff, not nitty-gritty. Urs effectively kicked the series off. Here’s my contribution.

I also want to take this opportunity to pay tribute to John’s incredible activities in higher-dimensional category theory. One of the first web pages I ever laid eyes on was one of his. It was just before I started my Ph.D., higher category theory had yet to enter my life, and Richard Thomas was showing me this thing called the ‘World Wide Web’. As I remember it, he typed ‘category theory’ into AltaVista and up came an issue of This Week’s Finds all about nn-categories.

For 15 years now, John’s been inspiring people to go and work on higher category theory; he’s been patiently explaining the basic ideas over and over again; he’s been making famous Hypotheses that shape the current mathematical landscape; he’s been categorifying everything in sight. Simply, he’s been an enormous influence on the subject. Now he’s moving on to other things. John, we salute you! Give that man a round of applause.

I’ll begin with a warning: this post is rambling and very long. Apparently I had a lot to say. For seven years I was obsessed with higher categorical structures. That obsession faded to an interest a while ago, and I spend most of my time now in other parts of mathematics. But still, higher category theory has played a large enough part in my life that questions such as ‘what do you think of the current state of higher category theory?’ inevitably have me asking myself larger questions such as ‘what do I think about category theory?’ and ‘what do I think about mathematics?’ Consider yourself warned!

What is category theory?

In my mind, ‘category theory’ has two meanings. The first is the obvious one. But sometimes I’ll be sitting in a category theory conference, listening to a category theorist talking about categories, and I’ll think to myself ‘that’s not category theory’. Then, I’m using ‘category theory’ in the second sense.

So what is this second meaning of ‘category theory’? I’ll explain with an example: Buffon’s noodle.

As you know, the Comte de Buffon’s château was floored with boards of equal width. The Comte had a needle that happened to be precisely as long as the floorboards were wide, which he enjoyed flinging to the floor at random. ‘What,’ he asked himself, ‘is the probability that my needle will land across a crack?’

There are many ways to solve the problem. I’m going to give you Barbier’s 1860 solution, which I learned from Klain and Rota’s book Introduction to Geometric Probability (1, 2, 3). This is one of the most beautiful pieces of mathematics I’ve learned in the last few years.

The first step is to generalize. Consider not just straight needles, but curved ones too.

The second step is to generalize again. Consider not just needles as long as the boards are wide — one unit, say — but curved needles of arbitrary length (‘noodles’).

The generalizations have to be steered in the right direction, though. Instead of asking ‘what is the probability that the needle crosses a crack?’, we ask ‘what is the expected number of times that the needle crosses a crack?’ This is a generalization, since a straight needle of unit length almost surely crosses at most one crack, and for an event that occurs at most once, the expected number of occurrences is simply the probability that it occurs at all.

So now we’ve got a much more general problem: for a curved needle of arbitrary length, what is the expected number of times that it crosses a crack?

To answer this, observe that if two curved needles are welded end to end, the expected number of crossings for the composite needle is the sum of the expected numbers of crossings for the two original needles. (The expectation of a sum of random variables is the sum of the expectations, even if the variables aren’t independent.) So by thinking of any given curved needle as the welding-together of a large number of small straight needles (and choosing a suitable definition of ‘curved’), we see that the expected number of crossings is proportional to the length.

So all that remains is to calculate the constant of proportionality. A circular needle of unit diameter always crosses the cracks exactly 2 times, and has length π\pi, so the constant is 2/π2/\pi. Hence for Buffon’s original, unit-length, needle, the probability that it crosses a crack is 2/π2/\pi.

I claim that this is a superb piece of category theory in the second sense. Of course, this is absurd: there’s not a category in sight. But consider the following properties of Barbier’s argument, and perhaps you’ll see why I want to call it category theory:

  • It’s highly conceptual, with almost no calculation. (Contrast other proofs.) You could easily explain it out loud to someone while walking in the park. The only point when you might be tempted to reach for pen and paper is the last step, calculating the constant of proportionality, but even that makes only a small demand on your powers of mental algebra.
  • It solves the problem by moving to a higher level of generality — and getting the right level of generality, at that.
  • The generalization wasn’t quite the most obvious one. Good generalization almost always begins by carefully preparing the ground. Here, this was the little observation that for events that occur at most once, expected value is the same as probability. A more obvious generalization would be to ask what the probability of crossing was for needles of arbitrary length.
  • We see uniform behaviour across a class of objects (curved needles of arbitrary length). Under other circumstances this would be called functoriality or naturality.
  • There is a piece of grit around which the pearl forms. You can never solve a specific problem by entirely general means: at some point, you have to make the link between the generalities and the specific situation that you started with. Here, the grit is the calculation of the constant 2/π2/\pi, got by thinking about a circular needle.

All these points are characteristic of (what I regard as) good category theory. Perhaps I should use a vague term such as ‘conceptual mathematics’ rather than ‘category theory’, if I want to apply it to non-categorical arguments such as this. I’m not too bothered about the choice of words; I only want to give an impression of a type of mathematics that I particularly value.

What isn’t category theory?

In a well-known piece of mathematical bitching, Miles Reid described the study of category theory for its own sake as

surely one of the most sterile of all intellectual pursuits

(Undergraduate Algebraic Geometry, p.116). I’ve become rather fond of that quotation, though not for the reason that Reid intended. ‘Sterile’ doesn’t only mean infertile or unproductive. It’s also what you want surgical instruments to be: clean, uncontaminated, disease-free. No one wants to be operated on with a dirty scalpel.

I’m going to clarify that point, because I can hear a sarcastic voice saying ‘oh, so you want to do category theory uncontaminated by real mathematics’. Category theory is at its best, I think, when it reflects what mathematicians actually do and shows how to do it better, when it functions as the mathematics of mathematics.

It doesn’t contradict this, though, that category theory has its own internal structure, concepts, and aesthetics. If a new categorical concept only seems natural from the point of view of one particular application, that’s dirtying the scalpel. (A categorical concept that only bears on one part of mathematics isn’t really a categorical concept at all.) If there’s a break in the conceptual trail — an ad hoc hypothesis or definition — that’s dirtying the scalpel too. I’ll give some examples in a moment.

These judgements are reversible: someone might come along later and show that the thing that looked so tailored to one particular application is, when viewed from the right angle, a completely natural concept. I’ll give an example of that, too. Category theory works because it’s clean, uncontaminated, sterile.

In other words, I’m about to explain just how fussy I am.

I’ll do this by listing some examples of what I call contamination. Since I don’t want to publicly criticize other people’s work, I’ll draw the examples either from my own work, or from parts of mathematics that seem old enough now that they’re common property.

A break in the conceptual trail  A good piece of category theory should be an unbroken conceptual trail that you can follow from beginning to end. There should be no conceptually dubious stretches. Barbier’s solution to the needle problem lives up to this gloriously. Here’s something that doesn’t.

For a few years now I’ve been very interested in notions of size. Central to this is the definition of the cardinality or Euler characteristic of a finite category. Now, on the one hand I’m almost certain that the definition is ‘right’: it satisfies all the criteria that I think it should, and it seems to produce satisfying results when used in several different areas of pure and applied mathematics (as demonstrated by ongoing joint work with Clemens Berger, Christina Cobbold, Mark Meckes, Catharina Stroppel and Simon Willerton.)

But on the other hand, the situation is not quite satisfactory. As a prelude to the definition of the Euler characteristic of a category, one has to define the notion of a ‘weighting’ on a category. Where does this definition come from? What is the conceptual justification? I don’t really know. My own motivation was from one particular application, on the combinatorics of finite sets. When I’m giving talks, the best I can do is to explain how I came to the notion of weighting (knowing full well that not everyone will be moved by this particular motivation) and draw some vague analogies. I can’t explain it by abstract considerations. It’s a conceptual gap.

A clever little thing  I’ve learned to be wary of anything in category theory that provokes the word clever. I don’t mean that I like my category theory stupid, of course. But if I ever catch myself thinking of something I’ve done as ‘my clever trick’ (and I have), then I know I’m in trouble. Each of those three words should ring alarm bells. It’s great for mathematics to be visionary, beautiful, etc., but if cleverness is the first quality that comes to mind then it suggests to me that something is insufficiently understood. When good conceptual mathematics has been done, everyone should feel that it belongs to them. It’s simply part of the general understanding, part of the landscape.

An unaesthetic notion  Here’s a demonstration of how fussy I am: I’m suspicious of the notion of subcategory. Actually, I feel like I’m compromising by calling that ‘fussy’, because for ages I assumed that anyone with a background in category theory would be equally suspicious. I assumed that was normal. But apparently it’s not.

So what’s wrong with subcategories? First there’s an almost sociological point: while subgroups are enormously important in group theory, and linear subspaces are enormously important in linear algebra and functional analysis, subcategories simply don’t feature so prominently in category theory. They’re a bit of a second-rate notion. Generally, instead of thinking about subcategories, we think about faithful functors, or functors reflecting isomorphism, etc. Sometimes you’ll start by contemplating some property of a subcategory, such as fullness or density, but then realize that the natural level of generality for the property isn’t ‘subcategory of a category’: it’s ‘functor’ — including, but not limited to, subcategory inclusions.

Then there’s a more mathematical point. Inclusion functors of subcategories are characterized by being faithful and injective on objects. When I say ‘injective on objects’, I mean injective up to equality. This is what is known in these parts as evil, and while I’m not totally on board with that usage, it should be enough to make one stop and think ‘am I really doing this right?’ If the condition were ‘full and faithful’ (as for full subcategories) or ‘faithful and reflects isomorphisms’, then at least these are wholesome conditions — though I’d still want to ask whether they can be shed. But ‘injective on objects’ is a pretty bizarre condition.

Here’s an example of how the notion of subcategory can lead to what I regard as poor-quality category theory. Again, the example is from my own work.

I spent a while developing a notion of ‘(strong) homotopy algebra’ for an operad — a non-algebraic notion, for those who know what I mean by that, in contrast to the previous, algebraic, notions such as Lada’s and Markl’s. (Now, at last, we’re edging into higher category theory.) I’ll explain it in the case of homotopy monoids, since that’s enough to get the point across.

It begins innocently enough. Let Δ\Delta be, as usual, the category of nonempty finite ordinals, and let D\mathbf{D} be the category of all finite ordinals. Thus, D\mathbf{D} has one more object than Δ\Delta, and is a monoidal category under ++. It’s easy enough to show that a simplicial set is the same thing as a (covariant) colax monoidal functor X:DSet. X: \mathbf{D} \to \mathbf{Set}. Colax monoidal means that XX comes equipped with structure maps ξ m,n:X(m+n)X(m)×X(n),ξ :X(0)1 \xi_{m, n}: X(m + n) \to X(m) \times X(n), \qquad \xi_\cdot: X(0) \to 1 satisfying some coherence axioms. The same is true if you replace Set\mathbf{Set} by any other category with products. In particular, it’s true for Top\mathbf{Top}.

Now, a monoid in Top\mathbf{Top} — a topological monoid — is a colax monoidal functor (X,ξ):DTop(X, \xi): \mathbf{D} \to \mathbf{Top} for which the maps ξ m,n\xi_{m, n} and ξ \xi_\cdot are isomorphisms. So, we might define a homotopy monoid in Top\mathbf{Top} to be a colax monoidal functor (X,ξ):DTop(X, \xi): \mathbf{D} \to \mathbf{Top} for which the maps ξ m,n\xi_{m, n} and ξ \xi_\cdot are homotopy equivalences. This turns out to be a good definition. In fact, it is equivalent to Segal’s much older notion of ‘special simplicial space’: ξ m,n\xi_{m, n} and ξ \xi_\cdot are homotopy equivalences if and only if the famous ‘Segal maps’ of the corresponding simplicial set are homotopy equivalences. But the advantage is that (unlike with Segal maps) the definition of homotopy monoid is sensible even when the monoidal structure on the category of ‘spaces’ is not cartesian. This happens, for instance, if you want a notion of homotopy differential graded algebra; then your monoidal category of ‘spaces’ is chain complexes, with the tensor product.

Good as this may seem, danger is looming. For what is the general definition of ‘homotopy monoid’ implicit in the last paragraph? Well, we have some category \mathcal{E}, thought of as the category of spaces, we have a distinguished class of maps in \mathcal{E} called ‘equivalences’, and we define a homotopy monoid in \mathcal{E} as a colax monoidal functor (X,ξ):D(X, \xi): \mathbf{D} \to \mathcal{E} for which the maps ξ m,n\xi_{m, n} and ξ \xi_\cdot are equivalences. So the context for the definition is: a category \mathcal{E} equipped with a subcategory, made up of all the objects but only the ‘equivalences’ between them. For everything to function correctly this subcategory should satisfy a couple of axioms, but the damage is already done. We’ve gone and dirtied the scalpel with a subcategory. I find this unnatural and unsatisfactory, and I never managed to re-do it to my satisfaction.

A suspended judgement  Sometimes a concept can be presented in a way that makes it look unnatural; but that doesn’t preclude the possibility that it really is natural.

For example, for a long time I couldn’t see where the notion of simplicial set came from. It was clear that simplicial sets played an important role in current mathematics, but what was the motivation for the concept? Sure, they’re presheaves on nonempty finite totally ordered sets, but why that particular combination of qualifiers, ‘nonempty’, ‘finite’ and ‘totally’? Why would you think of putting any kind of order on the vertices of a simplex? It was hard for me to believe that simplicial sets were an ad hoc notion; all the same, I couldn’t see where they came from abstractly.

But then I figured out a couple of things that convinced me that Δ\Delta was a canonical object of mathematics. I’ll tell this story very briefly. First, there is a canonical process that assigns to each nonsymmetric operad PP a monoidal category D P\mathbf{D}_P and a category Δ P\Delta_P. When PP is the terminal operad, D P=D\mathbf{D}_P = \mathbf{D} and Δ P=Δ\Delta_P = \Delta. (Specifically: D P\mathbf{D}_P is the free monoidal category on a PP-algebra, and contravariant functors from Δ P\Delta_P to a finite product category \mathcal{E} correspond to colax monoidal functors D P\mathbf{D}_P \to \mathcal{E}.) Second, there is a canonical process that assigns to each functor U:𝒜U: \mathcal{B} \to \mathcal{A} satisfying certain conditions a (full, dense) subcategory Δ U\Delta_U of \mathcal{B}. When UU is the forgetful functor from Cat\mathbf{Cat} to directed graphs, Δ U=Δ\Delta_U = \Delta. (Further details are here.)

So, while I really do care about keeping my categorical instruments scrupulously clean, I also recognize that something might initially look like a contaminant but not be one, after all. (The metaphor breaks down around here.) As with Δ\Delta, the most interesting situation is when all the social signs point to something being a clean or canonical notion, but there is as yet no good abstract explanation of where it comes from. That’s when work needs to be done.

What is an nn-category?

Now there’s a question I’d like to see answered.

Of course, lots of people have provided fragments of answers. In the second half of the 1990s there was a huge wave of proposals for how to define ‘nn-category’. One striking thing about this was the astonishing diversity of the contributors, and their reasons for contributing. There’s John Power, trying to develop nn-categories to advance computer science. There’s Todd Trimble, proposing a definition of nn-category that fits snugly into algebraic topology. There’s Michael Makkai, wanting to use \infty-categories as a logical foundation for mathematics. There’s Carlos Simpson from algebraic geometry; there’s John Baez from mathematical physics.

What matters in the long term, though, is not your motivation but the mathematics you leave behind. And on this score, too, the diversity was striking. The conceptions of higher category embodied by these proposals were highly varied, some of them very different from the conceptions of higher category prevalent today. For example, Penon elegantly defined the monad for \infty-categories in a few lines, characterizing it by a universal property. In doing so, he carefully positioned the theory of \infty-categories in the landscape of all possible algebraic theories. There is nothing overtly homotopical or simplicial in this conception. Instead, we have a notion of \infty-category motivated in terms of universal algebra, with beautiful conceptual clarity.

During this period, new proposed definitions of nn-category seemed to be popping up at a rate of one every few months, and some people started throwing up their hands and saying ‘Not another definition of nn-category! That’s the last thing we need!’ My own feeling was the opposite. I wanted to encourage the diversity, and I put a lot of energy into trying to persuade people that they shouldn’t view the various definitions as competing, but rather as complementary ways of accessing this important concept. That’s part of the reason why I wrote a survey of the proposed definitions that existed at the time. Gathering together many different ways of looking at one hard-to-reach concept — nn-category — can help us to discern the common factors. It’s only by coming at a concept from several different angles that we can gain the rounded understanding that is part of good category theory. We still don’t have a rounded understanding of what an nn-category is.

I need to say something about algebraic and non-algebraic notions of nn-category, because I’ll be using these words later. Here and everywhere else, n{}n \in \mathbb{N} \cup \{\infty\} and ‘nn-category’ means ‘weak nn-category’.

Roughly speaking, proposed definitions of nn-category fall into two groups. In the first, the algebraic, an nn-category is defined as an nn-globular set (nn-graph) equipped with extra structure. Other ways of saying more or less the same thing are that the category of nn-categories and strict nn-functors is monadic over nn-globular sets, or that nn-categories are the models in Set\mathbf{Set} for a finite limit theory. These three formulations aren’t precisely equivalent, and algebraic is only a semi-technical word, but I’ve never met a proposed definition of nn-category for which I’ve had trouble deciding whether it’s algebraic or not.

The second group is everything else — the non-algebraic notions of nn-category. Almost all such definitions are of the form ‘an nn-category is a presheaf on such-and-such a category satisfying such-and-such conditions’.

There’s a general principle telling us that any algebraic notion of nn-category can also be understood as a non-algebraic notion. (See the last two pages of these slides.) So the class of algebraic notions embeds into the class of non-algebraic notions. This, then, is the mathematical ‘non’: ‘non’ as in ‘non-commutative ring’, really meaning ‘not necessarily’. Just as commutative rings can be regarded as very special types of ring, the algebraic definitions can be regarded as very special notions of nn-category.

Progress in mathematics sometimes entails throwing away everything we thought we knew. Often, though, bodies of mathematics fall into obscurity through the forces of fashion or historical accident, not as a result of rational judgement. Mathematics has a constant tendency to fragment; mathematicians have a constant tendency to march off, with great speed and enthusiasm, in opposite directions to each other. One of the justifications for the existence of category theory is that it counteracts this fragmentation. Part of a category theorist’s job is to be a curator of unfashionable causes.

What is the Homotopy Hypothesis?

There’s an old joke:

Q.  How do you prove the Homotopy Hypothesis?

A.  Define an \infty-groupoid to be a Kan complex; define a space to be a Kan complex; done!

I’m going to explain what the Homotopy Hypothesis is (or how I think of it), and what would qualify as a satisfactory solution. Among other things, I’ll take that joke solution seriously and try to explain what, in fact, is wrong with it.

The Homotopy Hypothesis says, informally, that

\infty-groupoids are the same as spaces.

A more refined version is: for each n{}n \in \mathbb{N} \cup \{\infty\},

nn-groupoids are the same as nn-types.

The first version is the case n=n = \infty of the second, since ‘\infty-type’ is meant to mean the same as ‘space’.

All the terms in the Homotopy Hypothesis — ‘\infty-groupoid’, ‘same as’, and ‘space’ — are up for negotiation. Various interpretations of them have been suggested. Evidently, we can’t say that we fully understand the Homotopy Hypothesis until we fully understand the terms in its statement. This would mean, in particular, fully understanding what an \infty-groupoid is. Of course, this is closely related to the challenge of understanding what an \infty-category is, discussed in the previous section.

My mental picture of the Homotopy Hypothesis looks something like this:

Algebraic versus topological

I’ve already explained the contrast between algebraic and non-algebraic notions of \infty-groupoid. Actually, I explained it for nn-categories, but the issues are the same. For example, in Grothendieck’s Pursuing Stacks — one of the first written formulations of the Homotopy Hypothesis — the concept of \infty-groupoid is an algebraic one; Grothendieck specifies that they are to be a finite limit theory.

Now I need to say something about notions of space.

People can be a bit sniffy about general topological spaces. Personally, I think it’s remarkable that a single definition encompasses so many very different types of ‘space’. As some us were discussing recently at Math Overflow, topological spaces can very crudely be divided into three classes:

  1. the spaces that algebraic topologists think most about (CW-complexes, roughly)
  2. the ‘spectra’ (in various senses, but excluding that of homotopy theory)
  3. the infinite-dimensional spaces that arise in functional analysis (e.g. Banach spaces).

In algebraic topology there has been detailed study of the process of taking a space in Class 1 and moving it leftwards — away from topology and towards algebra — by equipping it with a combinatorial structure such as a triangulation or a cell decomposition. The contrast, then, is between purely topological notions of space (such as the general notion of topological space, or topological spaces satisfying some convenient condition) and more combinatorial or algebraic notions of space (such as simplicial set). This is what I mean by ‘algebraic’ and ‘non-algebraic’ notions of space. I’m using these terms pretty vaguely, and I don’t claim that there’s the kind of definite distinction that there is between algebraic and non-algebraic notions of \infty-groupoid. All the same, I hope it’s not so vague that you can’t see what I mean.

More vaguely still, the right-hand end of the line represents topological spaces as conceived by someone who is not thinking at all algebraically or combinatorially. It’s pure continuity; it’s the image of a sphere that forms in the mind of a child; it’s the feeling of nearness that we get as we move on a paint chart from ‘tropical aqua’ to ‘marine blue’ to ‘ocean azure’. The middle region of the line is where simplicial structures live: simplicial notions of space and simplicial notions of \infty-groupoid. For example, the algebraic Kan complexes that Thomas told us about recently are located in the right-hand part of the region labelled ‘algebraic notions of infinity-groupoid’. The left-hand end of the line represents \infty-groupoids as conceived by someone who is not thinking at all topologically. An \infty-groupoid is a globular set equipped with some operations satisfying some equations.

To make the Homotopy Hypothesis stand a chance of being true we need to restrict the word ‘space’ to mean something like ‘space of Class 1’, since these are the spaces that can be probed effectively by the real interval. I’ll use the word ‘space’ in this restricted way from now on.

A full-strength statement of the Homotopy Hypothesis will say, then, that

\infty-groupoids in a fully algebraic sense

are the same as

spaces in a fully non-algebraic sense

and similarly with ‘nn-groupoids’ in place of ‘\infty-groupoids’ and ‘nn-types’ in place of ‘spaces’. In other words, it matches up the leftmost and rightmost ends of the line; it shows that the whole thing can be collapsed to a point. Diluted statements of the Homotopy Hypothesis might use a non-algebraic notion of \infty-groupoid or an algebraic notion of space (or, indeed, a diluted notion of sameness). The extreme of dilution is the joke solution.

So what’s wrong with the joke solution? It’s a question worth taking seriously. In fact, someone asked me exactly this question, differently phrased, a couple of years ago at a conference in Split. Why not define an \infty-groupoid to be a Kan complex? Why not define a space to be a Kan complex?

My answer calls on the general points I made about category theory earlier. Good categorical definitions are not tailored to any particular application. Their strength is in their flexibility, their adaptability to multiple parts of mathematics, some as yet unknown. You could define an \infty-groupoid to be a Kan complex, but that would be to emphasize the topological and simplicial aspects of \infty-groupoids as the expense of other aspects. For example, if you want to think of \infty-groupoids as globular sets with structure then this definition is immediately un-ideal; even taking the underlying globular set of a Kan-complex-as-\infty-groupoid is a somewhat worrisome process. Similarly, you could define a space to be a Kan complex, but that would be to emphasize the simplicial aspects of topology above all others. We don’t always want to think of a space as a simplicial set. Kan complexes, and simplicial structures in general, do seem to be important; but not everything in life is naturally simplicial.

What would I like to see?

I spent a long time above explaining my taste in category theory. Here I’ll add a few things more particularly about higher category theory, and finish with some more personal comments.

The ultimate excitement would be a conceptually satisfying theory of categorification. This would include a conceptually satisfying theory of higher categories. At the time when I stopped focussing on higher category theory, I had a feeling that we — humanity at large — were missing at least one big idea about higher categories. I have that feeling still.

Really this encompasses everything else, but I’ll describe some aspects that I’d especially like to see.

One would be learning how to stop handwaving over the word ‘the’. This issue will be familiar to most people who’ve had much contact with category theory. We say ‘the product of two sets’, for example, and we reason as if it really were a definite set rather than an isomorphism class of sets, and we justify this by the fact that products are unique up to canonical isomorphism. Nevertheless, the act of choosing a product for each pair of sets is distasteful and sometimes inconvenient. The issue is both magnified and brought into focus in higher category theory. We all feel that we know what we mean by this ‘generalized the’, but handwaving is what it is. Here’s Rota again:

The simplest, and most remarkable, application of the definition of species is the rigorous combinatorial rendering of functional composition, which was formerly dealt with by handwaving — always a bad sign.

This is from Rota’s foreword to the book of Bergeron, Labelle and Leroux, Combinatorial Species and Tree-Like Structures. It’s no coincidence that the remedy for handwaving that Rota was talking about was also a piece of category theory (Joyal’s beautiful notion of species). Category theory has a long history of curing handwaving, from ‘natural transformation’ onwards.

I’d also like to see advances in our understanding of the nebulous concept of ‘sameness’. This is in the province of \infty-groupoid theory, and is related to the previous paragraph: any two products of a pair of objects are ‘the same’, for instance. I think the most precise expression of the idea that we’ve seen so far has been in work connecting higher category theory with type theory, such as the work of Awodey, van den Berg, Garner, Lumsdaine and Warren. I know regrettably little about this, and my outsider’s impression is that even those carrying out this work consider the story to be just beginning. (I hope they’ll correct me if I’m wrong.) But it would be great to see this go further and deeper.

I’ll finish with a couple of more personal things. For the seven years that I was working intensively on the subject, almost everything in higher category theory fascinated me. Now that the subject’s grown and I’ve moved on, it’s inevitable that that’s no longer the case. Some parts sing to me; some just don’t.

During that time I had a lot of energy for what you might call ‘refinement’. At the time (and maybe still) many papers on higher categorical structures contained excellent ideas, but obscured by what seemed to be hair-raising complications. I don’t mean this as a slur: original work often is messy. I put a lot into trying to refine this raw material, simplify, reshape, bring out the categorical essence. It’s a wholly worthwhile pursuit, I think, but it’s also very time-consuming. Nowadays I’m directing my energy elsewhere, so if a piece of work in higher category theory looks like a mess then I’ll probably just ignore it and think about something more appealing, rather than going in to try to clean things up as I would have done previously.

I don’t think I’m very mathematically goal-oriented. My view is that human beings know only epsilon about mathematics, and will still know only epsilon about mathematics on the day the universe dies. There’s an infinite amount of mathematics out there, and I’m more interested in getting things done right than getting them done fast. We’ll never get to the end anyway. Of course, ‘right’ is highly subjective, and probably just means ‘in a way that I find satisfying’. But this is roughly the reason why I’ve never had a problem with the idea of spending a long time making definitions without ever proving a theorem. I value the art of laying good foundations, and on most days I’m not in a hurry to go building on them before they’re ready. Other people are different; it’s a personal thing. And I now find myself thinking of Douglas Adams in The Hitchhiker’s Guide to the Galaxy:

Orbiting this at a distance of roughly ninety-eight million miles is an utterly insignificant little blue-green planet whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea.

Posted at March 8, 2010 12:22 AM UTC

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Re: A Perspective on Higher Category Theory

Very interesting! Thanks for continuing this idea.

I, too, like thinking of category theory as “conceptual mathematics” in your sense. I disagree, however, that “a categorical concept that only bears on one part of mathematics isn’t really a categorical concept at all,” because in practice, the worst we could ever say is that some categorical concept is only known to bear on one part of mathematics so far. You admitted that the judgments are reversible, but I don’t think we should exclude natural categorical concepts from category theory just because no one has yet found an application somewhere else. Of course I haven’t said what I mean by “natural,” and I probably can’t, but to me it definitely has a meaning that can still apply to a concept with only one known application.

I find your example of homotopy monoids, and an “unclean” use of subcategories, very interesting, especially in the light of what Urs had to say in his post about how (,1)(\infty,1)-categories are a category-theoretic conceptual reformulation of the old kludgy concept of model categories. A model category is surely just as bad as a category with only weak equivalences, from your point of view, and from the (,1)(\infty,1)-categorical perspective any sort of “homotopy monoid” is just a kludgy model-category-like way of getting at the underlying “natural” notion of a “monoid in a monoidal (,1)(\infty,1)-category.” But I’ve gathered from other things you’ve written that you aren’t as enthralled with the (,1)(\infty,1)-world — can you say anything about why this doesn’t appeal to you as a categorical version of “homotopy monoid”?

A totally different response to the question of homotopy monoids would be to replace the subcategory of weak equivalences by a functor which is faithful and essentially surjective. If all isomorphisms are weak equivalences, then you could also ask the functor to be pseudomonic. Does that idea run into problems?

We all feel that we know what we mean by this ‘generalized the’, but handwaving is what it is.

I don’t agree that the “generalized the” is handwaving. What does the ordinary “the” mean, anyway? In ordinary mathematics, when we say “let x be the y” we really mean “we’ve proven that there exists a unique y, so we might as well assume that we have a y, and let’s call it x.” This is really no different from saying “let x be a y” to mean “we’ve proven that there exists at least one y, so we might as well assume that we have a y, and let’s call it x,” except that in this case, then later on we can use the fact that any other y is equal to x, if we so desire. The “generalized the” is completely analogous: when we say “let C be the product of A and B,” we mean “we’ve proven that there exists a product of A and B which is unique up to unique specified isomorphism, so we might as well assume that we have a product of A and B, and let’s call it C,” additionally reserving for ourselves the right to use, later on, the fact that any other product of A and B is canonically isomorphic to C.

This is all completely precise, or can be made so in whatever precise-making formal theory you like. All that’s different is that we’re using the English word “the” to refer to a more general mathematical situation than previously.

Posted by: Mike Shulman on March 8, 2010 2:59 AM | Permalink | PGP Sig | Reply to this

Re: A Perspective on Higher Category Theory

Mike wrote:

I’ve gathered from other things you’ve written that you aren’t as enthralled with the (,1)(\infty, 1)-world

I think that’s a misunderstanding. OK, right now I haven’t been awake long and I’m hugging a cup of coffee with half-open eyes, so it would be a lie to say that I’m enthralled with anything. But I have absolutely no aesthetic or moral objection to the concept of (,1)(\infty, 1)-category, if that’s what you’re getting at. On the contrary, I think it’s both a canonical and a useful concept.

Posted by: Tom Leinster on March 8, 2010 9:48 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I think that’s a misunderstanding.

Okay, sorry! So, then, what do you think about “monoid in a monoidal (,1)(\infty,1)-category” as the “right” categorical notion underlying homotopy monoids?

Posted by: Mike Shulman on March 8, 2010 9:18 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Mike wrote:

What does the ordinary “the” mean, anyway?

I think “the X” is ordinarily understood to make sense if and only if there is precisely one object satisfying description X, and in that case, “the X” refers to that object.

For instance, the queen of England makes sense because there is precisely one person who is queen of England. The king of England does not make sense because there is no person who is king of England. The prince of England does not make sense because there are several people who are princes of England.

(I know, I know, she’s queen of a lot more than just England.)

Posted by: Tom Leinster on March 8, 2010 11:01 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

This is all completely precise, or can be made so in whatever precise-making formal theory you like. All that’s different is that we’re using the English word ‘the’ to refer to a more general mathematical situation than previously.

Is that formalisation really understood yet? Not in anything like the generality Tom was speaking about, I think.

Classical ‘the’ works because our language only lets us talk about properties/constructions that are invariant under equality, so if C1 = C2, we know it will never matter whether ‘the C’ is interpreted as C1 or C2.

But as long we’re using a language that lets us talk about things that aren’t invariant under isomorphism (or higher equivalences) then higher use of ‘the C’ needs to be restricted. And of course we do generally restrict it appropriately, but we don’t always justify that the contexts we’re using it in are safe. We could do so if asked (at least on a case-by-case basis), but we don’t — this is the big hand-wave!

Of course, there may hopefully be a good formalism which only lets us talk about suitably invariant constructions (while also not being problematically restrictive in any way). Your work on 2-/n-logic goes in this direction, as does the work by various of us on connections with intensional Martin-Löf equality; but as Tom says, this is far from a solved problem, even in principle, even in the case n=1, as far as I can see?

Posted by: Peter LeFanu Lumsdaine on March 8, 2010 3:01 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I think “the X” is ordinarily understood to make sense if and only if there is precisely one object satisfying description X, and in that case, “the X” refers to that object.

Yes. I’m not denying that the “generalized the” is, in fact, a generalization. But what I’m saying is that it’s a generalization which already has a completely precise mathematical meaning. Some mathematical formal systems have a basic “unique choice operator,” but in those which don’t, the ordinary “the” is interpreted as “a” together with an assertion of uniqueness. Likewise, the “generalized the” can be interpreted as “a” together with an assertion of uniqueness-up-to-isomorphism.

Now, the point Peter brings up is also interesting, namely, can we restrict our language so that we can only talk about things that are invariant under isomorphism? If so, this would enable us to use the “generalized the” in ways which are similar to the ways in which we use the ordinary “the,” such as freely allowing substitution of any two “the X”s for each other. But I don’t think isolating such a restricted language is necessary in order for the “generalized the” to make precise mathematical sense. We just have to remember that we can’t use it in exactly the same ways that we use the ordinary “the,” but the rules regarding how we can use it aren’t handwavy. I feel like I gave a completely precise interpretation of a generalized “the Y,” namely “let X be a Y,” additionally remembering the fact that any two Ys are uniquely isomorphic. The ordinary rules of ordinary logic then tell you exactly how this can be used. Of course, in practice mathematicians don’t check that they are using it in an acceptable way, but I don’t think that’s any different from any other part of “informal” language in mathematics, which no one ever bothers to actually translate into formal logic.

Going on to Peter’s question, it does seem to me that the problem is basically solved for 1-categories. It’s easy to show that if you reason about categories in a dependently typed logic without equality for the type of objects, then anything you can say is invariant under equivalence of categories, and also under replacing all objects along isomorphisms. I’m not sure if this is explicitly written down in so many words anywhere in print, but lots of people have rediscovered it. It can be extracted from Makkai’s paper on FOLDS, and he refers to some earlier work by Freyd (“Properties invariant within equivalence types of categories”) and Blanc (“Équivalence naturelle et formules logiques en théorie des catégories”) neither of which I’ve managed to read yet. Analogous facts for nn-categories in a suitable dependently typed logic, with equality only at the top level, should also be possible to prove once you pick a definition of nn-category; e.g. for bicategories and tricategories it should be quite easy. But perhaps I’m misunderstanding what you are asking for?

Posted by: Mike Shulman on March 8, 2010 9:32 PM | Permalink | PGP Sig | Reply to this

Re: A Perspective on Higher Category Theory

John, we salute you! Give that man a round of applause.

Hear, hear!

Posted by: Mike Shulman on March 8, 2010 3:02 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Hear, hear!

(Great post, btw.)

Posted by: James on March 8, 2010 6:30 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Hear, hear! Not just today, but every day for many years to come!

Posted by: Charlie C on March 8, 2010 2:06 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Thanks, guys!

I’ve been sort of ‘out of action’ lately, mainly because I couldn’t post nn-Café comments from home, which is the main place I like to do it. But now I’m back, thanks to a technical hack due to Mike Stay.

Thanks for a great post, Tom, and the interesting discussion it led to.

Posted by: John Baez on March 17, 2010 12:00 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Thanks for the post, Tom.

I share your feelings about clean categorical concepts very much. My last post that you refer to was an advertizement for the considerable de-contamination of concepts (in the sense you describe above) that has recently been achieved, in my opinion.

While I can’t quite know for sure what you think about the arguments I put forward – because I think you haven’t replied to them, possibly due to the desire not to critcize anyone’s current work – my impression is that you do disagree, and that the disagreement is related to remarks such as the following:

You ask:

Why not define an \infty-groupoid to be a Kan complex?

and suggest,

[because] good categorical definitions are not tailored to any particular application.

Here is probably where the crux is. Can you think of a definition of \infty-groupoid that does not tend to be tailored to any particular application? How would a Batanin-style \infty-groupoid be a less tailor-made model? The typical algebraic model for \infty-groupoids looks way more contrived than that of Kan complexes, already as far as the count of lines necessary for their definition goes.

I think we should take category theory seriously here: there will not be the single fundamental definition of \infty-groupoid. Instead, there will be a web of equivalent notions. It is the mutual consistency of this web that makes the notion of \infty-groupoid a good one, not any single one of its representatives, which is just a way for us mortals to bring the idea to paper.

And for that purpose it is good to have one representative of which one is reasonably sure that it is a correct one. This is the case for Kan complexes. Whatever definition of \infty-groupoid you come up with, if it does not form an (,1)(\infty,1)-category equivalent to that of spaces/Kan complexes, your definition will not be thought of as being correct. The homotopy hypothesis is less a hypothesis here, than a consistency condition.

From that perspective it is reasonable to hope for more and more algebraic definitions of \infty-groupoids, while at the same time feeling free and secure to use Kan complexes in proving theorems and developing theory.

All these theorems and theory will apply to your algebraic \infty-groupoids, too, once you have proven the homotopy hypothesis for them.

What makes this work is category theory: the insight that it is not the concrete definition of any object that matters, but only the web of relations that connects it to all other objects.

Posted by: Urs Schreiber on March 8, 2010 8:19 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Urs wrote:

Whatever definition of \infty-groupoid you come up with, if it does not form an (,1)(\infty, 1)-category equivalent to that of spaces/Kan complexes, your definition will not be thought of as being correct.

I can believe that that would be true in a social sense. But what’s the mathematical justification? Can you explain why Kan complexes deserve to be called \infty-groupoids without implicitly invoking the Homotopy Hypothesis?

Posted by: Tom Leinster on March 8, 2010 10:42 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

But what’s the mathematical justification? Can you explain why Kan complexes deserve to be called ∞-groupoids without implicitly invoking the Homotopy Hypothesis?

Yes,I think the widely used and accepted models for \infty-groupoids and (,1)(\infty,1)-categories justify themselves by the fact that they do support a higher category theory that has all the properties that one would expect it to have.

\infty-functors with values in \infty-groupoids constitute (,1)(\infty,1)-presheaf categories, from which we obtain notions of limits, adjunctions and Grothendieck construction that satisfy all the expected theorems: the Yoneda lemma, the adjoint functor theorem, the Barr-Beck theorem, the relation of \infty-sheaves to subtoposes.

At the same time, the models incorporate all the structures that one would have hoped they incorporate.

In short, we have a full working setup of higher category theory with all the expected properties and loads of expected applications and examples.

We also know well how various algebraic defintions of \infty-groupoids are equivalent to or special cases of this, such as strict grlobular \infty-groupoids, bigroupoids, trigroupoids, simplicial T-complexes, Nikolaus \infty-groupoids. Also Verity’s “weak complicial” \infty-groupoids are of this kind, of course.

So there is a big body of structures here that looks like (,1)(\infty,1)-category theory, smells like (,1)(\infty,1)-category theory and walks like (,1)(\infty,1)-category theory.

It would seem to me in light of this evidence that doubts that this is (,1)(\infty,1)-category theory would need very good justification. I haven’t seen that justifications yet.

What I have seen and keep seeing are remarks that certain constructions people consider in (,1)(\infty,1)-category theory are component based. To which I keep replying: yes, there are generators-and-relations presentations of \infty-groupoids and (,1)(\infty,1)-categories just as there is for about any mathematical structure. This doesn’t change that the objects themselves have intrinsic meaning. I would be surprised if you could specify a large number of specifc Batanin \infty-categories without presenting them somehow in terms of something.

Posted by: Urs Schreiber on March 8, 2010 12:46 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I wrote:

Can you explain why Kan complexes deserve to be called \infty-groupoids without implicitly invoking the Homotopy Hypothesis?

Urs wrote:

[Kan complexes] justify themselves by the fact that they do support a higher category theory that has all the properties that one would expect it to have.

OK, that’s one kind of answer. The foundations must be good because we’ve built a big building on top of them, and it hasn’t collapsed. That’s certainly persuasive in one sense.

But it also reminds me strongly of what I wrote under “A break in the conceptual trail”. I could (and more or less did) say that the concept of weighting justifies itself by the fact that it supports a theory of cardinality that has all the properties that one would expect it to have. And that’s great.

But, as I said, I’m still left unsatisfied. I don’t see a good conceptual explanation for weightings; the best I can say is that they seem to work. Similarly, I don’t see a good conceptual explanation for Kan complexes, and I understand your answer as saying (with scholarly detail) “they seem to work”.

I don’t claim that there isn’t a conceptual, intrinsic explanation for why Kan complexes deserve to be called \infty-groupoids. I just don’t know one. Given how much people are talking about Kan complexes these days, it would be great if someone could supply one.

Posted by: Tom Leinster on March 8, 2010 2:47 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Tom writes:

I don’t see a good conceptual explanation for Kan complexes,

and:

I don’t claim that there isn’t a conceptual, intrinsic explanation for why Kan complexes deserve to be called ∞-groupoids. I just don’t know one.

I feel like I must be missing some subtext of these statements, because I have trouble believing that you really mean that you don’t see how a Kan complex captures the notion of an \infty-category in which all morphisms are invertible. So I say what I would reply to a layman who asks me about the conceptual meaning of Kan-complexes, and then you can tell me what you really mean:

A Kan complex is a structure with kk-morphisms for all kk \in \mathbb{N}, such that whenever they are adjacent they have a composite and such that all have a weak inverse.

If you prefer equivalently we may talk about Nikolaus-Kan-complexes:

A Nikolaus-Kan-complex is a structure with kk-morphisms for all kk, together with a map that assigns composites to all composable kk-morphisms and such that every kk-morphism has a weak inverse under this composition.

This is precisely the conceptual description of an \infty-category that is an \infty-groupoid.

Posted by: Urs Schreiber on March 8, 2010 7:03 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I agree with Urs; to me it seems quite evident how a Kan complex is a notion of \infty-groupoid. The horn-filling conditions are (to me) clearly a sort of assertion that nn-cells can be composed for all nn. The only slightly funny thing is that the nn-cells are simplex-shaped rather than globular-shaped, which is how I more naturally think about an \infty-groupoid, but one thing the horn-filling conditions imply is that every simplex is equivalent to a “globular” one in which all the boundary is degenerate except for one “source” and one “target” – so we can view the non-globular simplices as “extra fat” that makes the definition work cleanly.

Posted by: Mike Shulman on March 8, 2010 8:50 PM | Permalink | PGP Sig | Reply to this

Re: A Perspective on Higher Category Theory

Why do we take for granted that the homotopy theory of spaces provides a good notion (among others) of \infty-groupoid? Why isn’t this an arbitrary choice (apart from the very fact that it works so well)? But we may also question the notion of \infty-groupoid: why should we expect everything to be enriched in higher groupoids?

After all, the good question might be something like: what is the mathematical structure over which everything is canonically enriched? And, last but not least, how can we formulate correctly such a question?

As a tribute to Grothendieck and Heller, here is an attempt to do so, and the good news are that the question even has an answer: everything is uniquely enriched in homotopy types of CW-complexes, and there is no other choice at all. The way I propose to do so consists to use only usual category theory, and to formulate the question through an enlargement of category theory.

I propose to assume we know what category theory is: categories, functors, natural transformations, Kan extensions, Grothendieck fibrations. I will also assume that we know enough about 2-categories to speak comfortably of the 2-category of categories.

So we have the 2-category of categories CatCat (categories are allowed to be large, which means their Hom’s might be classes, or non-small sets, depending on your favorite set theoretic preferences), and the 2-category of small categories catcat, which generate CatCat (in the sense that any category is a (large) filtered colimit of small ones). We would like to enlarge the theory of categories, because we know by experience that homotopical (or homological) algebra cannot be defined intrinsically in there. A striking example is the theory of triangulated categories: if AA is an abelian category, its (bounded) derived category D b(A)D^b(A) is not defined by a universal property: a natural statement would be that, given a triangulated category, the category of additive functors ATA\to T which send short exact sequences to distinguished triangles is equivalent to the category of triangulated functors D b(A)TD^b(A)\to T; but this statement is false, and in fact, does not even make sense (unless AA is semi-simple). But, in practice, everything behaves as if the above statement were meaningful and true. The `reason’ why this does not work is that the cone of a map in a triangulated category is not defined by a universal property. On the other hand, the cone of a morphism of complexes XYX\to Y is canonically defined: this is the homotopy colimit of the diagram 0XY.0\leftarrow X \to Y.

So, as the 22-category of categories does not seem to be large enough to express intrinsically our favourite structures, there is a need to enlarge it. An apparently naive way to do so consists to consider presheaves on catcat. If you allow me some short cuts, this leads us to the theory of Grothendieck derivators (which is the same as Heller’s theory of homotopy theories). As I already wrote, a natural way of enlarging CatCat is to consider presheaves on catcat, which Grothendieck calls prederivators. These are contravariant 22-functors from catcat to CatCat (contravariant means that it inverts 1-cells and 2-cells). Any category CC may considered a prederivator, by sending a small category XX to the category Hom(X op,C)Hom(X^{op},C) of presheaves on XX with values in CC. This defines an embedding of CatCat into the 2-category PDerPDer of prederivators. If CC is endowed with a subcategory of weak equivalences, we may consider the prederivator Ho(C) which sends a small category XX to the localization Ho(C)(X) of Hom(X op,C)Hom(X^{op},C) by termwise weak equivalences (as we allowed Hom’s to be large, these localization exist). You already get an example a non-representable prederivator. However, to do this, we don’t need anything else than genuine (2-)category theory. Now, if C is a model category, Ho(C) is a derivator. This means that, for any functor u:XYu:X\to Y in catcat, the inverse image functor u *:Ho(C)(Y)Ho(C)(X)u^*:Ho(C)(Y)\to Ho(C)(X) admits a left adjoint u !u_! and a right adjoint u *u_*, that any map in Ho(C)(X)Ho(C)(X) which is termwise invertible is invertible, and that we have nice formulas to compute u !u_! and u *u_*. In the case the weak equivalences of CC are just the isomorphisms, this derivator structure is just the theory of colimits and limits in CC (as far as they are indexed by small categories). For a general model category, this structure of derivator just encodes the notion of homotopy colimit and of homotopy limit. Note however that this way of seeing homotopy (co)limits does not use anything else than usual category theory.

We can consider now the 2-category DerDer of derivators: objects are derivators, 1-cells as (non-strict) morphisms of 2-functors F:DDF:D\to D' which commute with the functors u !u_! (i.e. with homotopy colimits), and 2-cells are the natural transformations (modifications) between them. Given two derivators, let us denote by Hom(D,D)Hom(D,D') the category of morphisms of derivators. Given a small category XX, there is a 22-functor, defined by evaluating at X opX^{op} DerCat,DD(X op).Der\to Cat \quad , \qquad D\mapsto D(X^{op}). Note that, for any (pre)derivator DD, the category D(X op)D(X^{op}) is canonically equivalent to the category Hom(X,D)Hom(X,D) of morphisms of prederivators from XX to DD (considering XX as a prederivator). This 22-functor is representable. This means that there exists a derivator X^\widehat{X}, endowed with a morphism of prederivators h:XX^h:X\to \widehat{X} (called the Yoneda embedding), such that, for any derivator DD, composing with h defines an equivalence of categories Hom !(X^,D)Hom(X,D)=D(X op).Hom_!(\widehat{X},D)\simeq Hom(X,D)=D(X^{op}). In other words, the map h:XX^h:X\to \widehat{X} is the “free completion of XX by homotopy colimits” in the sense of derivators.

Furthermore, X^\widehat{X} can be described rather explicitely: this is the derivator associated to the model category of simplicial presheaves on XX.

This provides a first argument that the usual homotopy theory of simplicial sets plays a central role (as *^\widehat{*}, where ** stands for the terminal category), and for this, we didn’t take for granted that homotopy types should be that important: its universal property is formulated with category theory only. From there, you can see that any derivator is canonically enriched in the derivator *^Ho(SSet)\widehat{*}\simeq Ho(SSet): as ** acts uniquely on any prederivator, *^Ho(SSet)\widehat{*}\simeq Ho(SSet) acts uniquely on any derivator (as far as we ask compatibility with homotopy colimits). The homotopy hypothesis might reformulated vaguely as: is there an algebraic model of *^\widehat{*}? And then, it looks like some notion of higher groupoid might do the job.

We might argue that the notion of derivator is much weaker than the one of (,1)(\infty,1)-category (the latter providing much more interesting structures, whatever model you choose), but that is precisely my point; derivators provide a truncated version of higher category theory which gives us the language to characterize higher category theory using only usual category theory, without any emphasis on any particular model (in fact, without assuming we even know any).

Posted by: Denis-Charles Cisinski on March 8, 2010 10:25 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Dear Denis-Charles,

thanks, that was very useful, indeed.

I have taken the liberty of feeding what you provided into the nnLab entry

You should add some of your papers in the references section there. Or I’ll do it later.

One question: when you wrote

Given two derivators, let us denote by Hom(D,D)Hom(D,D') the category of morphisms of derivators.

did you mean to write

Given two derivators, let us denote by Hom !(D,D)Hom_!(D,D') the category of morphisms of derivators.

?

Posted by: Urs Schreiber on March 9, 2010 1:05 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Dear Urs,

yes you are right: there should be Hom !(D,D){Hom_{!}(D,D')} instead of Hom(D,D){Hom(D,D')}.

Posted by: Denis-Charles Cisinski on March 9, 2010 10:19 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I started writing a paragraph

I am being told that the intuition identifying kk-cells in a simplicial set with kk-morphisms in an \infty-category varies in strength from person to person.

But luckily, we do not have to rely on our intuition here but can make this very precise: Ross Street’s orientals are the full combinatorial solution to the problem of how exactly a simplicial kk-simplex is to be read as a globular kk-morphism in an \infty-category.

In fact, all simplicial sets CC such as Kan-complexes and weak Kan-complexes/quasi-categories may be thought of as being the omega-nerve of an \infty-category. This manifestly and concretely identifies the kk-cells ϕC k\phi \in C_k with kk-morphisms.

Using this, it is pretty easy to see what the various horn filler conditions on a (weak) Kan complex encode: they assert precisely that given certain pasting diagrams of (k1)(k-1)-morphisms there is guaranteed the existence of certain kk-morphsims in the \infty-category, connecting the interior of these pasting diagrams.

This gives an immediate and concrete idebntification of simplicial sets with certain properties on the one hand and certain \infty-categories.

In fact, this is so immediate that one can turn this around and define more general \infty-categories in terms of simplicial sets with suitable properties. This is the approach of simplicial models for weak omega-categories as initiated by Ross Street and developed by Dominic Verity.

And using the orientals we see that this identification of simplicial sets with \infty-categories is not vague and unnatural, but is in fact very immediate: a kk-simplex is manifestly a kk-morphism whose source and target (k1)(k-1)-morphisms are presented as a certain pasting diagram. As encoded by the orientals.

Posted by: Urs Schreiber on March 11, 2010 8:23 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Tom wrote:

There’s an old joke:

Q. How do you prove the Homotopy Hypothesis?

A. Define an \infty-groupoid to be a Kan complex; define a space to be a Kan complex; done!

Hahaha!

Now I see why more people don’t come to the nn-category café — this is our idea of a joke.

But seriously…

My main complaint about using Kan complexes to ‘trivialize the homotopy hypothesis’ is this. Most of us feel Kan complexes are a good concept of ‘space’, at least as far as homotopy theory is concerned. But why?

Well, largely because Quillen showed the model category of simplicial sets is Quillen equivalent to that of topological spaces. And we now know this Quillen equivalence gives an equivalence of (,1)(\infty,1)-categories… very good.

But unfortunately, there’s a bit of a ‘cheat’ hidden in this result. Namely: Quillen’s construction of the model category of simplicial sets uses topological spaces. We say two simplicial sets are weakly equivalent if their geometric realizations are weakly equivalent as topological spaces!

So, it’s as if we secretly used our belief in the Homotopy Hypothesis to define a concept of equivalence for simplicial sets that makes the Homotopy Hypothesis come out true if we define an \infty-groupoid to be a Kan complex.

I don’t really mind this sort of cheating as long as one admits it’s a stopgap solution. But it would be very nice to have a construction of the usual model category of simplicial sets that did not refer to topological spaces!

My spies tell me that Joyal and Tierney have such a construction. Does anyone know what it is?

Posted by: John Baez on March 17, 2010 12:14 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

But it would be very nice to have a construction of the usual model category of simplicial sets that did not refer to topological spaces!

Here is one: cofibrations the monomorphisms, weak equivalences the morphisms that induce isos on simplicial homotopy groups.

Equivalently, weak equivalences those morphisms XYX \to Y that induce equivalences of \infty-groupoid enriched categories τ(X)τ(Y)\tau(X) \to \tau(Y) with τ\tau the left adjoint to the homotopy coherent nerve.

Notice also that the acyclic fibrations between Kan complexes are precisely the \infty-functors that are k-surjective for all kk \in \mathbb{N}. And that every weak equivalence between Kan complexes is a span of such acyclic fibrations, i.e. an \infty-anafunctor that is an equivalence.

No topological spaces here, just category theory.

Even better than all of this is the following: you don’t have to fall back to model category presentations in order to conceive Kan complexes as \infty-groupoids:

take the full sSetsSet-enriched subcategory of sSetsSet on Kan complexes. This is the (,1)(\infty,1)-category of \infty-groupoids conceived as Kan complexes.

Posted by: Urs Schreiber on March 17, 2010 1:15 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

A very thought-provoking piece. Thanks, Tom.

Just a few comments here. First, along with the applause due to John, some mathematical society like the AMS ought to award him a medal. Without him I would never have seen that a non-mathematician could understand something of cutting edge research.

Now, I was interested by your comment

There’s an infinite amount of mathematics out there, and I’m more interested in getting things done right than getting them done fast.

I’ll return to the first half of this below, and focus now on the second half. I’m very sympathetic to the idea of “getting things done right”. Indications that this has happened are, as you say, a simplicity in the account you can give, and some “adaptability to multiple parts of mathematics, some as yet unknown”. It’s interesting to think about how one assesses these indications, especially the former. Given a piece of mathematics which appears to XX as not being done right, as insufficiently simple, it could be that this is because it has indeed not been done right, or it could be a false appearance. A false appearance could be brought about by suboptimal expression, had they written it otherwise XX could have seen that it was done right (as in your ‘A suspended judgement’), or because of insufficiencies in XX’s background.

The example you gave of Buffon’s noodle is wonderfully clear and could be explained to a school child. (I’ll test that claim out tonight.) But there must be cases of what you take to be “done right” which will require a heck of a lot more background, where you couldn’t just explain it while walking through the park. I’m sure there must be some passage of topos theory that you would take to be natural but you’ll have a devil of a job explaining it to me. Surely Polanyi is onto something when he says

It is legitimate, of course, to regard simplicity as a mark of rationality, and to pay tribute to any theory as a triumph of simplicity. But great theories are rarely simple in the ordinary sense of the term. Both quantum mechanics and relativity are very difficult to understand; it takes only a few minutes to memorize the facts accounted for by relativity, but years of study may not suffice to master the theory and see these facts in its context. Hermann Weyl lets the cat out of the bag by saying: ‘the required simplicity is not necessarily the obvious one but we must let nature train us to recognize the true inner simplicity.’ In other words, simplicity in science can be made equivalent to rationality only if ‘simplicity’ is used in a special sense known solely by scientists.

(For the context of the quotation see here.)

Certainly there ought to be a much greater onus to make mathematics as accessible as possible to as many people as possible. John has led the way in showing what can be done, but there comes a point even in his exposition where you have to put the work in yourself, then you feel your neuronal connections growing, as he likes to say.

So now here’s my question, how does a mathematician know whether it’s something lacking in him or herself, perhaps some missing background, if they’re not seeing the naturality of some mathematics, or whether there’s something lacking in the mathematics itself? I suppose the answer is that one cannot know, but I’ve long hoped that a certain kind of critical discussion (as advocated by Imre Lakatos and Ronnie Brown) could help the situation, ‘critical’ in the sense of literary criticism. And I’ve long hoped that such discussion could happen on this blog, and indeed it does from time to time.

(It’s a frequent source of amusement when philosophy students don’t understand ‘critical’ in this sense, and so when they’re asked for a critical assessment of, say, Descartes’ cogito, begin “Descartes is hopelessly wrong for missing the obvious objection that…”)

If we’re all pursuing the truth, it shouldn’t matter to someone that someone else expresses their sincere belief that it hasn’t been done right. In fact they ought to be pleased since it offers the opportunity for one or both sides to enhance their understanding.

Finally, back to your “There’s an infinite amount of mathematics out there”. I suppose one could say that there’s an infinite amount of physics to do, yet if someone comes up with a neat quantum gravity, then I could sympathise with the idea that the rest is details, even if the detailed understanding of fluid flow and condensed matter physics were still well beyond us. If that were allowed, I wonder whether one might be able to contemplate ever saying we’d moved beyond knowing “an epsilon about mathematics”.

Anyway, thanks again for your very stimulating post.

Posted by: David Corfield on March 8, 2010 11:32 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

But on the other hand, the situation is not quite satisfactory. As a prelude to the definition of the Euler characteristic of a category, one has to define the notion of a ‘weighting’ on a category. Where does this definition come from? What is the conceptual justification? I don’t really know

I have no doubt this will eventually fall into the “reversible” category.

For what its worth, my thoughts about nn-category theory (after watching from the sidelines for more than a decade) is that it is very much like Newton in search of Einstein. Sure, it is believable that \infty-groupoids are somehow the same as spaces, but what really brings everything together is spacetime. Things that are complicated when projected to space suddenly become simple tautologies when viewed properly in spacetime. So rather than the homotopy hypothesis, what I’m hoping to see is the directed homotopy hypothesis, \infty-categories are somehow the same as directed spaces, i.e. spacetimes. Just as this change in perspective changed physics (and to a certain extent mathematics) forever, I suspect a similar simplification, unification, generalization to occur again (if it hasn’t already!)

To put a Lorenztian metric on a spacetime, you essentially need a way to “count”. I don’t think this is unrelated to Tom’s cardinality stuff.

Posted by: Eric Forgy on March 8, 2010 1:36 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

So rather than the homotopy hypothesis, what I’m hoping to see is the directed homotopy hypothesis, ∞-categories are somehow the same as directed spaces,

Yes, and we can have the analogous discussion here:

I think there is overwhelming evidence that the right notion of “directed space” with space in the sense of homtopy theory, is incarnated in the definition of quasi-category. To the extent that the homotopy hypothesis for Kan complexes is a tautology (I wouldn’t say it is quite, but I understand Tom’s point that it almost is) the directed homotopy hypothesis for quasi-categories could be taken to be a tautology.

Notice that it is isn’t quite. There are competing definitions of “directed space” out there in the literature, none of which so far however seems to be geared towards the homotopy-theoretic notion. By saying that a good homotopy-theoretic version of “directed space” is given by the notion of quasi-category, I am fixing a gauge at which to measure other definitions.

I think the notion of quasi-category has been proven to be a right carrier of (,1)(\infty,1)-category theory, such that every other definition should be equivalent to this one. This is already known to be the case for a bunch of definitions including the algebraic definition of Nikolaus-quasicategories.

All of these notions together establish a web of concepts that ecodes the notion of (,1)(\infty,1)-category.

Posted by: Urs Schreiber on March 8, 2010 2:29 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I had a go at the problem that Eric and Urs mention. I defined a simplicially enriched category for an directed space. The objects were the points of the space, and the n-simplices between two points were directed singular prisms. There are two obvious directed space structures on a simplex, in one it is basically unordered, in the other it is a subspace of the directed n-cube. For the first on using the product directed space structure you get a Kan enriched simplicial category. For the other which seems more natural, you don’t seem to. (I believe you do get a quasicategory enriched category, but I cannot remember for certain.)

This second form does seem to look at more structure, but it does not seem to give an (,1)(\infty,1)-category (unless I am wrong).

I thus am not 100% convinced that any directed homotopy hypothesis is going to be as suggested. The simple question is perhaps: should a directed homotopy between two directed paths be reversible?

(There is a discussion about some of these points in a preprint at http://drops.dagstuhl.de/frontdoor.php?source_opus=898. A shortened version was published in Theoretical Computer Science later.)

Posted by: Tim Porter on March 8, 2010 3:18 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I’m glad we have the nLab so I can easily look up quasi-category. Everything I hear you say about this stuff lately sounds good EXCEPT one thing…

Simplices…

Just as a Kan complex is a model in terms of simplicial sets of an ∞-groupoid – also called an (∞,0)-category – a quasi-category is a model in terms of simplicial sets of an (∞,1)-category.

This is actually a thought on my mind these days. Categories are inherently “simplicial”, which causes some problems when relating it to evolution of systems, causality, etc. In category theory, whenever there is a process α:xy\alpha:x\to y and another process β:yz\beta:y\to z, there is always a process βα:xz\beta\alpha:x\to z. Pictorially, with categories, this means you always “close the triangle”.

When you are thinking about what is fundamental in physics (which often translates to what is fundamental in mathematics), it is hard to get more fundamental than the “nature of light”. If α:xy\alpha:x\to y denotes a “ray of light” (or geodesic) from event xx in spacetime to event yy in spacetime, then given a second ray of light β:yz\beta:y\to z there is not generally a ray of light βα:xz\beta\alpha:x\to z.

Is this a clue? I think it is.

There are many clues, e.g. Zitterbewegung, Feynman checkerboard, etc that suggest that the fundamental morphism of nature is a “ray of light”.

If we think about this for a second (in my case, years) and try to put it into the context of “arrow theory” then given morphisms α:xy\alpha:x\to y and morphisms β:yz\beta:y\to z, then what would be more important than having a morphism βα:xz\beta\alpha:x\to z might rather be having a 2-morphism bounded by another pair of morphisms γ:xw\gamma:x\to w and δ:wz\delta:w\to z. This 2-morphism represents a 2-dimensional intersection of the “past of zz” and the “future of xx”.

I’m not suggesting anything as radical as throwing out simplices altogether, but I am suggesting that the canonical nature of (directed) simplices be interpreted as portions of the boundary of a directed space. The higher dimensional cells, at a fundamental level, represent causality and evolution. An nn-morphism is an nn-dimensional intersection of the future of one object and the past of another.

Posted by: Eric Forgy on March 8, 2010 3:58 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Everything I hear you say about this stuff lately sounds good EXCEPT one thing… Simplices…

The predominance of simplicial shapes is not an intrinsic phenomenon, but simply a reflection of the fact that these were historically preferred, for one reason or other.

It is known how to use cubical sets to model all weak \infty-groupoids: this is given by the model structure on cubical sets.

So you are perfectly entitled to think of a general \infty-groupoid as a cubical structure.

I suppose it can’t be too hard to generalize this to a model structure on cubical sets that models quasi-categories, and that the only reason it hasn’t been written down yet is that nobody has as yet seriously tried to.

Posted by: Urs on March 8, 2010 7:18 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Not only simplicial or cubical sets, but more generally, AA-sets for any test category AA in the sense of Grothendieck.

Posted by: Zoran Skoda on March 8, 2010 8:14 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

It’s really not at all clear to me that the homotopy hypothesis admits any sort of generalization to the directed world. I love (,1)(\infty,1)-categories, but I don’t get much intuition from thinking of them as “directed spaces,” and I haven’t seen anything I’d call a “directed space” that really looks very much like an (,1)(\infty,1)-category. Perhaps some kind of directed space can be used to model (,1)(\infty,1)-categories (probably with a specified collection of points representing the “objects”), but it’s by no means obvious to me that that particular kind of directed space is the one that people studying directed spaces are interested in.

Posted by: Mike Shulman on March 8, 2010 8:46 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

The Buffon’s needle argument is interesting because it doesn’t apply to two-dimensional “needles”. I wonder if this is further evidence that the “correct generalization” has been found in this case.

Posted by: Tom Ellis on March 8, 2010 2:00 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

You wrote that “a categorical concept that only bears on one part of mathematics isn’t really a categorical concept at all.” In this respect, I wonder how you think of the notion of opposite of a category. Indeed, on one hand it is general because it can be applied to any category. But on the other hand it is not an instance a general 2-categorical notion to the 2-category of categories (I would love here that someone proves me wrong.)

Posted by: David Leduc on March 8, 2010 3:02 PM | Permalink | Reply to this

Re: Opposites in a 2-category

… the notion of opposite of a category… is not an instance a general 2-categorical notion to the 2-category of categories (I would love here that someone proves me wrong).

The category C opC^{op} can be characterized, up to Morita equivalence, by the fact that two-sided discrete fibrations from AA to B×C opB\times C^{op} are naturally equivalent to two-sided discrete fibrations from A×CA\times C to BB, for any other categories AA and BB. That characterization is completely internal to the 2-category CatCat and can be imported into other 2-categories, although of course such an object may or may not exist in general. This approach is studied in Mark Weber’s paper “Yoneda structures on 2-toposes.”

Alternately, opposites can be constructed, up to equivalence, as follows. Given any category AA, let XX be the core of AA, with p:XAp\colon X\to A the inclusion. The comma object (p/p)X(p/p)\rightrightarrows X has the structure of an internal category object in CatCat, which in fact is a “2-congruence” (the 2-dimensional analogue of an internal equivalence relation) whose “quotient” is AA. But since XX is groupoidal, we can switch the two arrows (p/p)X(p/p)\rightrightarrows X to obtain another 2-congruence, and since CatCat is an “exact 2-category”, this switched congruence also has a quotient, which is precisely A opA^{op}. We can repeat this argument in any exact 2-category having cores, to thereby construct an opposite for any object and hence a duality involution 𝒦 co𝒦\mathcal{K}^{co}\to \mathcal{K}. More details are here as part of the 2-categorical logic project.

Posted by: Mike Shulman on March 8, 2010 9:11 PM | Permalink | PGP Sig | Reply to this

Re: A Perspective on Higher Category Theory

the notion of opposite of a category. Indeed, on one hand it is general because it can be applied to any category. But on the other hand it is not an instance a general 2-categorical notion to the 2-category of categories (I would love here that someone proves me wrong.)

This may be of interest: the operation of sending an (,1)(\infty,1)-category to its opposite (,1)(\infty,1)-category is up to equivalence the unique nontrivial automorphism on (,1)(\infty,1)Cat.

So it’s a perfectly intrinsic notion.

Posted by: Urs Schreiber on March 9, 2010 1:26 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

the operation of sending an (∞,1)-category to its opposite (∞,1)-category is up to equivalence the unique nontrivial automorphism on (∞,1)Cat.

Very interesting. (Of course, here (,1)Cat(\infty,1)Cat means the (,1)(\infty,1)-category of (,1)(\infty,1)-categories, not the (,2)(\infty,2)-category of them; () op(-)^{op} is not an automorphism of the latter.) Is that also true for 1-categories?

(The nLab page refers this result to 5.2.9.1 in HTT, but my copy of HTT has no section 5.2.9.)

Posted by: Mike Shulman on March 9, 2010 5:23 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Wasn’t it once suggested by John that there is a link between this unique nontrivial automorphism (at least in the 2-category of categories), time reversal and conjugation in \mathbb{C}?

Posted by: David Corfield on March 9, 2010 10:59 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Wasn’t it once suggested by John that there is a link between this unique nontrivial automorphism (at least in the 2-category of categories), time reversal and conjugation in ℂ?

Maybe you can give a reference for the precise suggestion. But generally, as we said, we may think of a category or (,1)(\infty,1)-category as a directed space – as the very definition of directed space, in fact – where in addition to reversible paths representing, if one wishes, gauge transformations there are other non-reversible morphisms that represent, if one wishes, time evolution .

As in the example of the category of future-directed paths in a principal bundle over a Lorentzian manifold: objects are points in the Lorentzian space together with a point in the fiber over them, to be thought of as the event of a charged particle being at some spacetime point with some internal state. Invertible morphisms are paths that don’t move in spacetime but just in one fiber, just changing the particle’s inner state by a gauge transformation. Non-invertible morphisms are paths in the bundle whose projection down to spacetime is future directed.

This is a simple model of the category of configurations and processes of a charged elementary particle. And evidently, applying the operation of passing to opposite categories to this example corresponds precisely to time-reversal, namely to switching the notions of future-directed and past-directed paths in the Lorentzian spacetime.

Posted by: Urs Schreiber on March 11, 2010 7:11 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

What about a category-like thing where objects are points of a Lorentzian manifold and instead of all future directed paths, you consider only light-like paths as I started to outline here?

I say “category-like” because light-like paths do not compose to give light-like paths (except when mass = 0). In this world, all particles move at the speed of light between any two adjacent points in spacetime, but the “average” velocity is the speed of light only if mass is zero. That is what I was alluding to when I mentioned Zitterbewegung and the Feynman checkerboard there.

This is probably the only remaining reason for me not to fully embrace (,1)(\infty,1)-categories. I can see that they are a model of directed space, but they are not the model I wanted. The model I want probably isn’t even a category, but is close. It is probably more like an nn-fold category.

I wonder if \infty-fold categories might be anywhere near as interesting as (,1)(\infty,1)-categories?

Posted by: Eric on March 11, 2010 9:55 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

What about a category-like thing where objects are points of a Lorentzian manifold and instead of all future directed paths, you consider only light-like paths as I started to outline here? I say “category-like” because light-like paths do not compose to give light-like paths

You can consider the category with morphisms being piecewise lightlike paths. And you can consider the nn-category whose nn-morphisms are piecewise lightlike nn-cubes. As in my AQFT article (I won’t provide the link right now, as I am on a miserable cell-phone connection).

As long as morphisms in all directions are of the same type (asis the case here), the distinction between nn-fold category and nn-category (with an \infty in front or not) is mainly just a technical question of little intrinsic relevance, I think (Mike? :-). In any case for your purposes it does not seem to matter to me.

Posted by: Urs Schreiber on March 11, 2010 11:53 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I was going to provide a service and give a link to your AQFT paper, but for all your stuff on the nLab, I could not easily find a list of papers!! :)

I think this is the one, but might I suggest you provide a concise list of papers somewhere? :)

Some young researchers might find it useful.

Posted by: Eric Forgy on March 11, 2010 1:58 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

As long as morphisms in all directions are of the same type (asis the case here), the distinction between n-fold category and n-category (with an ∞ in front or not) is mainly just a technical question of little intrinsic relevance, I think

If your n-fold category has a “connection,” then I think I agree. I’m not sure about otherwise, though.

Posted by: Mike Shulman on March 11, 2010 3:23 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I wrote:

As long as morphisms in all directions are of the same type (asis the case here), the distinction between n-fold category and n-category (with an ∞ in front or not) is mainly just a technical question of little intrinsic relevance, I think (Mike? :-).

Mike commented

If your n-fold category has a “connection”, then I think I agree. I’m not sure about otherwise, though.

Yes, that’s what i have in mind. Effectively, I am just saying that the nn-fold category that I think Eric has in mind is one that comes from an nn-category (its cubical kk-cells are the commuting cubical kk-diagrams in an nn-category) and that for these it makes no intrinsic difference whether we think of them as nn-fold categories or nn-categories.

I think there is an interesting question here, which I feel I have a partial but still no fully satisfactory answer to: given

or more generally

  • an internal poset Σ\Sigma in TopTop or DiffDiff

or more generally

  • an (,2)(\infty,2)-sheaf Σ\Sigma on suitably nice version of TopTop or DiffDiff

or more generally

(ΠLConstγ):Sh (,n)(C)(,n1)Cat (\Pi \dashv LConst \dashv \gamma) : Sh_{(\infty,n)}(C) \to (\infty,n-1)Cat

then we want to call Π(Σ)\mathbf{\Pi}(\Sigma) for

Π:=LConstΠ:Sh (,n)(C)Sh (,n)(C) \mathbf{\Pi} := LConst\circ \Pi : Sh_{(\infty,n)}(C) \to Sh_{(\infty,n)}(C)

the internal fundamental homotopy (,n1)(\infty,n-1)-category of Σ\Sigma.

And we might want to see something like that the “endomorphism coporesheaf” of a morphism Π(Σ)A\mathbf{\Pi}(\Sigma) \to A as in my AQFT article is a local net of monoids on XX, where local is Lorentzian local with as seen by the non-invertible morphisms in Σ\Sigma.

And we probably want to obtain such beast by starting with a differential cocycle Π(X)A\mathbf{\Pi}(X) \to A on some target space object (an (,1)(\infty,1)-sheaf) and form the push-forward of differential cocycles along the extended configuration space projection X×ΣΣX \times \Sigma \to \Sigma.

I can see a good bit of the two ends of this story. But I am still lacking someting in between…

Posted by: Urs Schreiber on March 11, 2010 8:03 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

You can consider the category with morphisms being piecewise lightlike paths. And you can consider the nn-category whose nn-morphisms are piecewise lightlike nn-cubes.

Interesting. That is not the way I interpreted things the first time I saw your paper, but this is the kind of category I could live with :)

But I’d point out the obvious fact that such a category is not “smooth”, which I think is important. Each piecewise light-like path could even represent a stochastic process, for example.

Now that I think of it, “stochastic” is not a term I hear a lot around the nn-community, but I think there is a vast potential for lots of interesting work in that direction.

Note: Tom, I actually do hope to relate this back to the theme of your original post, which I thoroughly enjoyed (as usual) by the way :)

Posted by: Eric Forgy on March 11, 2010 10:24 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

By the way, a little Googling of “stochastic” and “functor” turned up:

Stochastic relations: foundations for Markov transition systems

By Ernst-Erich Doberkat

Book Overview:

Collecting information previously scattered throughout the vast literature, including the author’s own research, Stochastic Relations: Foundations for Markov Transition Systems develops the theory of stochastic relations as a basis for Markov transition systems.After an introduction to the basic mathematical tools from topology, measure theory, and categories, the book examines the central topics of congruences and morphisms, applies these to the monoidal structure, and defines bisimilarity and behavioral equivalence within this framework. The author views developments from the general theory of coalgebras in the context of the subprobability functor. These tools show that bisimilarity and behavioral and logical equivalence are the same for general modal logics and for continuous time stochastic logic with and without a fixed point operator.With numerous problems and several case studies, this book is an invaluable study of an important aspect of computer science theory.

Posted by: Eric on March 12, 2010 12:59 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

But I’d point out the obvious fact that such a category is not “smooth”,

The category itself can well be smooth: you can have smooth families of piecewise lightlike paths, smoothly varying with respect to some parameters.

You can also think of each lightlike path as a smooth path, if you demand only its union of points to be lightlike. Then the path itself could smoothly decelerate at the kinks with respect to its parameter, have a “sitting instant” there and then continue smoothly in the other direction.

More efficiently, you can encode a piecewise lightlike path simply by its collection SS of kink points SXS \to X. Then a smooth family of such over a smooth parameter space UU is simply a smooth map U×SXU \times S \to X. This makes your category of piecewise lightlike paths a category internal to diffeological spaces. With a little care taken even internal to smooth manifolds.

How exactly you go about this is not really important. The category does not care about how you identify any one of its morphisms, it only cares about the rule by which morphisms compose.

Posted by: Urs Schreiber on March 12, 2010 8:16 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Thanks for explaining that. I guess I should have said the “paths were not smooth”. Who knew a path category could be smooth even though the paths themselves are not? :) Obviously, I still have no intuition for this stuff yet…

Posted by: Eric on March 12, 2010 10:02 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Consider spaces of polygonal paths in R^n.

Posted by: jim stasheff on March 12, 2010 1:12 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

As Andrew says here at MathOverflow, piecewise smooth paths form a terrible manifold.

Posted by: David Roberts on March 12, 2010 1:00 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

The nnLab page refers this result to 5.2.9.1 in HTT, but my copy of HTT has no section 5.2.9.

Section numbers changed in the arXiv update from July 2008. In the current version, section 5.2.9 starts on page 307.

Is that also true for 1-categories?

It is true for automorphims of the category of posets. This is prop. 5.2.9.14.

From this the statement is deduced for (,1)(\infty,1) -categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, prop. 5.2.9.11, which means that posets CC are characterized by the fact that

π 0(,1)Cat(D,C)Hom Set(π 0(,1)Cat(,D),π 0(,1)Cat(,C)) \pi_0 (\infty,1)Cat(D,C) \to Hom_{Set}( \pi_0 (\infty,1)Cat(*,D) , \pi_0 (\infty,1)Cat(*,C) )

is an injection for all D(,1)CatD \in (\infty,1)Cat.

This is preserved under automorphisms of (,1)Cat(\infty,1)Cat, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or () op(-)^{op} there, by the above statement for posets.

Now finally the main point of the proof is to see that (,1)Cat(\infty,1)Cat is strongly generated by the linear posets ΔCat\Delta \subset Cat, in that the identity transformation of the inclusion functor Δ(,1)Cat\Delta \hookrightarrow (\infty,1)Cat exhibits Id (,1)CatId_{(\infty,1)Cat} as the left Kan extension

Δ (,1)Cat Lan=Id (,1)Cat. \array{ \Delta &\hookrightarrow& (\infty,1)Cat \\ \downarrow & \nearrow_{Lan = \mathrlap{Id}} \\ (\infty,1)Cat } \,.

Posted by: Urs Schreiber on March 9, 2010 7:04 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Section numbers changed in the arXiv update from July 2008.

Thanks. Could we establish a convention that when referring to a work that has been so modified, we say explicitly to which version a reference refers (if it matters)? It’s disconcerting to be told to look at HTT 5.2.9.1, and open the copy on my desk to find that there is no section 5.2.9! Now I know, and hopefully I will remember, but we will hopefully continue to acquire new readers who don’t yet know. I’ve added such comments to the nlab pages HTT and (∞,1)Cat.

It is true for automorphims of the category of posets…. From this the statement is deduced for (,1)(\infty,1)-categories

Presumably the same proof then applies to the 1-category of 1-categories, or even the (2,1)-category of 1-categories. Of course, the (3,1)-category of 2-categories has (at least) three nonidentity automorphisms, namely () op(-)^{op}, () co(-)^{co}, and () coop(-)^{coop}, and presumably so does the (,1)(\infty,1)-category of (,2)(\infty,2)-categories. Does the (,1)(\infty,1)-category of (,n)(\infty,n)-categories always have exactly 2 n2^n automorphisms?

(,1)Cat(\infty,1)Cat is strongly generated by the linear posets Δ(,1)Cat\Delta\subset (\infty,1)Cat, in that the identity transformation of the inclusion functor Δ(,1)Cat\Delta\hookrightarrow (\infty,1)Cat exhibits Id (,1)CatId_{(\infty,1)Cat} as the left Kan extension [of the inclusion along itself].

I think that everyone else calls that property being dense; being a strong generator refers instead to the weaker property that maps out of the subcategory in question are jointly strong-epic. (However, the two are equivalent when the big category is locally presentable; I presume that (,1)Cat(\infty,1)Cat is a locally presentable (,1)(\infty,1)-category?)

Posted by: Mike Shulman on March 11, 2010 3:26 PM | Permalink | PGP Sig | Reply to this

Re: A Perspective on Higher Category Theory

I presume that (∞,1)Cat is a locally presentable (∞,1)-category?)

Ahm, no, it is not, is it?

It would be a locally presentable if it were, equivalently,

  • a reflective sub(,1)(\infty,1)-category of an (,1)(\infty,1)-presheaf (,1)(\infty,1)-category;

  • presented by a combinatorial simplicial model category.

So Grpd\infty Grpd is, being \infty-presheaves on the point and presented by sSet QuillensSet_{Quillen}.

But (,1)Cat(\infty,1)Cat would not seem to be. sSet JoyalsSet_{Joyal} is enriched over itself and hence not a “simplicial model category” which implicitly means sSet QuillensSet_{Quillen}-enriched.

But likely (,1)Cat(\infty,1)Cat as an (,2)(\infty,2)-category is a locally presentable (,2)(\infty,2)-category, where likely these are the reflective subcategories of (,2)(\infty,2)-presheaf (,2)(\infty,2)-categories presented by sSet JoyalsSet_{Joyal}-enriched model categories. And possibly the statement you mention is true after all. But I don’t know, I’d need to think about this.

Even more urgently, though, I need to jump on a train to Standed airport and reply to some other emails…

Posted by: Urs Schreiber on March 11, 2010 4:38 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Sorry (,1)Cat(\infty,1)Cat is of course presented also by the model structure on marked simplicial sets , which is combinatorial and sSet QuillensSet_{Quillen}-enriched. So yes, that should make it locally presentable.

Posted by: Urs Schreiber on March 11, 2010 5:00 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Sorry (∞,1)Cat is of course presented also by the model structure on marked simplicial sets, which is combinatorial and sSet Quillen-enriched. So yes, that should make it locally presentable.

Good. Cat is a locally presentable 1-category, and I would find it weird if that broke in higher dimensions.

Posted by: Mike Shulman on March 11, 2010 7:27 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

The main post is very beautiful; Tom is as usually to the point. This sharpening of the viewpoint inside the community of people inclined to categorical thinking is however far from the ear of the rest of the mathematicians. In a representation theory seminar few days ago somebody introduced the path algebra of a quiver/directed graph, and I just thought to help the speaker saying to the audience: take the free category on a quiver and then take the category algebra of it. Nobody wanted to listen; and I protested saying that it is more clear if one separates the definition into these two parts: taking all paths (with bonus: composition of paths) and then linearizing. But the answer was: “it is more clear to YOU, categorists”. From very sympathetic audience, appreciating category theory much more than average mathematician. But even those who appriciate and sympathise it, usually do it in a wrong way. Like category theory is a “good language to compactly express something by functoriality” or naturality condition and alike; sometimes a little better, including few universal properties, adjoint functors and Yoneda arguments (these belong to the culture in algebraic geometry at least), what is all of course short of true self-aware categorical thinking and sometimes indispensable level of abstraction.

Posted by: Zoran Skoda on March 8, 2010 5:25 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

What I find a bit frustrating about the discussion we are having here and had in previous entries is that there is not much of an exchange of arguments between the “criticism” that is being voiced and the replies that are being given.

We heard the complaint that Kan complexes and quasi-categories are not an algebraic model for higher categories. In reaction to that Thomas Nikolaus wrote an entire article that explains that there is a very nice algebraic model easily obtained here

So what’s the reply to that reply now? Do those who complained about the non-algebraicity of models of \infty-groupoids and (,1)(\infty,1)-categories agree that this is one good algebraic model? If not, it would be very useful to hear why not.

Then we heard the complaint that it is unclear even intuitively why a Kan complex should be anything like an \infty-groupoid. Replies to that were given and some lines were written into an nnLab entry at

Does this answer the question? If not, it would be useful to know how that fails to answer the question.

More generally, this series of posts was started with me listing three points where modern higher category theory drastically serves to clean up concepts and make them more categorical. Now the main point made in this entry here was that higher category theory should have clean categorical concepts. So we all agree on that. It would be nice to know therefore, what disagreement there is, as there must be, I suppose, with the three points I talked about.

Posted by: Urs Schreiber on March 17, 2010 1:56 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

You are politely not naming any particular person here, saying instead things like “criticism that is being voiced”, “those who complained”, “we heard the complaint”, and so on. I think this makes the discussion unnecessarily mysterious. Maybe I’m just paranoid, but when you don’t name names, it makes me start worrying that you’re worrying that I’m worrying about simplicial methods in nn-category theory.

So, let me just say: I agree with all your points here. I think the concept of a Kan complex captures a lot of intuitions about \infty-groupoids in a very elegant way. I would love an approach to \infty-groupoids that laid out all the coherence structures in a neatly dissected form — so that, for example, I could easily survey all the homotopy types with π 3=/2\pi_3 = \mathbb{Z}/2, π 5=/2\pi_5 = \mathbb{Z}/2 and all other homotopy groups trivial. Kan complexes aren’t that. But that doesn’t stop me from loving Kan complexes, or simplicial methods in general.

So, I think you must be talking to Tom or someone.

Posted by: John Baez on March 18, 2010 4:43 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Urs noted

…there is not much of an exchange of arguments between the “criticism” that is being voiced and the replies that are being given.

As I wrote in my book,

mathematicians, unlike philosophers, have no particular inclination to engage in sustained argumentative activity. I am inclined to believe that this is due to a deficiency in mathematical training, rather than because it is unnecessary. (p. 214)

Come on, people. Prove me wrong. One of my motivations for starting the Café was to provide an arena for such discussions. And I’m only encouraging you to become more civilized:

Being civilized means living so far as possible dialectically, that is, in constant endeavour to convert every occasion of non-agreement into an occasion of agreement. A degree of force is inevitable in human life, but being civilized means cutting it down, and becoming more civilized means cutting down still further. (Collingwood, The New Leviathan, 39.15)

Perhaps the intellectual sphere is not quite what Collingwood had in mind, but when a group believes that another group has taken over an intellectual field for reasons they don’t agree with, it is experienced as being done by force.

[Originally posted on the temporary site at 2:03 pm on 18 March 2010.]

Posted by: David Corfield on March 25, 2010 5:12 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

How does agreeing to disagree fit into Collingwood’s description of civilization? My feeling is that mathematicians may disagree on particular points, but nevertheless agree that everyone’s time is more productively employed in doing mathematics than in arguing, especially when the arguments seem unlikely to produce any more agreement. More generally, I don’t think it follows from “civilization means cutting down on force” that civilization necessarily means trying to make all disagreements into agreements; aren’t some disagreements likely to prove impossible to rectify? Civilization just means not fighting even when you disagree.

[Originally posted on the temporary site at 9:00 pm on 18 March 2010.]

Posted by: Mike Shulman on March 25, 2010 5:16 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

This is probably a naive comment, but I’ve always thought that argument (in the sense of trying to convince someone about something by semi-rigorous logical and “empirically suggestive” points) is pretty much inversely proportional to the ability to do “definitive” experiments (or in mathematics constructs concrete theories and calculations). No one argues much about the validity and domain of applicability of Newtonian mathematical physics because the people who care can do experiments to resolve the issue, whereas the areas of physics where people are primarily arguing are the ones where there’s little experimental accessibility. So it’s surely where there’s no direct “experiment” one can do where one should see arguments (in the above sense). And so, take for example Arnold Neumaier’s posts on FMathML: my (possibly incomplete) understanding, was that everything can be acheived with the “representation” of either side, the debate is about which one has “nicer” properties which are somewhat personal and difficult to define. This is precisely where one would expect to see “argument”.

(Originally posted on the temporary site at 1:33 pm on 19 March 2010.)

Posted by: bane on March 25, 2010 5:20 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Mike, yes nobody’s going to believe that all disagreements are resolvable to the satisfaction of all parties. And one shouldn’t underestimate the time and effort required to re-enact the other side’s thought sufficiently well so as to bring two bodies of thinking together to be in a position to be able to understand where the fundamental disagreement lies. It often feels that that time would be better spent developing your own point of view with the hope that ‘truth will out’.

So it’s a question of a balance with dangers from weighing down either pan. My thought is that the ‘getting to the heart of the difference’ pan is more often neglected than the ‘blast ahead with my own ideas’ pan. It is easy to pick up on discontent from those who feel their own position has been paid insufficient attention. Of course, it may be right to overlook theor work, or they may be partially to blame for failing to explain it, but I dare say we can dig up examples of unfairly neglected work.

Bane, I agree some of the considerations as to which approach is faring better are subtle and often inconclusive. You can see why the ability to prove longstanding problems is taken to be an important indication, even if it can be argued that the importance of a particular problem has been overrated. But still some people tend to steer mathematics down eventually more fruitful paths than others do.

(Originally posted on the temporary site at 2:38 pm on 19 March 2010.)

Posted by: David Corfield on March 25, 2010 5:24 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Ok. I agree that in this particular case, there is more “getting to the heart of the difference” that ought to be done.

(Originally posted on the temporary site at 3:13 am on 20 March 2010.)

Posted by: Mike Shulman on March 25, 2010 5:26 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I’d be happy to agree to disagree, but was still trying to find out what the disagreement actually is.

I was hoping we could have a productive discussion on the disagreement that moves us all forward. But if my attempts at this are regarded as an attempt to pick a fight I’ll better stop.

(Originally posted on the temporary site at 10:00 pm on 18 March 2010.)

Posted by: Urs Schreiber on March 25, 2010 5:29 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I tried posting this yesterday, but obviously failed.

No one on the skeptical side of this discussion seems particularly interested in talking details — probably for the obvious reason that these objections pretty clearly point to dissatisfaction with things that a lot of people here are interested in — so I thought I would throw out some thoughts as a response to Urs. I will preface this by saying that I have nothing against simplicial or homotopical methods in category theory as should be obvious from the research I do, but I am still a category theorist first and I think that means that deep down I have certain preferences. I also want to say that nothing here means I think the work that other people have done is bad or in some way wrong, but that does not stop me from saying that some results are not the ones that I would want to be the final story.

I want to point out three ways in which I think people could be skeptical of the current trend of doing everything quasicategorically: a philosophical point on things vs models of things, the question of how algebraic something is, and the question of how algebraic the totality of some objects is.

First, people should realize that saying ‘an (,1)(\infty,1)-category is a simplicial set such that something’ is quite different from saying ‘there is a model for (,1)(\infty,1)-categories given by simplicial sets such that something’ even if the theorems all come out the same. In both cases there is a standard for knowing what it means to be talking about (,1)(\infty,1)-categories, but they are very different standards. I think a major source of discomfort with how these issues get discussed is that the first of these statements seems to imply something like ‘the right definition of (,1)(\infty,1)-category is the one that proves the most theorems in the shortest amount of time’ while in fact most people that have been thinking about higher categories for a while probably already have something in mind, and that something is almost certainly not a simplicial definition. The key is that no one is arguing about whether some theorem is true, but instead that some use of language goes against our basic notions about what higher categories are. This is nit-picky and might even seem silly, but is something that I think is lurking underneath the discussion.

My second point is to respond directly to Urs on the topic of how algebraic these things are. Personally, I would say not very. In particular, I would say that the algebraic Kan complexes of Thomas Nikolaus, while definitely being interesting, do not really feel that algebraic to me. I think the issue there is that for me, algebraic structures have two parts: operations and axioms. On the other hand, the reason that using quasicategories is so successful is that the axioms are pushed off to infinity, and I don’t think that changes by adding specified fillers. I liked Tom’s spectrum of how algebraic things are because it points out something really important — there are different grades of being algebraic, and there are probably a lot of people that will not be happy to call something algebraic until it is about 99.99% algebra. There is not anything wrong with that, it gives people that like both sides of the story something pretty hard to work towards, namely taking non-algebraic structures and making them really, extremely algebraic.

Third, I would like to make very explicit something that I consider important and that John almost touched upon at the beginning of this discussion — if you take some algebraic objects and then declare them to be the same by reference to some external notion of sameness, then you are quite possibly doing something non-algebraic. For instance, we can declare two categories to be ‘the same’ if they have homotopy equivalent nerves. This is an extremely useful thing to do for certain purposes, but it is not algebra. That is why I think the Homotopy Hypothesis is a really serious thing: to prove it, you should really show that something completely algebraic is somehow the same as something completely non-algebraic.

(Originally posted on the temporary site at 10:38 am on 19 March 2010.)

Posted by: Nick Gurski on March 25, 2010 5:38 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Nick Gurski wrote:

would throw out some thoughts as a response to Urs.

Thanks, Nick! Very nice, I appreciate it a lot.

I’ll reply briefly point-by-point:

Re “quasicategories are just one model”.

Yes, certainly! In fact it looks like it is not even going to be the preferred model among those who you might think prefer it. Most recent development revolve around the model given by complete Segal space. This is a pretty immediate model of the idea “category weakly enriched in \infty-groupoids”.

But quasicategories are a model that accomplish one important aspect: the proof of existence of (,1)(\infty,1)-category theory. from the results of Joyal and Lurie it seems clear that whatever other definition of (,1)(\infty,1)-category you come up with, it should have a simplicial nerve that identifies it with a quasicategory.

I think we should take category theory seriously: the way we think of our objects (here: (,1)(\infty,1)-categories) as being presented is irrelevant. What matters is the interrelation between all possible such objects: the category they form.

Would you agree with that?

Apart from this I can’t quite see what is so ugly about Kan complexes and quasi-categories. If you just hold two pages next to each other, one carrying the definition of quasi-category, the other carrying the definition of a more algebraic, say operadic, definition, the second one will look way more involved. For one, it will likely not fit on that one page. So I am not sure I see the intuition for why that is somehow intrinsicaly better. But also, by the above remark, I don’t care too much, what matters is not the presentation of the objects, but their interrelation.

Re: algebraicity of Nikolaus’ algebraic quasi-catgeories:

Maybe it serves to unwind a bit what it actually is that Thomas has there: I think it is really exactly the typical description of bicategory, tricategory, tetracategory, etc, fully systematized for the (n,1)(n,1)-case.

Here is what I mean: what is called a “choice of fillers” is really: “an algebraic specification of composition with its coherences”.

Let’s look at it starting in low dimensions: First for each pair of composable morphisms, you are to choose a 2-morphism with these as source. The target of that 2-morphism is the chosen composite. The collection of all these chosen 2-morphisms is the composition operation.

Then for every triple of composable morphisms, you are to choose a 3-morphism going between the corresponding four composition 2-morphisms. The choice of that 3-morphism is a choice of associator.

Nect, for sequences of five composable morphisms, you are to chose a 4-morphism between the corresponding five associators. That’s the corresponding pentagonator.

And so on. If you decide to look at an object where this is truncated at some point, this means in the next step you assume to be able to choose these fillers to be identity nn-morphisms. The fact that these exist is then the coherence law for all the structure that you have chosen before.

So one “algebraic quasi-category” in Thomas Nikolaus’s sense is really essentially exactly a choice of weak (n,1)(n,1)-category in the style of bicategory, tricategory, etc.

Do you see what I mean?

Re point 3:

I am maybe still not sure if I see what these arguments (those you give and those you refer to) are supposed to say. If we are talking about pure catgeory theoretic higher category theory, we shouldn’t care much about topological spaces. I don’t care much about them.

What I care about is that Kan complexes and quasi-categories are sensible in themselves have been proven to be models for (,0)(\infty,0)-category theory and (,1)(\infty,1)-category theory in every conceivable sense of what this should mean. So for me the important point here seems to be that whatever other model for (n,1)(n,1)-category that one comes up with, it will be very helpful to check if it matches the behaviour of quasi-catgeories under its notion of simplicial nerve. For if it does, it immediately proves that good (,1)(\infty,1)-category theory will also exist for your model. If it doess not, then, conversely, this is a bit worrisome, since it is unlikely that there will be two inequivalent good “(,1)(\infty,1)-category theories”.

Concenring algebraic definition of equivalence. if you look at what equivakence of Nikolaus-quasi-categories means, you will see that it means precisely the kind of equivalence that we are used to from bicategories, tricategories, etc.: you map all the kk-moprhism over such that all composition is respected up to corresponding images of corresponding compositors.

Or so I think. Let me know what you think.

(Originally posted on the temporary site at 12:39 pm on 19 March 2010.)

Posted by: Urs Schreiber on March 25, 2010 5:48 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

what is called a “choice of fillers” is really: “an algebraic specification of composition with its coherences”.

that’s essentially true but highly nontrivial to prove in detail: it took Duskin 100 pages to fully work out even the case of bicategories!

That said, I love Kan complexes, and they are still my favorite model for infinity-groupoids, non-algebraicity notwithstanding. In fact, if one is interested, as I am, in higher groupoid objects in categories other than Set (e.g. smooth manifolds – so, higher Lie groupoids), requiring the existence of global horn fillers is unrealistic: more often than not the fillers exist only locally. For instance, one must admit such non-algebraic Lie 2-groupoids in order to obtain a finite-dimensional model for the String Lie 2-group, due to Shommer-Pries. I am only mentioning this so long after the thread is over because this point doesn’t seem to have been made.

Posted by: Dmitry Roytenberg on June 2, 2010 10:05 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

it took Duskin 100 pages to fully work out even the case of bicategories!

I feel like this should be a lot easier if you ask only for an equivalence between the theories of locally groupoidal bicategories and “homotopically 2-truncated” quasicategories, rather than asking for a characterization of precisely those simplicial sets which arise as nerves of bicategories up to isomorphism. For instance, a locally groupoidal bicategory is the same as a simplicial category whose hom-spaces are nerves of groupoids, and the equivalence between simplicial categories and quasicategories should restrict to an equivalence between “2-truncated” ones.

requiring the existence of global horn fillers is unrealistic: more often than not the fillers exist only locally. For instance, one must admit such non-algebraic Lie 2-groupoids in order to obtain a finite-dimensional model for the String Lie 2-group, due to Shommer-Pries.

One response to that may be that what you’re really constructing is a stack with values in higher groupoids, but if you insist on representing this stack by an internal groupoid, then the composition operation may not be globally representable.

Posted by: Mike Shulman on June 3, 2010 2:52 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Mike writes:

I feel like this should be a lot easier if you ask only for an equivalence between the theories of locally groupoidal bicategories and “homotopically 2-truncated” quasicategories, rather than asking for a characterization of precisely those simplicial sets which arise as nerves of bicategories up to isomorphism.

True, it is certainly much clearer conceptually. Besides, composing Duskin’s nerve with the left adjoint to the homotopy coherent nerve (does it have a name?) lands you in strict locally groupoidal 2-categories, so you get coherence as a byproduct. However, I understood that Urs was talking about the latter: interpreting horn-filling conditions as compositions and coherences, which takes quite a bit of work once you get down to the details.

what you’re really constructing is a stack with values in higher groupoids, but if you insist on representing this stack by an internal groupoid, then the composition operation may not be globally representable.

That seems to be the situation. The “insistence”, however, is not a whim but the difference between formulating a moduli problem (defining a stack) and solving it (proving the stack is representable): the former is easy while the latter is usually quite hard. There’s value in representability: if I understand correctly, the reason stacks were introduced in the first place was that some moduli problems didn’t have a solution as sheaves but did as stacks (most famously the Deligne-Mumford stacks of stable curves). In any case, I don’t think this lack of algebraicity is bad: it just means compositions are not straight maps but bimodules, which in the world of (higher) groupoids is the more natural notion anyway.
Posted by: Dmitry Roytenberg on June 3, 2010 11:45 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Oh, absolutely representability is important. I didn’t mean to imply that I thought it wasn’t. I just meant that it’s not as if algebraic notions become useless once you internalize; the notions are still equivalent when you are in the right context.

Posted by: Mike Shulman on June 5, 2010 1:13 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Agreed.

Posted by: Dmitry Roytenberg on June 6, 2010 10:00 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Dmitry writes:

In fact, if one is interested, as I am, in higher groupoid objects in categories other than Set (e.g. smooth manifolds – so, higher Lie groupoids), requiring the existence of global horn fillers is unrealistic: more often than not the fillers exist only locally. For instance, one must admit such non-algebraic Lie 2-groupoids in order to obtain a finite-dimensional model for the String Lie 2-group, due to Shommer-Pries.

Mike replies:

One response to that may be that what you’re really constructing is a stack with values in higher groupoids, but if you insist on representing this stack by an internal groupoid, then the composition operation may not be globally representable.

To just add a perspective on this, which is likely obvious to you all:

I (also) think entities like the string 2-group are best thought of as objects in a gros (,1)(\infty,1)-topos of \infty-Lie groupoids, which means to think of them as \infty-stacks on a suitable site of smooth test spaces. Such as smooth manifolds or just Cartesian spaces.

In terms of the non-algebraic model for \infty-groupoids given by Kan complexes, this may of course be modeled by a model structure of presheaves with values in the standard model on simplicial sets, and the fibrant objects here will have test-object-wise fillers, which will however not fit together into an internal Kan complex.

But now we can apply Thomas Nikolaus’ theorem to this model. The theorem says that the standard model on simplicial sets for Kan complexes is Quillen equivalent to one for Kan complexes with chosen fillers. This implies in turn that there is an equivalent model for our \infty-stacks in terms of presheaves with values in these algebraic Kan complexes.

This guarantees that for whatever \infty-Lie groupoid you have, there is an algebraic model for it by an \infty-stack with values in algebraic \infty-groupoids, i.e. assigning \infty-groupoids with chosen composition and inverse operation on each test object, and such that restriction along test maps respects these composition and inverse operations.

(By the way: if we use the projective model structure on presheaves with values in some model for \infty-groupoids here, then by a general result on cofibrant objects in that model it follows that every object has a model which is degreewise a coproduct of representables. So if we work over manifolds, this says that every \infty-Lie groupoid is guaranteed to have a model that is degreewise a manifold. It may have a large number of components of differing dimension, though.)

Posted by: Urs Schreiber on June 3, 2010 6:11 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Urs, It is certainly true what you’re saying: on the sheaf level, algebraic models always exist. It’s representability that’s at stake here: you may have it in non-algebraic models but not in algebraic ones, and the string 2-group is a basic example of this. I believe that’s the kind of “grit” Tom talks about in his original post. Which leads me to your parenthesized comment: it’s a neat construction, but let’s see what it produces in a simple concrete case, such as the mapping space between two manifolds, viewed as a sheaf on Euc, constant in the simplicial direction. As 0-simplices, you’re supposed to take the coproduct of the domains of all the plots, as many times as there are such plots. Are you prepared to accept such a thing as a finite-dimensional smooth manifold? :)
Posted by: Dmitry Roytenberg on June 4, 2010 12:17 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I wrote in parenthesis:

(By the way: if we use the projective model structure on presheaves with values in some model for ∞-groupoids here, then by a general result on cofibrant objects in that model it follows that every object has a model which is degreewise a coproduct of representables. So if we work over manifolds, this says that every ∞-Lie groupoid is guaranteed to have a model that is degreewise a manifold. It may have a large number of components of differing dimension, though.)

Dmitry replies:

it’s a neat construction, but let’s see what it produces in a simple concrete case, such as the mapping space between two manifolds, viewed as a sheaf on Euc, constant in the simplicial direction. As 0-simplices, you’re supposed to take the coproduct of the domains of all the plots, as many times as there are such plots.

Yes, inded, that’s what the general construction yields. That’s why I said: “it may have a large number of components of differing dimension”.

Maybe I should have italicized “large”. ;-)

Are you prepared to accept such a thing as a finite-dimensional smooth manifold? :)

Well, I could ask back: did you tell me beforehand that this is not what you mean by a finite dimensional smooth manifold?

But, sophistery aside, of course I agree that this alone is not what one wishes to see in concrete applications. But also, this is just the very general construction that by design is as large as possible in order to apply in all cases.

I think it is still noteworthy that this is guaranteed to exist. In concrete cases you can then try to further trim this down.

For instance: let XX be a paracompact manifold. Then Dugger’s construction produces this cofibrant replacement in [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov} of it which is an immense hypercover that is degreewise a vast coproduct of contractibles.

But now this can be reduced drastically and still be a cofibant replacement: we can take the Cech nerve of a good open cover of XX. This is still degreewise a coproduct of representables and its degenerate simplices degreewise split off as a corpoduct. So by Dugger’s theorem in Universal homotopy theories it is cofibrant. Also, by using the characterization of local epimorphisms (generalized covers) in Dugger-Hollander-Isaksen, it is still weakly equivalent to XX.

So that’s now a nice and small cofibrant replacement. Of course we needed to do a bit of work by hand to obtain it, but the general formula with the huge hypercover at least did put an upper bound on how bad it could have been: at worst a degreewise coproduct of contractible manifolds. That’s pretty good already.

Posted by: Urs Schreiber on June 4, 2010 12:59 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

Urs,

I agree with everything you say, as long as it is not claimed that Dugger’s construction solves the representability problem (I doubt a Lie theorist would be happy if offered this as a proof of Lie III, for example). So let’s tentatively call this “pararepresentability” instead, then we get that all simplicial presheaves are pararepresentable. This is not a vacuous statement (unlike the “homotopy hypothesis” in Tom’s joke), since we don’t have that in the world of presheaves of sets, so I agree with you that it is valuable.

Still, the objects the construction yields do fall short of being finite-dimensional manifolds. For one thing, they are not second-countable. For another, they can’t be said to be finite-dimensional, since they not only have pieces of different dimensions, but all dimensions occur. (There’s also the small issue that coproducts of representables are not representable even in subcanonical sites with coproducts, but that can be easily fixed by sheafification.)

So, to make the representability problem interesting, one must impose some restriction on size (second countability), as well as some version of connectivity to make the dimension well-defined. For instance, one can define the dimension of a simplicial manifold to be the Euler characteristic of its normalized tangent complex at a total degeneracy; this is well-defined if it is finite and independent of the base point (trying this on Dugger’s construction doesn’t yield anything meaningful, since the degeneracies split off and all dimensions occur – not surprising for a construction that applies to all presheaves in the world). Lastly, I would like the horn (and hence the face) maps to be submersions.

Anyway, I am new here and not sure about all the etiquette, so please let me know if it is not appropriate to continue this discussion here.

Posted by: Dmitry Roytenberg on June 6, 2010 9:56 PM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

I’ll respond more later, but briefly to one of Nick’s points, it seems to me that the idea of “pushing axioms off to infinity” is not special to quasicategories, but is a general feature of all kinds of weak omega-categories. Why are quasicategories special?

(Originally posted on the temporary site at 3:16 am on 20 March 2010.)

Posted by: Mike Shulman on March 25, 2010 5:51 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

When I started to work with Cordier on the Homotopy Coherent Nerve (and hence on quasi-categories) and to think of Kan complexes as one of a possible set of models for weak infinity groupoids, I was concerned by exactly the sort of points that Nick is making. Ronnie Brown and his students had been introducing T-complexes and it was clear that giving a specification of a filler for each horn, satisfying certain axioms (just three which was marvellous), was completely algebraic. It was clear also that the very pretty T-complex structure on simplicial groups (satisfying one condition on the Moore complex) was encoding a lot of structure in a very compressed form. Laxifying the algebra, relaxing the conditions hence allowing a set of fillers but only in certain dimensions, and adjusting the axioms accordingly, is a very fruitful route to follow. Note that in a simplicial group or groupoid the filling algorithm (given in May for instance) is explicit and gives very nicely structured fillers. (Some work by Ali Mutlu and myself may be useful if you want to see what I mean here. I think this should be considered algebraic, but there is a combinatorial side that adds in something as well. )

If you have a Kan complex with extra axioms and or structure in this way you can algebraicise it by passing to the Wbar of the loop groupoid, and can see to what extent this gives you new filler information. (For instance if GG is a simplicial group satisfying the Moore complex condition mentioned above then Wbar(G) is a T-complex in a natural way and you can manipulate things explicitly. (The working can be tedious and is fraught with problems of a human nature… slips are extremely easy to make and difficult to find… but it can be done.) SImplicial groups also have explicit formulae for Whitehead products and other operations, and these can be interpreted both algeraically and geometrically.

If one relaxes the T-complex axioms you can still get somewhere algebraically. For instance asking for unique T fillers above a certain dimension but not worrying below that. (This needs careful handling and is easiest to see in the group T-complex case with the Moore complex condition.)

Finally returning to Kan complexes, and a side comment, it always seemed to me that they could be very useful for modelling things outside our normal area. A Kan complex has fillers,… but are they all equally probable!!!!! Suppose one laxifies things from T-complex by considering a probability distribution on each set of fillers, ‘the composite of two composable arrows is probably this one …, with probability whatever.’ There do seem to be situations (e.g. in Tom’s area of interest), where this may be worth looking into.

(Originally posted on the temporary site at 7:45 am on 20 March 2010.)

Posted by: Tim Porter on March 25, 2010 5:56 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

introducing T-complexes and it was clear that giving a specification of a filler for each horn, satisfying certain axioms (just three which was marvellous)

Can you say why we need any extra axioms at all? Just picking fillers seems to do the job.

(Originally posted on the temporary site at 10:05 am on 20 March 2010.)

Posted by: Urs Schreiber on March 25, 2010 5:58 AM | Permalink | Reply to this

Re: A Perspective on Higher Category Theory

The axioms gave the algebraic nature of things. For instance the third T-complex axiom implies associativity of the composition given by the fillers. This gave strict structures.

I am just saying that the ‘strict’ algebraic case corresponded to those three axioms. If you just have fillers then it is a bit like in universal algebra, fine for theory but without the equations governing the objects the calculation is more or less empty of meaning. It gives the operations only. If you take a free resolution of a group, it presents the group exactly BUT with a lot of redundancy. Pragmatically the free fillers added are going to give objects that will need a new set of tools for developing their calculation.

(Originally posted on the temporary site at 12:11 pm on 20 March 2010.)

Posted by: Tim Porter on March 25, 2010 6:01 AM | Permalink | Reply to this
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