March 2, 2010

Algebraic Models for Higher Categories

Posted by Urs Schreiber

[ guest post by Thomas Nikolaus ]

A few months ago we had here a discussion about $\infty$-groupoids and the homotopy hypothesis. In this discussion I had the idea to use algebraic Kan complexes as a model for $\infty$-groupoids and to endow the category of algebraic Kan complexes with a model stucture. After a fair amount of work I finally could prove that these things really do work out.

Notes on Algebraic Kan Complexes (pdf)

Abstract. We establish a model category structure on algebraic Kan complexes. In fact, we introduce the notion of an algebraic fibrant object in a general model category (obeying certain technical conditions). Based on this construction we propose algebraic Kan complexes as an algebraic model for ∞-groupoids and algebraic quasicategories as an algebraic model for (∞,1)-categories.

Simplicial sets have been introduced as a combinatorial model for topological spaces. It has been known for a long time that topological spaces and certain simplicial sets, called Kan complexes, are the same from the viewpoint of homotopy theory. To make this statement precise Quillen introduced the concept of model categories and equivalence of model categories as an abstract framework for homotopy theory. He endowed the category $\Top$ of topological spaces and the category $\sSet$ of simplicial sets with model category structures and showed that $\Top$ and $\sSet$ are equivalent in his sense. He could identify Kan complexes as fibrant objects in the model structure on $\sSet$.

Later higher category theory came up. A 2-category has not only objects and morphisms, like an ordinary (1-)category, but also 2-morphisms, which are morphisms between morphisms. A 3-category has also 3-morphisms between 2-morphisms and so on. Finally an $\infty$-category has $n$-morphisms for all $n \geq 1$. Unfortunately it is very hard to give a tractable definition of $\infty$-categories.

An interesting subclass of all $\infty$-categories are those $\infty$-categories for which all $n$-morphisms are invertible. They are called ∞-groupoids. A standard construction from algebraic topology is the fundamental groupoid construction $\Pi_1(X)$ of a topological space $X$. Allowing higher paths in $X$ (i.e. homotopies) extends this construction to a fundamental ∞-groupoid $\Pi_\infty(X)$. It is widely believed that every $\infty$-groupoid is, up to equivalence, of this form. This believe is called the homotopy hypothesis.

There is another important subclass of $\infty$-categories, called (∞,1)-categories. These are $\infty$-categories where all $n$-morphisms for $n \geq 2$ are invertible. Thus the only difference to $\infty$-groupoids is that there may be non-invertible $1$-morphisms. In particular the collection of all $\infty$-groupoids forms a $(\infty,1)$-category. Another example of a $(\infty,1)$-category is the category of topological spaces where the $n$-morphisms are given by $n$-homotopies. In the language of $(\infty,1)$-categories a more refined version of the homotopy hypothesis is the assertion that the fundamental $\infty$-groupoid construction provides an equivalence of the respective $(\infty,1)$-categories.

From the perspective of higher category theory Quillen model structures are really presentations of $(\infty,1)$-categories, see e.g. HTT appendix A.2 and A.3. Hence we think about a model category structure as a generators and relations description of a $(\infty,1)$-category. A Quillen equivalence then becomes an adjoint equivalence of the presented $(\infty,1)$-categories. Thus the classical Quillen equivalence between topological spaces and simplicial sets really encodes an equivalence of $(\infty,1)$-categories.

Keeping this statement in mind, it is reasonable to think of a simplicial set $S$ as a model for an $\infty$-groupoid. The $n$-morphisms are then the $n$ simplices $S_n$. And in fact there has been much progress in higher category theory using simplicial sets as a model for $\infty$-groupoids. This model has certain disadvantages. First of all a simplicial set does not encode how to compose $n$-morphisms. But such a composition is inevitable for higher categories. This problem is usually adressed as follows:

The model structure axioms on $\sSet$ imply that in the corresponding $(\infty,1)$ category each simplicial set is equivalent to a fibrant object i.e. a Kan complex. It is possible to interpret the lifting properties of a Kan complex $S$ as the existence of compositions in the $\infty$-groupoid.. Although the lifting conditions ensure the existence of compositions for $S$, these compositions are only unique up to homotopy. This makes it sometimes hard to work with Kan complexes as a model for $\infty$-groupoids. Another disadvantage is that the subcategory of Kan complexes is not very well behaved, for example it does not have colimits.

The idea of this paper to solve these problems is to consider a more algebraic version of Kan complexes as model for $\infty$-groupoids. More precisely we will consider Kan complexes endowed with the additional structure of distinguished fillers. We call them algebraic Kan complexes. We show that the category of algebraic Kan complexes has all colimits and limits. Furthermore we endow it with a model structure and show that it is Quillen equivalent to simplicial sets. The name algebraic will be justified by identifing algebraic Kan complexes as algebras for a certain monad on simplicial sets. The fact that algebraic Kan complexes really model $\infty$-groupoids will be justified by a proof of the appropriate version of the homotopy hypothesis.

We will generalize this notion of algebraic Kan complex to algebraic fibrant objects in a general model category $\mathcal{C}$, which satisfies some technical conditions. In particular we show that $\mathcal{C}$ is Quillen equivalent to the model category $Alg\mathcal{C}$ of algebraic fibrant objects in $\mathcal{C}$. We show that $Alg\mathcal{C}$ is monadic over $\mathcal{C}$ and that all objects are fibrant. In addition we give a formula how to compute (co)limits in $Alg\mathcal{C}$ from (co)limits in $\mathcal{C}$.

Finally we apply the general construction to the Joyal model structure on $\sSet$. This is a simplicial model for $(\infty,1)$-categories. The fibrant objects are called quasicategories. We propose the category $Alg\mathcal{Q}$ of algebraic quasicategories as our model for $(\infty,1)$-categories. One of the major advantages is that the model structure on algebraic quasicategories can be described very explicitly, in particular we will give sets of generating cofibrations and trivial cofibrations. Such a generating set is not known for the Joyal structure.

Posted at March 2, 2010 9:47 PM UTC

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Re: Algebraic Models for Higher Categories

Around proposition 2.11 I think there are some references to AlgKan and sSet which are misplaced.

Awesome paper by the way!

Posted by: David Roberts on March 3, 2010 4:02 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Hi David,

here . Unfortunately I can not edit the entry, so I try to keep the version on the nLab up to date.

Posted by: Thomas Nikolaus on March 3, 2010 9:44 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I’m very excited by this development. Even if it doesn’t turn out to be technically useful, I think the conceptual connection between algebraic and non-algebraic notions of higher category that this builds is very nice. But I’ve also been thinking idly about whether algebraic Kan complexes or quasicategories would be any easier to compare with things like Batanin $\omega$-categories.

I also think the conclusion that any model category in which all acyclic cofibrations are monic is Quillen equivalent to one in which all objects are fibrant is pretty cool.

Another question that just occurred to me reading this:

One of the major advantages is that the model structure on algebraic quasicategories can be described very explicitly, in particular we will give sets of generating cofibrations and trivial cofibrations. Such a generating set is not known for the Joyal structure.

Is there any (previously known) model structure purporting to describe $(\infty,1)$-categories for which there is a concretely describable generating set of acyclic cofibrations? Most of them seem to be constructed as Bousfield localizations or using other cardinality arguments.

Posted by: Mike Shulman on March 3, 2010 6:17 AM | Permalink | PGP Sig | Reply to this

Re: Algebraic Models for Higher Categories

But I’ve also been thinking idly about whether algebraic Kan complexes or quasicategories would be any easier to compare with things like Batanin ω-categories.

I have been thinking this: my vague impression – which may be wrong – has been that Todd Trimble produced his description of tetracategories by applying some general machine that starts with a more operadic definition, such as that of a Trimble $\omega$-category. (Is that right, Todd?)

Now, the axioms for tricategories, tetracategories, etc. should pretty much be the characterization of systems of chosen fillers in a Nikolaus-$(\infty,1)$-category (in the relevant special cases): all those coherence axioms state which properties a system of chosen fillers will have, once it is chosen.

So I was thinking that if Todd knows a machine to go from Trimble $4$-categories to tetracategories, he might know a machine to go from Trimble-$(\infty,1)$-categories (i.e. Trimble $\omega$-categories satisfying the evident extra conditions) to algebraic quasi-categories. Or maybe at least have a good guess for how one might proceed.

Posted by: Urs Schreiber on March 3, 2010 6:40 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I’ll try to say a few words about these machines before touching upon your question.

First, there are general ‘machines’ for writing down the polyhedral shapes of higher categorical data, which were directly inspired by Stasheff’s classic work on the combinatorics of $A_\infty$-spaces and $A_\infty$-maps. A difference is that units (and how weak maps are supposed to interact with units) don’t appear in Stasheff’s work, but they are not too hard to work in.

Before I describe one such machine, let me mention Stefan Forcey as someone who has also thought a lot about the nature of these shapes and their geometric descriptions as subsets of Euclidean space defined by specified linear inequalities.

If you allow me to talk not about the shapes directly but about their barycentric subdivisions, which are simplicial sets, then I can say very quickly what those are in operadic language. Let $O$ be the monad for the monadic underlying functor

$U: Operad \to Set^{\mathbb{N}}$

from nonpermutative (aka ‘Stasheff’) operads to graded sets. Let $t$ be the terminal operad. Then the bar resolution of $t$, which in the language of two-sided bar constructions is denoted

$B(O, O, t),$

is a simplicial operad, a universal cofibrant replacement of $t$, and the elements of this operad $M$, which I call the ‘monoidahedral operad’ (after the associahedral operad $K$ of Stasheff), are exactly the simplices which appear in the subdivisions of all those classical shapes like Stasheff polytopes. In other words, if you carefully work through the combinatorics, then out pops the ten triangles in the barycentric subdivision of the pentagon; they reside in the 2-dimensional part of the simplicial set which consists of operations of arity 4, just as one would expect from Stasheff’s work.

The subdivisions of the polyhedral shapes which show up for $A_\infty$ maps can also be described by a bar resolution or universal cofibrant replacement, this time starting with the terminal graded simplicial set $t$ as bimodule over this monoidahedral operad $M$.

Since these techniques are simplicial, I’ll bet that it wouldn’t be that hard to make explicit contact with the algebraic Kan complexes being explored by Thomas. The guiding idea is to consider universal cofibrant replacement of the terminal operad, which is contractible. The guiding philosophy I had all along (since 1995) is that a reasonable (algebraic) theory of weak $n$-categories should be governed by a universal contractible operad in some sense. This seems to be close to Batanin’s philosophy, who worked it all out nicely in terms of globular operads.

I was about to write some paragraphs on why my own progress in implementing this simple philosophy was blocked, and I can still do that if desired, but you asked about the connection with so-called ‘Trimble $n$-categories’, and to what extent Trimble 4-categories correspond to tetracategories. I still feel odd using my name here; I originally called them “flabby $n$-categories” and I’ll do so again for this comment.

As I think you know (Urs), flabby $n$-categories were introduced to give a rigorous algebraic meaning to the term ‘fundamental $n$-groupoid’. The original definition was based on topology, and took advantage of a specific curious feature of $Top$ itself and the interval object therein: that there’s a contractible operad $E$ whose $n$-ary operations are maps

$I \to I^{\vee n} = I \vee \ldots \vee I$

(where $\vee$ is a join on bipointed spaces, basically tensor product of cospans). How is this specific to $Top$? Because in a lot of model categories, there might not be any such maps (as maps between bipointed objects)! For example, there are no such maps for the interval object in simplicial sets.

The relevance of this particular operad $E$ is that we have, for any space $X$, a natural action

$E(n) \times P X(x_0, x_1) \times \ldots \times P X(x_{n-1}, x_n) \to P X(x_0, x_n)$

where $P X(x, y)$ is the space of paths $\alpha: I \to X$ such that $\alpha(0) = x$ and $\alpha(1) = y$. (Full details have been written up by Tom Leinster and by Eugenia Cheng & Aaron Lauda; see for instance here.) Given that, we define

$\Pi_n(X) := \Pi_{n-1}(P X)$

and note that by applying the product-preserving functor $\Pi_{n-1}$ to this natural action, $\Pi_n(X)$ carries an action by $\Pi_{n-1}(E)$, which is a pithy way of inductively defining what a flabby $n$-category is.

The ‘machine’ for defining flabby $n$-categories can be generalized away from $Top$, under certain conditions. We could work with other model categories $V$ (for example), provided that a few basic ingredients are in place:

• There’s a product-preserving functor $\Pi_0: V \to Set$, to get the induction started;
• There’s a contractible operad $M$ valued in $V$;
• $M$ acts naturally on “enough” path spaces $P X$ to push the induction through.

I’m not sure what I mean by this last; I need to think about it more. But if some such set of conditions applied to the monoidahedral operad $M$ in simplicial sets, then the connection between, on the one hand, my hoped-for approach to things like tetracategories, and on the other hand, flabby $n$-categories, could be made a great deal tighter I think. (There are however strict interchange laws for flabby $n$-categories, so this is not a “fully weak” notion of $n$-category, which is what I was originally hoping to describe in my various approaches to $n$-categories.)

If this comment is hard to follow, then I’m happy to try to make it clearer.

Posted by: Todd Trimble on March 3, 2010 10:19 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Enlightening! You wrote:
I still feel odd using my name here; I originally called them “flabby n-categories”

Reminds me of Steenrod’s dislike for `Steenrod’s reduced powers’ ;-)

Thanks, Todd

Posted by: jim stasheff on March 4, 2010 1:22 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I was about to write some paragraphs on why my own progress in implementing this simple philosophy was blocked, and I can still do that if desired

I’d like to hear it!

Posted by: Mike Shulman on March 5, 2010 8:57 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I also think the conclusion that any model category in which all acyclic cofibrations are monic is Quillen equivalent to one in which all objects are fibrant is pretty cool.

Thanks. But I am still not sure how mild the assumption that “all trivial (acyclic) cofibrations are monic” really is. Mike and I had a discussion about this before.

Does anyone know a model category where this assumption is not satisifed???

In most of the standard model categories all cofibrations are per definition monic. As an example where not all cofibrations are monic, Mike came up with the model structure on small categories. There the cofibrations are the functors that are injective on objects. Nevertheless, since the weak equivalence are the equivalences of categoris, the trivial cofibrations are injective on objects and morphisms and thus monic.

Posted by: Thomas Nikolaus on March 3, 2010 10:20 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Let me slightly modify that question: does anyone know an interesting and easily describable model category where this assumption is not satisfied? I feel like there must be trivial examples as well, but I’m also guessing that it will fail in some Bousfield localizations. Note in particular that in any such example, it must be the case that not all objects are fibrant, since any acyclic cofibration with fibrant domain is necessarily split monic.

Posted by: Mike Shulman on March 3, 2010 6:34 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Here’s a pretty trivial example. There’s a model structure on the category of groupoids which Bousfield localizes the usual model structure and which is Quillen equivalent to $Set$ (with its trivial model structure where the only weak equivalences are isomorphisms). The cofibrations are the functors which are injective on objects, the weak equivalences are the functors inducing bijections on $\pi_0$, and the fibrations are the isofibrations which additionally induce isomorphisms on $\pi_1$ with all basepoints. The fibrant objects are the disjoint unions of cliques, and there can be lots of acyclic cofibrations (with non-fibrant domain) that are not injective on arrows.

Posted by: Mike Shulman on March 4, 2010 1:02 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Of course trivial examples are easy to obtain. Just take any weak factorization system (L,R) where the maps in L are not all monic and take the corresponding trivial model structure where L are the cofibrations, R are the fibrations and all maps are weak equivalences.

Posted by: Marc Olschok on March 4, 2010 9:22 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Ah I see. Okay, these are nice counterexamples. But they are degenerate in the sense, that the homotopy cateogries (or better: The simplicial Localizations) are trivial. Thus they don’t encode a homotopy theory one might be interested in…

But however, thank you for these nice counterexamples.

Posted by: Thomas Nikolaus on March 5, 2010 4:57 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I have just glanced at the paper, but would note that there are several developments that should be mentioned and are not. I am thinking of the ideas of Nick Ashley on simplicial T-complexes and group T-complexes. Of course, the work of Dominic Verity would seem very closed related and yet gets no mention. Also two papers by Gary Nan Tie, a student of Duskin, are relevant.

Some of these only refer to crossed complex / strict cases, but surely they deserve mention and comparison.

Posted by: Tim Porter on March 3, 2010 7:35 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

the work of Dominic Verity would seem very closed related and yet gets no mention.

I am less familiar with Dominic Verity’s work than I wish I were, but as far as I can see his definition of weak complicial sets is a geometric simplicial definition of higher categories that generalizes the definition of quasi-categories to a notion of simplicial nerves of general weak $\infty$-categories.

In particular, a weak complicial set is, unless I am misremembering, a simplicial set with the property that for certain horns – the complicial horns – fillers are guaranteed to exist.

So no fillers are chosen and the model is not algebraic.

What Thomas is doing here goes in a different direction.

But of course one can turn ths around: Thomas has shown how to make Kan complexes and quasi-categories become algebraic by choosing fillers. The obvious next question is: can one do the analogue with weak complicial sets?

I mean, it is clear that one can consider the notion of a weak complicial set with a collection of chosen complicial horn fillers. But what can one say about the collection of objects obtained this way?

Posted by: Urs Schreiber on March 3, 2010 8:34 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

In a simplicial T-complex, there is a requirement that each horn has a unique ‘thin’ filler. The properties of thin elements are described (3 axioms), thus there is a choice of thin filler specified by the choice of the thin elements, thus a certain ‘algebraic’ quality about the notion.

In a group T-complex, there is even an algorithm to specify the fillers. The strictness of the model is obtained by the T-complex condition which is simply that in each dimension the subgroup generated by the degenerate elements intersects the Moore complex trivially - that simple. Again this is algebraic.

In a complicial set (not the weak version) there is a very similar condition on the thin elements and again this amounts to a choice of composite via a choice of filler. Dominic’s work predates the re-emergence of quasi-categories, as it feeds off Ross Streets earlier work and John Robert’s notion of a model for non-Abelian cohomology.

In Jack Duskin’s hypergroupoid again the condition is algebraic, given by an isomorphism from a space of horns. (The situation is slightly different as in the usual form the models are often truncated.)

Gary Nan Tie’s work explored the links between Duskin’s models and Nick Ashley’s simplicial T-complexes. I should also mention Pilar Carassco’s hyper-crossed complex.

I do not say that what Thomas has done is not different, no, merely that I am very surprised not to see some mention of this extensive earlier work.

(I can provide references for the above articles if people would like, but in fact most are referenced in versions of the Menagerie, and can be sought there.)

Posted by: Tim Porter on March 3, 2010 8:57 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

In a simplicial T-complex, there is a requirement that each horn has a unique ‘thin’ filler.

Oh, I see.

So after we have chosen all fillers in a Kan complex as Thomas considers, if we declare all chosen fillers as thin, do we get a simplicial T-complex?

(Have to run now, will look into this in more detail later.)

Posted by: Urs Schreiber on March 3, 2010 9:38 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I think the answer is no to your question. The point being that in simplicial T-complexes thin elements satisfy some ‘categorified’ equations corresponding to the axioms listed in the n-Lab entry for instance.

Posted by: Tim Porter on March 3, 2010 7:06 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Lets see:

In a T-complex each horn must have a _unique_ thin filler. That means in particular that each thin n-simplex $\Delta(n)$ is “the” filler for all of its $n+1$ different horns $\Lambda^k(n)$. In an algebraic Kan complex there might be diffrent fillers for those horn. It seems to me that this is implicitly a certain relation that fillers in T-complexes have to satisfy. I thought about such a implicit relation for algebraic Kan complexes. But then the category of them would not have had these nice properties it now has.

So to conclude: Each T-complex determines an algebraic fibrant object, but not the other way around. Thus algebraic fibrant objects are “weaker” or more “lax” versions of $\infty$-groupoids then T-complexes.

Posted by: Thomas Nikolaus on March 3, 2010 7:37 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

there are several developments that should be mentioned and are not. I am thinking of the ideas of Nick Ashley on simplicial T-complexes

So a simplicial T-complex is like a stratified simplicial set (which may be that underlying a weak complicial set) with unique thin fillers, right?

I think it would be nice if what Thomas did for quasi-categories could be generalized to weak complicial sets. Is there any work on simplicial T-complexes that goes in this direction?

Posted by: Urs Schreiber on March 3, 2010 9:02 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

Tim Porter wrote:

I have just glanced at the paper, but would note that there are several developments that should be mentioned and are not.

First of all thank you for looking at my paper. Sorry that I have not mentioned those works. The reason is simply the fact that I was not aware of it. I hoped to get some reply, like yours.

I am thinking of the ideas of Nick Ashley on simplicial T-complexes and group T-complexes. Of course, the work of Dominic Verity would seem very closed related and yet gets no mention. Also two papers by Gary Nan Tie, a student of Duskin, are relevant.

I can not say somethink about a connection at the moment, I first have to read what they exactly do. The only thing I did up to now was looking at Dominic Veritys work. I thought that he mainly considered weak complicial sets, where the fillers are only guaranteed to exist. Furthermore he also equipt his stratified simplicial sets with a model structure. Thus it would be easy to apply the methods from section 2 of my paper to it. And of course you are right, this should be done and mentioned.

At the moment I do not see the link to non-weak complicial sets…

I can provide references for the above articles if people would like, but in fact most are referenced in versions of the Menagerie, and can be sought there.

Could you maybe give a hint, what would be good references to start reading about these things?

Posted by: Thomas Nikolaus on March 3, 2010 9:59 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

That is why I replied! There are nice results that I go through in the Menagerie. (I think that the present version in the Lab has that stuff in but if not I can send you the on going one (which is a lot longer!))
I will put some more references on the Lab. Both simplicial T and complicial are poorly represented. The Menagerie treatment is meant as an introduction, so any comments as to where more detail or examples or …. would be welcome.

I have yet to read your paper so don’t thank me for that yet! One obvious question is whether the algorithmic algebraic fillers given for a simplicial group in May’s book for instance, give it the structure of an algebraic Kan complex, and if they do, does thaat structure give an induced one on the classifying space. (This may be in your paper in which case I should have read it before asking the questions!)

One thought that you probably have some ideas about (and I ask this of all readers of the café postings) if one has any version of T-complex, or algebraic Kan complex, then it is natural to ask for the T-fibrations (and there may be an algebraic Kan fibration in your paper). Has anyone a candidate for such a thing? I have asked Ronnie Brown and he does not remember seeing such an idea. The point is that the classifying space of a crossed complex classifies fibre bundles with some thinness conditions in the fibres …, but what form do they most naturally take?

Posted by: Tim Porter on March 3, 2010 11:23 AM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I have edited some of the entries in the n-Lab (simplicial T-complex, group T-complex, etc) that might be useful here. It occurs to me that these categories may be varieties in the category of algebraic Kan complexes. That seems sort of ‘right’.

Posted by: Tim Porter on March 3, 2010 5:47 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

I have edited some of the entries in the n-Lab (simplicial T-complex, group T-complex, etc) that might be useful here.

Thanks, Tim!

And let me also link to

Posted by: Urs Schreiber on March 3, 2010 6:31 PM | Permalink | Reply to this

Re: Algebraic Models for Higher Categories

My congratulations to Nikolaus for not only beautiful and nontrivial work, but also for I think the choice of an important and conceptually needed problem in the big picture!

Posted by: Zoran Skoda on March 3, 2010 2:19 PM | Permalink | Reply to this
Read the post Confessions of a Higher Category Theorist
Weblog: The n-Category Café
Excerpt: Why might we continue to search for more models for higher categories, aside from the simplicial ones that have proven so useful?
Tracked: April 29, 2010 3:03 AM
Read the post Algebraic model structures
Weblog: The n-Category Café
Excerpt: Algebraic model categories and weak factorization systems.
Tracked: June 16, 2010 10:18 PM

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