2-Toposes

January 12, 2008

Posted by David Corfield

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As 2-toposes seem to be cropping up a bit, here and here, let’s see if we can attract some experts to teach us about them.

On p. 36 of Mark Weber’s Strict 2-toposes, a 2-topos is defined as a finitely complete cartesian closed 2-category equipped with a duality involution and a classifying discrete opfibration. Cat is a good example of a 2-topos. Are there other familiar ones?

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A 1-topos is a kind of 1-category. The 1-category of sets is a paradigmatic example, in which $1 \to 2$ (the set of truth values) is the classifier.

A 2-topos is a kind of 2-category. The 2-category of categories is a paradigmatic example, in which Pointed Set $\to$ Set is the classifier.

Hmmm, so what’s a 0-topos? It ought to be a kind of 0-category or set. The set of truth values should be a paradigmatic example. What kind of set should it be?

If in Set we have $A \to 2^{A}$ , and in Cat we have $C \to {Set}^{C^{op}}$ , is there a 0-Yoneda?

If it is possible to extract an internal language from a topos, what results from a similar process applied to a 2-topos? Do we find that it supports a higher order categorified logic?

Posted at January 12, 2008 4:13 PM UTC

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Re: 2-toposes

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David asked:

a lot of questions.

Let me make some guesses, to get the ball rolling.

Cat is a good example of a 2-topos. Are there other familiar ones?

Trivially, finite categories must be an example. Then, since sheaves on a site give an important class of 1-toposes, surely stacks (of categories) on a site give an important class of 2-toposes.

I think Urs asked somewhere whether the 2-category of toposes is a 2-topos. I would have thought not, since Mark requires a duality involution. In the case of $Cat$ this must be $(-)^{op}$ , but this isn’t going to work for toposes: the opposite of a topos isn’t a topos.

I don’t think I know any examples of toposes except for sheaves on a site, finite sheaves on a site, and trivial variants thereof. I think it’s the case that the effective topos, and probably other realizability toposes, provide further examples. Also, one has the classifying topos for any geometric theory; I don’t know if that’s always a topos of sheaves. Maybe there are analogues of these things for 2-toposes. And maybe there are 2-toposes that are not the analogue of anything in the 1-dimensional world!

Hmmm, so what’s a 0-topos? It ought to be a kind of 0-category or set. The set of truth values should be a paradigmatic example. What kind of set should it be?

My guess: a 0-topos is just a set. Agreed: the set $2 = {true, false}$ should be a paradigmatic example.

If in Set we have $A \to 2^{A}$ , and in Cat we have $C \to {Set}^{C^{op}}$ , is there a 0-Yoneda?

Well, a $(- 1)$ -category is a truth value in the classical sense (i.e. $true$ or $false$ ), and there’s meant to be just one $(- 2)$ -category, somehow corresponding to the truth value $true$ . (I suppose I mean “corresponding” in the same way that the $(- 1)$ -categories $true$ and $false$ correspond to the $0$ -categories $0$ and $1$ , and that a $0$ -category corresponds to the discrete $1$ -category on it.) Also, the exponentiation of truth values is surely implication: $β^{α}$ means $(α \Rightarrow β)$ . So I think the $0$ -Yoneda embedding is the statement $α = (α \Rightarrow true),$ where $α \in 2 = {true, false}$ . How come that’s an equality? Well, both $α$ and $(α \Rightarrow true)$ are elements of $2 = (- 1) Cat$ , which is a $0$ -category, so they’re either equal or not — there’s no chance of mapping between them.

Now I have a question of my own. Cantor’s theorem says that $not (A ≅ 2^{A})$ , for sets $A$ . I spent a little while trying to imitate this for categories, to show that $not (C ≃ {Set}^{C^{op}})$ for categories $C$ (small if you like). But I couldn’t, the sticking point being that ‘op’. Can anyone (dis)prove this?

Posted by: Tom Leinster on January 12, 2008 5:10 PM | Permalink | Reply to this

Re: 2-toposes

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In my last paragraph, I asked for someone to prove or disprove the statement that for categories $C$ , we never have $C ≃ {Set}^{C^{op}}$ . I’d still like someone to do that.

But I added in brackets that $C$ could be ‘small if you like’, which I now regret. For small $C$ , it’s easy to prove. If $C ≃ {Set}^{C^{op}}$ then $C$ has all small limits, which if $C$ is small implies that $C$ is a preorder (equivalent to a complete lattice). So ${Set}^{C^{op}}$ is a preorder, which (by considering constant functors) implies that $C$ is the empty category $0$ . But ${Set}^{0^{op}}$ isn’t empty.

Posted by: Tom Leinster on January 12, 2008 5:58 PM | Permalink | Reply to this

Re: 2-toposes

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It’s also easy to extend this result to locally small categories. If $C ≃ {Set}^{C^{op}}$ and $C$ is locally small, then ${Set}^{C^{op}}$ is also locally small, and therefore $C$ must be small. Then apply your argument.

(By the way, it seems that an alternate argument when $C$ is small is: if $C ≃ {Set}^{C^{op}}$ , then ${Set}^{C^{op}}$ is essentially small. But this is absurd in any case, because there is a proper class of non-isomorphic constant presheaves on any nonempty category.)

On the other hand, when $C$ is not locally small, it has no Yoneda embedding $C \to {Set}^{C^{op}}$ , so the question becomes somewhat stranger.

Posted by: Mike Shulman on January 12, 2008 11:36 PM | Permalink | Reply to this

Re: 2-toposes

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Hmm, OK. I agree that you’ve proved it, but — and this is going to sound ungrateful — I’m disappointed. I was hoping to see a proof resembling Cantor’s diagonal argument for sets, but with an exciting extra ingredient that handled the contravariance.

Let me try an a priori harder question, in the hope of provoking an answer that I like. Cantor actually showed that for sets $S$ , no function $S \to 2^{S}$ is surjective. Can it be shown that for categories $C$ , no functor $C \to {Set}^{C^{op}}$ is essentially surjective on objects?

Digression: I like thinking of Cantor’s theorem in a way that makes it look like a kind of opposite of the Riesz Representation Theorem. Given a Hilbert space $H$ (real, say), we have a bilinear map $〈 -, - 〉 : H \times H \to ℝ$ and Riesz tells us that every map $H \to ℝ$ is of the form $〈 x, - 〉$ for some $x \in H$ . Cantor’s theorem says that if we take a set $S$ and a map $〈 -, - 〉 : S \times S \to 2$ then not every map $S \to 2$ is of the form $〈 x, - 〉$ for some $x \in S$ . My new question can be phrased as: given a category $C$ and a functor $〈 -, - 〉 : C^{op} \times C \to Set,$ can it be shown that not every functor $C \to Set$ is isomorphic to $〈 X, - 〉$ for some $X \in C$ ? (A functor $C^{op} \times C \to Set$ is usually called a $(C, C)$ -module, or $(C, C)$ -bimodule, or a profunctor or distributor from $C$ to $C$ .)

This formulation of Cantor’s theorem is in Lawvere and Schanuel’s book, and maybe in Lawvere and Rosebrugh’s book too.

Posted by: Tom Leinster on January 13, 2008 2:21 AM | Permalink | Reply to this

Re: 2-toposes

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Well, I don’t have an answer (yet), but I just want to clarify a notational point that confused me briefly, in case it confuses anyone else. The Riesz theorem is about the particular bilinear map $〈 -, - 〉$ , namely the inner product of the Hilbert space. The inner product can be thought of as analogous to the hom-functor of a category, in some sense which to my knowledge no one has yet succeeded in making precise.

Thus, the analogue of this particular question for sets would be to consider only the “equality” function, corresponding to asking whether the `singleton’ functor $S \to 2^{S}$ is surjective (obviously not!) and for categories it would be to consider only the hom-functor and ask whether the Yoneda embedding is essentially surjective (obviously not!). What you’re asking instead in these two cases is whether there is any function $S \times S \to 2$ or functor $C^{op} \times C \to Set$ whose slices can represent any function $S \to 2$ or functor $C^{op} \to Set$ . Thus the use of the same notation $〈 -, - 〉$ in the three cases confused me briefly, but now I’m set straight.

By the way, I can’t find that version of the Riesz theorem in either Lawvere-Schanuel or Lawvere-Rosebrugh; can you give a page number?

Posted by: Mike Shulman on January 13, 2008 6:33 AM | Permalink | Reply to this

Re: 2-toposes

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I take your point, though all I was trying to do was to point out a kind of familial resemblance. I suppose I could have made the resemblance look a little stronger by phrasing things like this:

Let $X$ be a vector space. Then any bilinear form $〈 -, - 〉$ on $X$ that makes $X$ into a Hilbert space also has the ‘Riesz property’.
Let $S$ be a set. Then there is no $〈 -, - 〉$ whatsoever on $S$ with the Riesz property.

I can’t find that version of the Riesz theorem in either Lawvere-Schanuel or Lawvere-Rosebrugh; can you give a page number?

In Lawvere and Schanuel it’s p.305 (Session 29, Cantor’s Contrapositive Corollary). They don’t specifically mention the Riesz theorem, understandably given their target audience, but it seems likely that they had it in mind. As for Lawvere–Rosebrugh, I’ve lent my copy to someone.

Posted by: Tom Leinster on January 13, 2008 7:06 AM | Permalink | Reply to this

Re: 2-toposes

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In Lawvere-Rosebrugh, it begins page 129. As they explain :

Cantor’s method for proving this theorem is often called the “diagonal argument” even though the diagonal map $δ_{X}$ is only one of two equally necessary pillars on which the argument stands, the second being a fixed-point-free self-map $τ$ (such as logical negation in the case of the set 2).

In the case of a category $C$ , you need a diagonal $δ_{C} : C \to C \times C^{op}$ , which is problematic due to the contravariance, and a fixed-point-free functor $Set \to Set$ .

Posted by: Mathieu Dupont on January 13, 2008 2:00 PM | Permalink | Reply to this

Re: 2-toposes

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Ah, one of my favorite topics to rant about! (-:

If you look at my appendix to John’s Lectures on n-Categories and Cohomology, you’ll find some arguments to the effect that a 0-topos should actually be a complete Heyting algebra. A Heyting algebra can be defined to be a poset which is cartesian closed as a category. The paradigmatic example is the set ${true, false}$ where false is less than true. Let me summarize the relevant points here.

Of course, if an $n$ -topos is a kind of $n$ -category, then in order for this to work we need to allow 0-categories to be not sets but posets. There are other reasons this is a useful thing to do. For example, $n$ -categories are supposed to be categories enriched over $(n - 1)$ -categories, but if $(- 1)$ -categories are truth values, then a category enriched over truth values is exactly a poset. It’s (equivalent to) a set if it’s also a groupoid. Thus a set might be more appropriately called a 0-groupoid. It’s also logical to extend the ` $(n, k)$ -category’ terminology to the case $k = n + 1$ to describe poset-enriched things. Thus a set is a (0,0)-category while a poset is a (0,1)-category.

Why should a 0-topos be a Heyting algebra? Well, if a 1-topos is a universe of sets (0-categories or 0-groupoids), then a 0-topos should be a universe of truth values ( $(- 1)$ -categories or $(- 1)$ -groupoids). But you can’t do very much with truth values if you have only a set of them; you need to know which ones imply which other ones, and have operations like and, or, and implies. For instance, as Tom pointed out, you definitely want implication to be the exponentiation of truth values. A complete Heyting algebra has all this; it is the constructive-logic version of a Boolean algebra, and as such is also from a different point of view the correct notion of a `generalized collection of truth values’.

Every 1-topos comes with a canonical internal Heyting algebra: its subject classifier, which describes the internal truth values of that 1-topos. Conversely, any complete Heyting algebra $H$ has a 1-topos of sheaves, from which you can recover $H$ as the poset of subobjects of the subobject classifier. If you take $H$ to be the open-set lattice of a topological space, which is always a complete Heyting algebra, you recover the topos of sheaves on that space. In fact, this is a contravariant full embedding (bicategorically speaking) of complete Heyting algebras into toposes. The opposite category of complete Heyting algebras is called the category of locales, and the 1-toposes arising from them are called localic. So in fact a more accurate statement would be that a 0-topos is a locale.

Another reason why locales are a good notion of 0-topos is that while first-order geometric theories have classifying 1-toposes (which are, in fact, always toposes of sheaves on some site), propositional geometric theories have classifying locales (and thereby localic toposes). So propositional logic, which deals with truth values only ( $(- 1)$ -groupoids), has the same relationship to locales that predicate logic, which deals with sets (0-groupoids), has to 1-toposes.

Now what about 2-toposes? By analogy, a 2-topos should be a universe of categories. So the 2-category of toposes shouldn’t be a 2-topos, just like the category of locales is not a 1-topos. One can easily leap ahead to conjecture that the 2-category of stacks (of categories) on a site (and in particular, on a 1-topos) will be a 2-topos, that the classifying whatever in a 2-topos will be an `internal 1-topos’, and that there may be a notion of higher order logic for which geometric theories have classifying 2-toposes.

However, there is a sticky point here: the question of size. The power set $2^{A}$ (or power object $Ω^{A}$ in any topos) of a set $A$ , while larger than $A$ , is completely determined by $A$ ; thus we can find a category of sets which contains the power set of all its objects. In fact, we can find many such categories, such as the category of sets of cardinality less than $κ$ for any strong limit cardinal^* $κ$ , and all of these are elementary toposes. It is customary, however, to fix some $κ$ , which is not only a strong limit but inaccessible^**, and call the category of sets smaller than $κ$ `the’ category of sets $Set$ . (If we then redefine `set’ to mean `set smaller than $κ$ ’ and define `proper class’ to mean `set not smaller than $κ$ ’, then we obtain the usual way of thinking about $Set$ as the category of `all’ sets.)

However, when we move up a level, the presheaf category ${Set}^{C^{op}}$ of a category $C$ depends not only on $C$ , but on what category of sets we choose to call $Set$ . Weber’s notion of 2-topos, which is a strengthening of Street’s notion of `fibrational cosmos’, corresponds to choosing two cardinals $κ$ < $λ$ , letting $CAT$ be the 2-category of categories smaller than $λ$ , and $Set$ the category of sets smaller than $κ$ . Since $κ$ < $λ$ , $Set$ is an object of $CAT$ , and then every object of $CAT$ has a presheaf category ${Set}^{C^{op}}$ .

In general, to make a nice 2-category into a 2-topos, you pick an object $Ω$ like $Set$ together with a `classifying discrete opfibration’ over it, and think of $Ω^{A^{op}}$ as the `presheaf object’ of $A$ . You can develop a good deal of category theory in this way, and there are definitely analogies to the theory of 1-toposes. However, there are some definite divergences as well.

Firstly, not every object of $CAT$ has a Yoneda embedding. The only categories that do are those whose hom-sets are smaller than $κ$ . This is not a huge problem in practice, since most categories that arise in the real world are locally small—but on the other hand, that makes one feel (at least, it makes me feel) that the inclusion of all the non-locally-small categories in $CAT$ is somewhat superfluous. (This is also problematic from an enriched point of view, where the analogue of non-locally-small categories is less clear.)

Secondly, while being a complete Heyting algebra is a property of a poset, and being a 1-topos is a property of a 1-category, being a 2-topos in Weber’s sense is a structure on a 2-category, because you need to pick the object $Set$ . For fixed $λ$ , different choices of $κ$ will give different 2-toposes.

For example, the 2-category of finite categories corresponds to taking $λ = ω$ . In order to make this a 2-topos, we need to choose a natural number $n = κ$ so that $Set$ can be the category of sets smaller than $n$ . As far as I can tell, any such choice does give a 2-topos, but it’s perhaps not what one intuitively feels `the universe of finite categories’ should be.

Thirdly, as far as I can tell, there is no `internal’ sense in which the classifier $Ω$ is a 1-topos. There seems to be nothing in the definition which would force $κ$ to be a strong limit, let alone inaccessible.

To be clear: I’m not saying that there’s anything wrong with Weber’s and Street’s definitions. It’s just that the extension of 1-dimensional topos-theoretic ideas to the 2-dimensional case is less straightforward than you might naively guess, due primarily to the question of size.

* Being a strong limit just means that if $A$ is smaller than $κ$ , so is $2^{A}$ .

** Being inaccessible means essentially that the sets smaller than $κ$ are closed under all the ordinary constructions of set theory.

Posted by: Mike Shulman on January 12, 2008 11:29 PM | Permalink | Reply to this

Re: 2-toposes

First I’d like to remark that once you accept non-locally small categories, that the analogous objects in enriched category theory are pretty natural to consider also. Suppose we have “Set” a category of small sets, and “SET” a larger universe containing Set as an object as discussed in earlier posts. When doing enriched category theory one initially takes the V you enrich over to be locally small (homs in Set). Then you can define a universe enlargement V’ = [V^op,SET] with monoidal structure given by Day convolution. The yoneda embedding is strong monoidal so you can regard V as a monoidal subcategory of V’. So just as one considers small, locally small and non-locally small categories, one can consider their V-categorical analogues:

small = collection of objects is an object of Set, and hom objects are in V.
locally small = hom objects are in V, though in general, collection of objects is an object of SET. These are the objects of the 2-category “V-Cat” as usually defined.
non-locally small = hom objects are in V’ but not V.

Second I’d like to remark that section 8 of my 2-topos paper (now called “Yoneda structures from 2-toposes”) makes a start on exhibiting “Omega” as an internal topos. The result is that when “Omega” is cocomplete, plus some other mild conditions, then “Omega” is in fact cartesian closed as an object of the ambient 2-category K.

Third, I’d like to say that I think that the notion of 2-topos is preliminary. I think that the real notion will involve further conditions including that “Omega” be cocomplete in the sense of its induced yoneda structure. A basic fact from topos theory to lift to 2-toposes is that the slices of a topos form a topos. I think that the analogue of this is that if K has a 2-topos structure and X an object of K, then Spl(X), the 2-category of split fibrations over X, should have an induced 2-topos structure. The idea is that Spl(X) is the correct analogue of slice in the 2-categorical setting. This result is proved for K=CAT in my paper, but I don’t know the general formulation, and feel that it requires further conditions on K, so that the free fibrations monads (on naive slices K/X) are sufficiently well-behaved to allow the proof in the case K=CAT to be generalised.

Posted by: Mark Weber on January 13, 2008 11:06 AM | Permalink | Reply to this

Re: 2-toposes

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Mark Weber writes:

Third, I’d like to say that I think that the notion of 2-topos is preliminary. I think that the real notion will involve further conditions including that “Omega” be cocomplete in the sense of its induced yoneda structure.

Regarding conditions which allow one to exhibit $Ω$ as a ‘1-topos’, do you foresee some use being made of the concept of lex totality? (Cf. Street’s article on Notions of Topos.)

Posted by: Todd Trimble on January 13, 2008 3:11 PM | Permalink | Reply to this

Re: 2-toposes

For the sake of other readers it’s worth spelling out the background here a little. Street and Walters observed that a short abstract definition of “Grothendieck topos” is that it’s a locally small category A for which the yoneda embedding A–>PSh(A) has a left exact left adjoint. Following from this, various authors then started to study the notion of totality, that is, locally small categories for which yoneda has a left adjoint (not necessarily left exact). Totality in CAT is a strong cocompleteness condition, which in fact implies also completeness. There are many examples: locally presentable categories are total, but so is the category of topological spaces and continuous functions and also the category of group objects therein. The theory of total categories even works well in the enriched setting, especially when the V over which you enrich is itself total.

So the only obstruction to giving the definition of “Grothendieck topos object” in a 2-topos (as currently defined), is knowing what a left exact arrow is. This doesn’t seem like a huge obstruction. I think it might be very interesting to internalise various aspects of the theory of Grothendieck toposes in this way.

Posted by: Mark Weber on January 13, 2008 11:14 PM | Permalink | Reply to this

Re: 2-toposes

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Street and Walters observed that a short abstract definition of “Grothendieck topos” is that it’s a locally small category $A$ for which the yoneda embedding $A \to PSh (A)$ has a left exact left adjoint.

Isn’t it true that you still also need some sort of small generator condition? Street’s “Notions of Topos” gives a proof due to Freyd that a “moderate” generator (= the size of the universe) suffices; is there a stronger result that does away with this condition entirely?

Posted by: Mike Shulman on January 15, 2008 12:35 AM | Permalink | Reply to this

Re: 2-toposes

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Mark Weber said:

Suppose we have “Set” a category of small sets, and “SET” a larger universe containing Set as an object as discussed in earlier posts. When doing enriched category theory one initially takes the V you enrich over to be locally small (homs in Set). Then you can define a universe enlargement V’ = [V^op,SET] with monoidal structure given by Day convolution. The yoneda embedding is strong monoidal so you can regard V as a monoidal subcategory of V’.

Mark, are you saying that you can get around having to use the change-of-universe axiom (which I hate) if you enrich your idea of categories so that your Hom sets are allowed to be Hom presheaves? Is that right? In SGA 4, Grothendieck and Verdier say that the only known universe is the one consisting of finite sets built out of the symbols “{” and “}”, such as {{},{{}}}. Can you then use this known universe and the procedure you mentioned above to avoid ever having to postulate the existence of a universe? It looks like you require two universes to get started, which seems a little odd to me, so I’m kind of worried I’m misunderstanding you.

Posted by: James on January 13, 2008 6:02 PM | Permalink | Reply to this

Re: 2-toposes

No, all I’m saying is that once you have two universes of sets, so that it makes sense to speak of categories that are small, locally small or otherwise, then for locally small monoidal V, you are able to consider V-categories that are small, locally small or otherwise. In other words “universe enlargement” is no harder for enriched category theory than for ordinary category theory.

Posted by: Mark Weber on January 13, 2008 11:24 PM | Permalink | Reply to this

Re: 2-toposes

Yeah, I guess you did say that! Thanks for the clarification!

Posted by: James on January 14, 2008 3:00 AM | Permalink | Reply to this

Re: 2-toposes

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I’m aware of the Day embedding construction for enlarging the enriching category $V$ , but I have to say it feels feels artificial to me, rather than natural. I’m not especially fond of non-locally small categories, although they do have their uses, but at least in that case $SET$ is really the only sensible candidate for an enlargement of $Set$ .

On the other hand, while $[V^{op}, SET]$ is one possible enlargement of $V$ , in any particular case there are usually other `natural’ candidates. For example, if $V = Top$ , then maybe the enlargement should be $TOP$ . If $V$ is a Grothendieck topos, so the category of (small) sheaves on a (small) site, then maybe its enlargement should be the category of large sheaves on that site. And of course, even $SET$ is not the same as $[{Set}^{op}, SET]$ ! What I don’t like is having to choose some non-canonical enlargement of $V$ just in order to be able to do ordinary $V$ -category theory.

Re: slices, can you say anything about why you think split fibrations are the appropriate analogue? Since most fibrations in nature don’t split, it seems more natural to me to allow all fibrations.

Posted by: Mike Shulman on January 13, 2008 10:51 PM | Permalink | Reply to this

Re: 2-toposes

I agree that there are other ways to imagine universe enlargement for enriched category theory. Day convolution just gives one recipe.

As for slices, ultimately you probably do want to regard Fib(X) – the 2-category of fibrations over X an object in a finitely complete 2-category K – as your real slice. Also I think that ultimately the notion of 2-topos I isolated should be relaxed so that it’s invariant under biequivalence.

I’ve been working with the strict notion of 2-topos and split fibrations because of applications that I have in mind – to organise the general theory of operads in a 2-categorical fashion.

This is a project to do all the formal operadic constructions of Michael Batanin’s “Eckmann-Hilton argument and higher operads” at the level of generality of my “Operads within monoidal pseudo algebras”. The initial reason for doing this is so that Batanin’s techniques would then be usable for studying the stabilisation hypothesis in the globular setting. In view of the applications of Batanin’s work to loop spaces, one should expect other applications of such a general operad theory.

The effort to make Batanin’s operad theory 2-categorical is what led me to the notion of 2-topos I consider, and to considering split fibrations instead of non-split ones.

Posted by: Mark Weber on January 13, 2008 11:52 PM | Permalink | Reply to this

Re: 2-toposes

Mike wrote

…there may be a notion of higher order logic for which geometric theories have classifying 2-toposes.

Is ‘higher order’ the best choice here? That’s already used for the logic of a 1-topos. Why not n-logic?

In your paper with John you wrote

As far as I know, there has been very little work on notions of n-theories for higher values of n.

So is there anything we know about what a 2-topos supports?

Posted by: David Corfield on January 13, 2008 4:10 PM | Permalink | Reply to this

Re: 2-toposes

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David wrote:

Is ‘higher order’ the best choice here? That’s already used for the logic of a 1-topos. Why not $n$ -logic?

I’m afraid I never got around to talking to Mike about 2-logic as modal logic, and the vague dreams of generalizations to higher $n$ . So, you could be ahead of him when it comes to pondering ‘ $n$ -logic’, and its relation to Klein $n$ -geometry and the like.

It’s time for people to get serious about this stuff!

So is there anything we know about what a 2-topos supports?

You know already: modal logic, but with an intuitionistic constructive tinge to it.

But if you’re talking about theorems, I guess Mark Weber is the one to ask. Or maybe Ross Street, whose ‘cosmoi’ are a version of 2-topoi.

Posted by: John Baez on January 14, 2008 6:59 AM | Permalink | Reply to this

Re: 2-toposes

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I’m afraid I never got around to talking to Mike about 2-logic as modal logic, and the vague dreams of generalizations to higher $n$ .

Has anyone writen anything about this anywhere that I can read? I’ve always wished that I could find a way to make modal logic make sense to me….

Posted by: Mike Shulman on January 15, 2008 12:10 AM | Permalink | Reply to this

Re: 2-toposes

Mike wrote:

Has anyone written anything about this anywhere that I can read?

Nope. Maybe David Corfield has some notes he could scan in… I’ve forgotten half of what I once knew about this.

I’ve always wished that I could find a way to make modal logic make sense to me…

Modal logic as ordinarily done is a bit silly, which is why it hasn’t developed many interesting connections to mathematics, the way some other forms of logic have.

In particular, people in modal logic often experiment with different axioms without thinking hard enough about what a model should be. Too much free-wheeling syntax, not enough semantics to ground it.

The work of Awodey and Kinshida is one of the few counterexamples to my claim. Some might argue that Kripke’s semantics for modal logic is mathematically interesting, but I think it got interesting right around the time it morphed into the Kripke–Joyal semantics for topos logic, and stopped having much to do with modal logic.

So: I think that instead of trying to understand modal logic as it exists now, it would be more efficient to develop it yourself based on this principle: a theory in modal logic should be precisely the sort of thing that has a 2-groupoid of models, just as a theory in predicate logic is precisely the sort of thing that has a 1-groupoid of models.

Of course, it would help to get the connection between ordinary predicate logic and groupoids really clearly worked out first. Todd Trimble has done a lot of work on this. So, it would be very good if someone could categorify everything Todd said here and here.

Posted by: John Baez on January 15, 2008 2:00 AM | Permalink | Reply to this

Re: 2-toposes

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So: I think that instead of trying to understand modal logic as it exists now, it would be more efficient to develop it yourself based on this principle: a theory in modal logic should be precisely the sort of thing that has a 2-groupoid of models, just as a theory in predicate logic is precisely the sort of thing that has a 1-groupoid of models.

That’s a terribly pregnant comment. I’m betting Jim Dolan for one has thought a lot about this (cf. his pioneering connections between the stuff-structure-properties paradigm, factorization systems for groupoids, Postnikov systems, etc.).

Posted by: Todd Trimble on January 15, 2008 2:23 AM | Permalink | Reply to this

Re: 2-toposes

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There is of course the topos of $G$ -sets. One would hope that for a 2-group $𝒢$ there is a 2-topos of $𝒢$ -categories (or groupoids). Since this is nothing but $Hom (Σ 𝒢, Cat)$ where I’m not being fussy about size, though I should, we are in the realm of presheaves on 2-groupoids. Personally I feel there should be a notion of 2-site. Such a categorification shouldn’t be too difficult, should it?

Posted by: David Roberts on January 15, 2008 1:46 AM | Permalink | Reply to this

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Urs wrote in another thread:

Is there any good intuiitive way to think of set-valued truth values? Is that what you meant by “variable truth values”? I must say I do not understand what you mean by that.

This is one of the important ideas of topos theory: that truth is often a more complicated (and interesting!) affair than ‘true or false’.

The example I mentioned before was ‘is it raining?’ The answer depends on where you are in space and time. It’s not the same in all places and all times: it varies. To give a complete answer, you’d have to specify all the pairs $(point on the surface of the earth, point in time)$ at which it is raining. In other words, you’d have to specify a subset of $S^{2} \times ℝ$ . So in this context, you can view the truth values as the subsets of $S^{2} \times ℝ$ .

I got interested in this recently when I was teaching a course on category theory. If you want to read my ramblings about toposes, variable truth values and variable sets, see p.110-114 of the course notes here.

Posted by: Tom Leinster on January 12, 2008 5:25 PM | Permalink | Reply to this

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Tom wrote:

The example I mentioned before was ‘is it raining?’ The answer depends on where you are in space and time. It’s not the same in all places and all times: it varies. To give a complete answer, you’d have to specify all the pairs $(point on the surface of earth, point in time)$ at which it is raining. In other words, you’d have to specify a subset of $S^{2} \times ℝ^{2}$ . So in this context, you can view the truth values as the subsets of $S^{2} \times ℝ$ .

I do follow this reasoning, but I am not sure how this applies to truth values with values in $Set$ . It seems that in your example it is crucial that I known which set my truth-value set is a subset of.

But in the 2-topos $Cat$ , the truth value of any proposition will just be a set, by itself.

So if I am asking: “Is it raining?” and you reply in the 2-topos Cat: “The three element set!”

What am I to make of that? It doesn’t tell me that there are three points on the surface of earth, where it is raining at some instant of time. It just tells me: “the three element set.”

What am I missing?

Posted by: Urs Schreiber on January 14, 2008 6:12 PM | Permalink | Reply to this

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I wrote:

What am I missing?

and only afterwards started reading this thread. Now I realize that apparently I had been under the wrong impression that a truth value in the 2-topos Cat is a functor ${•} \to Set$ while apparently it is instead a functor $pointedSet \to Set .$

Hm…

Posted by: Urs Schreiber on January 14, 2008 6:18 PM | Permalink | Reply to this

Re: 2-toposes

I think this thought is quite helpful. Well, it was a mini-revelation for me, even if Todd has known it since Kindergarten.

I suppose in a very primitive way, we could take a discrete category, and take a generalised truth value as a functor to Set. So, we ask not of a set of people do you have friends? But of the discrete category of people, which friends do you have?

It’s the shift from ‘whether’ to ‘how’. Not whether here is connected to there, but how.

Posted by: David Corfield on January 14, 2008 6:27 PM | Permalink | Reply to this

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I think this thought is quite helpful.

Okay. Could you unwrap that thought further, for me? Over the weekend I didn’t do much blog reading, and when I came back everybody was already familiar with all the details of 2-topoi. I apparently don’t even know the basics yet.

If you could spell out for me what “Mark Weber’s classifying discrete opfibration in the 2-topos of categories” is, and how the weak quotient $X / / G$ is a pullback over that, I’d be most grateful!

Posted by: Urs Schreiber on January 14, 2008 6:58 PM | Permalink | Reply to this

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Let’s look at a $G$ -set $X$ for some group $G$ . So we have an arrow (functor) in the 2-topos of categories: $G \to Set$ .

We want to pull back over the forgetful functor Pointed set $\to$ Set. The pullback is $X / / G$ . Let’s see what happens to a morphism in this groupoid

$(x, g) : x \to g \cdot x .$

All objects $x$ get mapped to the single object in $G$ , and morphism $(x, g)$ is mapped to $g$ .

Now horizontally, i.e., the functor from $X / / G$ to Pointed set. $x$ is sent to pointed set $(X, x)$ . Morphism $(x, g)$ is sent to the obvious map from $(X, x)$ to $(X, g \cdot x)$ .

Another illustration would be to think of a functor from the category Th(Gp), the theory of groups, to Set. The pullback in this case has as objects tuples of elements from the set which is hit. Above $m : G \times G \to G$ , would be arrows such as $(g, h) \to g \cdot h$ .

Yet another case, this time from the category which is the fundamental groupoid of the circle. Take a functor to Set. Let’s have the one which maps each object to the 2 element set, so as to permute them every turn of the circle. Then the pullback is the fundamental groupoid of the border of the Möbius strip.

Posted by: David Corfield on January 15, 2008 9:43 AM | Permalink | Reply to this

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If I may, I’ll add a few more remarks to what David has already said. (They may have already been said, but that’s okay.)

Let’s start with Set, as a 1-topos. One way of describing the subobject classifier is this: suppose given a set $X$ and a subobject $i : A ↪ X$ . This means we have a set $X$ together with a fibering by $(- 1)$ -categories (i.e., by elements 0, 1: 0 if the fiber over $x$ is empty, 1 if the fiber is occupied). To each such fibering we have, tautologically, a map

χ_{i} : X \to 2 = (- 1) - Cat

which specifies that fibering.

We can figure out the universal subobject that we are pulling back against, by asking which fibering over $2$ would correspond to the identity map $2 \to 2$ . (In asking the question of the identity, note we are making implicit appeal to the Yoneda lemma.) Well, the fiber over 1 should be 1, and the fiber over 0 should be 0. We obtain ${1} ↪ {0, 1}$ as the universal subobject.

Now let’s bump up one dimension, working now in the 2-topos $Cat$ , and by following our noses imagine that $Set = 0 - Cat$ just has to be the “classifier” (whatever that may mean). By analogy, we expect that a functor

F : B \to Set

classifies “something”, a functor $p : E \to B$ , so that for an object $b$ the set $F (b)$ is precisely the fiber $p^{- 1} (b)$ . So the fibers of such “somethings” are discrete categories; putting aside precise definitions for now, the somethings are called discrete fibrations.

Proceeding a little further, we ask, “what is the universal discrete fibration?” Here, we ask what fibering over $Set$ corresponds to the identity $Set \to Set$ . This means the fiber over a set $S$ is $S$ itself: the objects of the fiber are points $s \in S$ . Thus, objects of the total category of the universal discrete fibration are ordered pairs $(S, s \in S)$ , that is to say, pointed sets. We arrive at the underlying functor

(Pointed sets) \to Set

as the universal discrete fibration.

The discrete fibration $p : E \to B$ that is classified by $F : B \to Set$ is obtained by pulling back the universal discrete fibration along $F$ . Officially, this gives what is called the category of elements $p : E \to B$ of $F$ : the objects of $E$ are ordered pairs $(b, x \in F (b))$ , and morphisms $(b, x) \to (c, y)$ are morphisms of $f : b \to c$ of $B$ such that $F (f) (x) = y$ (this last condition corresponds to the fact that morphisms in $(Pointed sets)$ preserve the chosen basepoints.

The rest goes as David explained: the category of elements of a functor $X : G \to Set$ (for $G$ a group) has, for its objects, just elements $x \in X (1)$ (where I am using 1 to denote the sole object of $G$ ). The morphisms $x \to y$ are elements $g \in G$ such that $X (g)$ sends $x$ to $y$ . It’s now pretty clear that we are describing the translation or action groupoid $X / / G$ of $X$ .

Ain’t it cool?

Posted by: Todd Trimble on January 15, 2008 4:47 PM | Permalink | Reply to this

Re: 2-toposes

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So it shouldn’t be too hard to move up a step and look in the 3-topos of 2-categories for a classifier. It ought to be $Cat = 1 - Cat$ . (Or should it be Grpd?)

Then what fibring over Cat corresponds to the identity $Cat \to Cat$ ? I thought it should be the forgetful 2-functor from Pointed Cat, but now I’m not so sure.

With that in place we could run through my examples:

a) Map a 2-group into Cat.

b) Map the 2-category Th(2-gp) (theory of 2-groups) into Cat.

c) Map the fundamental 2-groupoid of a space, such as the 2-sphere, into Cat.

In each case the pullback should be interesting.

Posted by: David Corfield on January 16, 2008 9:03 AM | Permalink | Reply to this

Re: 2-toposes

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David wrote:

what fibring over Cat corresponds to the identity $Cat \to Cat$ ? I thought it should be the forgetful 2-functor from Pointed Cat, but now I’m not so sure.

Given a $Cat$ -valued functor $X : A \to Cat$ , the corresponding (op)fibred category $E$ over $A$ is defined as follows. An object of $E$ is a pair $(a, x)$ with $a \in A$ and $x \in X (a)$ . A map $(a, x) \to (b, y)$ in $E$ is a map $f : a \to b$ in $A$ together with a map $(X f) (x) \to y$ in $X (b)$ .

In the special case $A = Cat$ , $X = id$ , an object of $E$ is indeed a pointed category $(C, c)$ , but a map $(C, c) \to (D, d)$ is a functor $F : C \to D$ together with a map $F (c) \to d$ in $D$ . Hence the fibre over $C \in Cat$ is indeed $C$ itself, as you hoped.

So the classifying thing isn’t really “pointed Cat”, because the maps are different.

Posted by: Tom Leinster on January 16, 2008 5:19 PM | Permalink | Reply to this

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This is really bugging me trying to think of the universal classifying thingie in the 3-topos 2-Cat. We would expect the classifying 2-category to be the 2-topos Cat.

Now, if I try to replicate what you did one level down with Set, it would seem that fibred above a category, $C$ , would be $C$ , just as fibred above set $X$ was $X$ .

So let’s take $C$ as the two object category with a single nontrivial arrow $f : a \to b$ . We want the fibre above it to be $C$ , so we might think of pointed categories $(C, a)$ and $(C, b)$ sitting up there. But then we’d need a pointed functor between these pointed categories to correspond to $f$ . There is a pointed functor which sends all of $C$ to $b$ and ${id}_{b}$ .

Something doesn’t seem right. Might we learn from a level below? Back down in the mundane world of 2-toposes, is there a recipe to the passage from Set to Pointed set, which would work for 2-toposes other than the 2-topos of categories? What happens to other classifying objects?

Posted by: David Corfield on January 16, 2008 4:24 PM | Permalink | Reply to this

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I’m not sure either, and I’m guessing there are various interesting options one could pursue, but one thingie I’m looking at is where the total 2-category $E (Cat)$ over $Cat$ has pairs $(C, c)$ for $0$ -cells, pairs $(F : C \to D, ϕ : F c \to d)$ for $1$ -cells $(F, ϕ) : (C, c) \to (D, d)$ , and pairs $(η : F \to G, α : ϕ \to ψ \circ η c)$ for $2$ -cells $(η, α) : (F, ϕ) \to (G, ψ)$ .

Here, if you look at the fiber of $E (Cat)$ over a category $C$ (whose 1-cells map to $1_{C}$ and whose 2-cells map to $1_{1_{C}}$ ), you get $C$ back. Also, the homomorphism down to $E (Cat) \to Cat$ is given by projection to the first components of pairs. These feel “right” to me. However, there is some asymmetry in the definition; for example, you could have $(F : C \to D, ϕ : d \to F c)$ for $1$ -cells instead. There are variations based on reversing directions in $2$ -cells as well.

I am guessing Mike Shulman would have a lot to say about this… I would like to study more carefully what he has written here.

[Edit: Tom Leinster has already anticipated this construction, above.]

Posted by: Todd Trimble on January 16, 2008 5:46 PM | Permalink | Reply to this

Re: 2-toposes

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Sorry, my previous comment had a dumb level slip in the description of 2-cells: I think those $α$ ’s are just identities.

Posted by: Todd Trimble on January 16, 2008 6:06 PM | Permalink | Reply to this

Re: 2-toposes

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When I first saw Mark Weber’s definition of a 2-topos I also wondered why the classifier in $Cat$ was given by ${Set}^{*} \to Set$ and not by $1 \to Set$ . And in fact Todd Trimble’s remark David mentioned clarified for me the situation : both are classifiers, it depends on how you classify.

The first possibility is to classify with a pullback. Then you use ${Set}^{*} \to Set$ : in Cat, any discrete opfibration $E \to B$ is the pullback of an arrow $B \to Set$ along ${Set}^{*} \to Set$ .

The second possibility is to classify with a comma object. Then you use $1 \to Set$ : any discrete opfibration $E \to B$ is the comma object of an arrow $F : B \to Set$ along $1 \to Set$ (i.e. the forgetful functor from the category of elements of $F$ to $B$ ).

The two constructions are equivalent because in the following diagram, the right-hand square is a comma object. This implies that the left-hand square is a pullback iff the whole rectangle is a comma object.

$\begin{matrix} E & \to & {Set}^{*} & \to & 1 \\ ↓ & ↓ & ⇓ & ↓ \\ B & \overset{F}{\to} & Set & = & Set \end{matrix}$

Posted by: Mathieu Dupont on January 14, 2008 11:23 PM | Permalink | Reply to this

Re: 2-toposes

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Obviously one of the interesting things about Pointed Set $\to$ Set is that above each set is a fibre of that cardinality. So all sized fibres are covered.

Posted by: David Corfield on January 14, 2008 6:33 PM | Permalink | Reply to this

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January 12, 2008

2-Toposes

Posted by David Corfield

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