The n-Category Café

John ends his TWF week 190 with a remark on operads in the context of “categorification of C”. It would be great if one could read more about that (or have I just missed where to look?).

I’ll say Merry Christmas, Eugenia!

Some years back I made a pact with my immediate family: no presents between the adults. (There’s really no getting around giving presents to the children.) I do exchange gifts with my wife, but my parents and my brother and I are very happy not having to deal with buying gifts for each other.

As long as we’re on the topic of the annoyances with Christmas, I’ll mention the constant barrage of Christmas songs one hears while going about one’s business. (Is that a US thing?) Some of the traditional carols are lovely, but the majority of songs are just plain grating! (I really can’t stand “It’s the most wonderful time of the year”!)

I like the expository gift idea; hadn’t heard of that one.

Why thank you!

The world wants to know (and I speak on its behalf): how can we get your introduction to operads?

Thanks for the comments, Jon. TQFT is “topological quantum field theory”; I’ll probably expand that. I think PROP stands for “PROducts and Permutations”, or something like that, but it’s one of those words like “radar” where the unabbreviated form has been mostly forgotten. So I think I’ll leave that one as it is.

I don’t understand why it suffices to restrict to representable $U$. An extension of $S \rightarrow X$ along $f$ is determined by a cone of maps from the representables in the colimit diagram for $U$, but I don’t see how this helps.

A related question, how could I have copied the relevant paragraph from the above post into my comment? Is the procedure to TeX it by hand and then use the html “blockquote” tag?

how could I have copied the relevant paragraph from the above post into my comment?

There’s no way to copy a paragraph with math in it unless the person writing the paragraph was considerate enough to sign their comment with PGP. If they did that, then you can click on “PGP Sig” and the blog software will also show you the itex source of the page. For instance, I’ve signed this comment, and $(h^{er})_e \; \mathbf{is} \; \sum\; \mathbb{M}\alpha\theta$.

I think usually what people do when quoting text with math is to manually re-itex the math; at least, that’s what I do. But you don’t need to use html tags, if you choose a “Markdown with itex2mml” text filter you can just put a “>” in front of the text you copied and it will get blockquoted automatically.

Here’s something I currently understand at a mechanistic level, but would like to understand at a conceptual level.

It’s about subtoposes and Lawvere-Tierney topologies. The theorem mentioned in my notes is that given a topos $E$, subtoposes of $E$ correspond one-to-one with Lawvere-Tierney topologies on $E$.

A subtopos of $E$ is a subcategory of $E$ with certain properties.

A Lawvere-Tierney topology on $E$ is a subobject of $\Omega_E$ with certain properties. (It’s usually presented as a map $\Omega_E \to \Omega_E$ with certain properties, but that’s the same thing.)

So:

a subcategory of $E$ with certain properties

is the same as

a subobject of $\Omega_E$ with certain properties.

Is there a nice story explaining this?

Thanks for this. The question of which toposes really deserve to be thought of as generalized spaces is interesting, I think.

I wonder how one decides what constitutes a good answer to that question. Of course, one can always employ one’s mathematical nose: ‘this smells right’. But I don’t know what criteria are used by people who work seriously on these things.

I guess you know this, but just to be clear: I meant the slogan ‘a topos is a generalized space’ in its most common, primitive sense—the class of (sober topological) spaces embeds into the class of toposes. In the same way, if I was explaining the definition of monoid to someone who knew what groups were, I might tell them that monoids were a generalization of groups. Later, if they get interested in monoids, they’ll discover that some monoids behave a lot more like groups than others.

(As you’ll have guessed, I’m not going to put anything about cohesion and big/little toposes into the article. I’ve already failed to resist the irresistible temptation to write more than I actually said in the talks.)

I’ve been following the unfolding axiomatization of cohesive toposes from a distance (i.e. interested but not wholly paying attention). Let me see if I can understand something simple.

Take a small category $\mathbf{C}$ with both an initial and a terminal object. Then according to the definition in your comment, $Psh(\mathbf{C}) = Set^{\mathbf{C}^{op}}$ is a topos of cohesion ($=$ cohesive topos?).

Proof:  the unique functor $\mathbf{C} \to \mathbf{1}$ has both a left and a right adjoint. This chain of three adjoint functors between $\mathbf{C}$ and $\mathbf{1}$ gives rise, by Kan extension, to a chain of five adjoint functors between $Psh(\mathbf{C})$ and $Psh(\mathbf{1}) = Set$. Throw away the leftmost functor. What remains is a chain of four adjoint functors (going in the correct direction), with the leftmost one preserving all limits, and in particular finite products. QED.

On the other hand, it’s not true for arbitrary small categories $\mathbf{C}$ that $Psh(\mathbf{C})$ is a topos of cohesion. Maybe someone knows exactly which $\mathbf{C}$ it is true for. (I couldn’t see it on the $n$Lab page.)

My questions: why does $Psh(\mathbf{C})$ deserve to be thought of as a category of generalized spaces for any small category $\mathbf{C}$ with initial and terminal objects? And why not for general $\mathbf{C}$?

Take a small category $C$ with both an initial and a terminal object. Then according to the definition in your comment, $PSh(C) = Set^{C^{op}}$ is a topos of cohesion

Yes. The special minimal case of the interval category $C = \{\emptyset \to *\}$ was already discussed at cohesive site in the section Families of sets.

For $X \in PSh(C)$ we may interpret the value

$$X(\emptyset \to *) = (\Pi_0(X) \leftarrow \Gamma(X))$$

as a $\Pi_0(X)$-indexed family of sets. The functor $\Gamma$ produces the disjoint union of all sets in the family, the functor $\Pi_0$ produces the indexing set. So this is the simplest notion of cohesive lumps of points : we take a set and partition it into subsets.

For general $C$ with initial and final object this interpretation remains true, only that now in addition for every object $c \in C$ there is a notion of “hierarchy of lumps”

$$X(\emptyset \to c \to *) = (\Pi_0(X) \leftarrow X(c) \leftarrow \Gamma(X)) \,,$$

where first the collection $\Gamma(X)$ of all points is partitioned into cohesive pieces $X(c)$ which are then further partitioned into cohesive pieces $\Pi_0(X)$.

I have added this more general discussion to the entry now.

Notice, however, that while these examples have a quadruple of adjoint functors with the leftmost one preserving (in particular) finite products, they fail to satisfy two extra axioms that Lawvere introduced later in

Bill Lawvere, Axiomatic cohesion.

One of them demands that the canonical morphism $\Gamma(X) \to \Pi_0(X)$ is an epimorphism for all $X$. This says that “every cohesive piece of points indeed has at least one point”. This fails in the above class of examples: there are cohesive pieces with no points.

topos of cohesion (= cohesive topos?).

Yes, that’s how we ended up using the terminology, deviating a bit from Lawvere’s usage. I think “cohesive topos” fits better with “connected topos”, “local topos”, “locally connected topos”. Which is good, because a cohesive topos is all three of this.

On the other hand, it’s not true for arbitrary small categories $C$ that $PSh(C)$ is a topos of cohesion.

Yes, a counter-example emphasized by Lawvere are the presheaf toposes $G Set \simeq PSh(\mathbf{B}G)$ for $G$ a finite non-trivial group.

Maybe someone knows exactly which $C$ it is true for. (I couldn’t see it on the $n$Lab page.)

I don’t know. What’s the most general condition that ensures that taking the limit over a presheaf has a right adjoint?

What we have on the $n$Lab is a notion of cohesive site $C$ that encapsulates some sufficient conditions for $Sh(C)$ to be cohesive. This is taylored towards examples of cohesive topose that one wants naturally in practice, such as smooth toposes modelling (synthetic) differential geometry.

(A question that I want to get the answer to is: which derived geometry gives a cohesive $(\infty,1)$-site?)

My questions: why does $Psh(C)$ deserve to be thought of as a category of generalized spaces for any small category $C$ with initial and terminal objects? And why not for general $C$?

This is asking: How can we tell that the axioms for a cohesive topos are good ? How can we see that indeed they do formalize the notion “category of generalized geometric spaces” in a suitable sense.

This is a good question, in particular since there is room to fine-tune the axioms. Lawvere himself has slightly modified the axioms over the years.

Questions like these find their answers by checking against loads of examples: in all the cases where we have categories that we certainly want to consider as categories of geometric spaces: do the axioms admit them? And are they sufficient to abstractly deduce all the properties of and constructions on geometric spaces that we are interested in?

It is questions like this that made me become interested in cohesive toposes: I was looking for contexts of geometry internal to which we have good general abstract notions of Lie theory and differential cohomology .

From looking at examples, I have come to the following conclusion: we should keep the core axiom (quadruple of adjoints over the base topos, with the leftmost preserving products) and discard the extra axioms (cohesive pieces have points and pieces of powers are powers of pieces). Instead, we should strengthen the core axiom by demanding that quadruple of functors not on the topos over a given site, but on the corresponding $(\infty,1)$-topos.

If we do this, the leftmost adjoint $\Pi$ enhances from a functor that assigns connected components $\pi_0$ to a functor that assigns the full fundamental $\infty$-groupoid of a generalized geometric space. Together with the fact that this has threefold right adjoints, this alone implies a pretty large amount of general structure on a cohesive $(\infty,1)$-topos, that seems to fully justify to thinking of this definition indeed as correctly encapsulating the notion of a category of generalized geometric spaces. A fairly long list of such structres is given at Structures in a cohesive $(\infty,1)$-topos.

I think this is a good proof of concept that the axiomatics of cohesive toposes is good in that it well captures the notion of categories of generalized geometric spaces.

I don’t quite agree with the title of Lawvere’s article; I think that every Grothendieck topos can be regarded as a generalized space. In particular, if a topos classifies some geometric theory, then it can be regarded as the space of all models of that theory. It’s just that some toposes (those which classify a suitable type of theory, such as the theory of local rings or local $C^\infty$-rings) can fruitfully also be regarded as categories of generalized spaces.

i haven’t followed this discussion closely enough yet to understand the concept of “cohesive topos” very well or to know how much i sympathize with the attempt to develop it (or to know whether the question that i’m about to ask has already been answered), but i have a naive question about it. if cohesiveness is supposed to be somewhat antagonistic to localicness, then might this antagonism actually amount to some sort of factorization system for geometric morphisms in which the cohesive morphisms come before the localic ones?

(i’ve thought a bit about various sorts of “moore-postnikov”-like factorization systems for toposes and higher toposes, and i’m vaguely aware that other people have worked on such ideas too but i haven’t read much of their work yet.)

A fairly long while ago, Tom had remarked on the term cohesive topos :

I prefer “topos of cohesion” to “cohesive topos”. Actually, I’m not super-keen on “topos of cohesion”; to my ears it sounds a bit clumsy. But I find “cohesive topos” actively misleading. Surely it’s the objects of the topos that are cohesive, not the topos?

(I had mentioned some possible replies.)

This here is to point out that, as I only discovered just now, Bill Lawvere himself did after all use the term “cohesive topos” previously, in 2008, after it had evolved from category of Being in his 1990 Como contribution to category of cohesion in his last 2007 article: 18 years later, back in Como, he speaks about

• Cohesive Toposes – Combinatorial and Infinitesimal Cases

By the way, behind the above link is also a video of his first of three lectures – but only of the first handful of minutes.

Does anyone have more information about this 2008 Como lecture?

Hi Scott,

Thanks for your suggestions. To take the second one first:

The second suggestion is in your examples, put the definition of a presheave as a contravariant functor between a POSet and a ?

Sorry, I don’t understand. By “examples”, do you mean Examples 1.6 (examples of toposes)? Also, I’m having trouble parsing the rest of your sentence.

I thought the definition of presheaf on a category was given (albeit slightly implicitly) in Example 1.6(iv), and the definition of presheaf on a space was in Example 1.6(v). But if something’s unclear, I’ll fix it.

Right after you introduce the word “monic” give the definition

I find this very interesting. First: it’s no trouble to put the definition of monic in—but that’s not the interesting thing. The interesting thing, to me, is that it never crossed my mind that someone might (1) be comfortable with the Yoneda lemma, pullbacks, etc, and (2) be interested in a categorical introduction to topos theory, but (3) not know the definition of monic.

In case it needs saying—and this being the internet, perhaps it does—I don’t regard the definition of monic as ‘so basic that everyone should know it’ or anything snooty like that. Indeed, I wouldn’t describe any mathematical definition that way. We know what we know, and we’re interested in what we’re interested in, often for random reasons. It’s just the mixture of (1), (2) and (3) that surprises me.

But maybe I shouldn’t be surprised. I know that my own mathematical knowledge forms a strangely-shaped space, so I shouldn’t be surprised at anyone else’s.

Thanks. Those are nice facts. The one about the classifying topos of a theory as the space of models is definitely going in, and the other ones, I’ll think about.

I’m very conscious of my own ignorance here. I probably don’t know much about how the parts of the elephant are connected. To use a less surreal metaphor, maybe it’s more like a city that you’ve visited several times but don’t know well, so that you’re familiar with the layout of several neighbourhoods but don’t have a good sense of how they join together.

You may wish to try

L’Enseignement Mathématique.

It publishes expository articles. The caveat is that the article should be of interest to a general mathematical audience and be accessible. Of course what that means is in the eyes of the editors.

I think your article is interesting and well written. The category theory background, on the other hand, may be a bit too demanding for a typical mathematician (in other words you assume that your readers have seen some category theory). Maybe if you add a two page appendix on adjoint functors and Yoneda, you’d be good to go.

If I’m comfortable handling the notation of set theory and I now want to switch my attention to a topos other than SET, what do I need to bear in mind? What new rules do I have to follow when manipulating and interpreting those familiar-looking formulas in this new context, and what can I no longer do as before?

Essentially the single logical rule that fails is that of excluded middle: the internal logic of a general topos is intuitionistic logic.

Some discussion on how to reason inside a topos (or in a general category) is at internal logic.

The examples-section describes basics of internal logic in a presheaf topos and highlights how the failure of excluded middle arises.

Alex Simpson has a spectacular application of locale theory to the theory of random sequences.
He constructs a (pointfree) locale of random sequences, sequences satisfying all probabilistic (=measure 1) laws. It is easy to see that the locale has no points, still the locale itself has many interesting properties.

the category of “locales internal to the topos Sh(C)”, for some locale $X$, is equivalent to the slice category of (external) locales over $X$.

Is there an analogous statement for locales internal to a non-localic topos?

There is the “noncommutative Gelfand theorem” for a noncommutative $C^*$-algebra

$A$ which realizes $A$ as an internal commutative $C^*$-algebra in the presheaf topos $PSh(CSub(A))$ over the semilattice of commutative subalgebras of $A$ and finds a locale internal to the topos that is Gelfand-dual to the internal $A$.

Mike wrote

…every (localic) subgroup of a localic group is closed! Thus, since $\mathbb{Q}$ is not a closed sublocale of $\mathbb{R}$, it cannot be a localic subgroup. I believe that non-closed subgroups of topological groups are often regarded as pathological – for instance, in equivariant algebraic topology, the orbit category of a topological group is defined only using closed subgroups.

Is this linked to the definition of a Klein geometry as a pair, a Lie group and a closed Lie subgroup? So with a localic Lie group (smooth localic group?) we could do Klein geometry without worrying about closedness of subgroup?

Mike wrote (lots of very interesting things and then)

Probably it would take you too afar afield to say even a fraction of this in the paper, but it might be nice to convey some idea of the difference between locales and sober spaces.

Yes… it’s always a subjective judgement, how much to include. One can of course decide that one is always going to include that interesting thing just over the horizon, but in that case one’s text is named after a pachyderm.

I probably feel that it’s enough to have introduced locales, pointed out the adjunction between locales and toposes, pointed out the adjunction between spaces and locales, and said a little about the equivalence to which the latter adjunction restricts. That seems like enough to absorb at a single sitting. But perhaps I’ll add a sentence tempting the reader to go and read about the kind of things you mentioned in your comment.

I already have the feeling that I included slightly too much. In Latex it’s 24 pages, and I would have liked to have kept it strictly under 20. But as I wasn’t willing to either condense the writing style or cram more onto each page, that would have meant chopping some material. So I decided to leave it.

Mike wrote:

I believe that non-closed subgroups of topological groups are often regarded as pathological

The quotient of a topological group by a subgroup is Hausdorff if and only if the subgroup was closed. (Reference here.) Given that people often see non-Hausdorffness as pathological, it would be natural if non-closed subgroups were also seen as pathological.

Is there a name for the kind of generalized smooth space that completes the analogy topological space : locale :: topological manifold : ???

Yes… “topological manifold”. (-: Since open subsets of $\mathbb{R}^n$ have enough points, if you glue them together along opens you will still get something having enough points. More generally, any locale that admits a surjection from a topological space is itself topological.

(Interestingly, that means that “non-Hausdorff manifolds”, like the line with a doubled point, give some nice examples of sober but non-Hausdorff spaces.)

Note, though, that if you work constructively, then even the locale $\mathbb{R}$ need not have enough points! The topological space of real numbers can have some poor properties (e.g. the Heine-Borel theorem can fail) but this is not true if you work with the locale instead. So in that case it might be the “localic manifolds” which are what you want to work with instead. I don’t know whether anyone has tried putting Lorentzian or Kahler structures on them.

Submitted now. It’s this Latex version that I’ve submitted for peer review. Andrew Stacey has written a script for converting Latex to Instiki, and he’s going to run that on my revised version (thanks, Andrew!), so please don’t spend time cleaning up the version that’s currently on the Lab: it’ll be replaced.

Urs mentioned coauthorship. I’m not sure what he had in mind, but in my mind this isn’t a possibility except in the case that a referee has a large amount of input. The paper submitted for peer review is single author, and while of course I’m extremely grateful to those who are helping with the conversion process, I don’t think this is a reason to be named as a coauthor.

(For comparison: every now and then, Reprints in Theory and Applications of Categories runs a project whereby a team of people type up some out-of-print classic category theory text. I’ve been involved in a couple of these, and it’s a fair amount of work. But the typists aren’t named as authors.)

I’m aware that this doesn’t sound particularly generous, but I think it’s fair; and the system will collapse if it’s not perceived to be fair.

If I understand the process correctly, we’re going to convert the Latex document into wiki format while the peer review is going on. The wiki page should be an exact replica of the Latex document, modulo formatting. (In some ways it will be better, because it’ll be hyperlinked.) If the paper passes peer review then—after the necessary revisions according to the referee report(s)—that replica page will be frozen, stamped “approved”, and write-protected. A duplicate page will also be created, which will be like any other $n$Lab page: anyone can edit it, and it takes on a life of its own, beyond my control. But it’s important to me to have complete control of the peer-reviewed version of the text, as in an ordinary journal submission.

Urs mentioned coauthorship. I’m not sure what he had in mind

By $n$Lab coauthor I meant what is maybe better called contributor . I was just trying to sell the idea that people might help with converting the source, not that they become coauthors of the peer-reviewed version of your article that will be moved to the $n$Journal.