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January 12, 2008

2-Toposes

Posted by David Corfield

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?

A 1-topos is a kind of 1-category. The 1-category of sets is a paradigmatic example, in which 121 \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 A2 AA \to 2^A, and in Cat we have CSet C opC \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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50 Comments & 3 Trackbacks

Re: 2-toposes

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 CatCat this must be () op(-)^{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}2 = \{true, false\} should be a paradigmatic example.

If in Set we have A2 AA \to 2^A, and in Cat we have CSet C opC \to Set^{C^{op}}, is there a 0-Yoneda?

Well, a (1)(-1)-category is a truth value in the classical sense (i.e. truetrue or falsefalse), and there’s meant to be just one (2)(-2)-category, somehow corresponding to the truth value truetrue. (I suppose I mean “corresponding” in the same way that the (1)(-1)-categories truetrue and falsefalse correspond to the 00-categories 00 and 11, and that a 00-category corresponds to the discrete 11-category on it.) Also, the exponentiation of truth values is surely implication: β α\beta^\alpha means (αβ)(\alpha \Rightarrow \beta). So I think the 00-Yoneda embedding is the statement α=(αtrue), \alpha = (\alpha \Rightarrow true), where α2={true,false}\alpha \in 2 = \{ true, false\}. How come that’s an equality? Well, both α\alpha and (αtrue)(\alpha \Rightarrow \true) are elements of 2=(1)Cat2 = (-1)Cat, which is a 00-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(A2 A)not(A \cong 2^A), for sets AA. I spent a little while trying to imitate this for categories, to show that not(CSet C op)not(C \simeq Set^{C^{op}}) for categories CC (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

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

But I added in brackets that CC could be ‘small if you like’, which I now regret. For small CC, it’s easy to prove. If CSet C opC \simeq Set^{C^{op}} then CC has all small limits, which if CC is small implies that CC is a preorder (equivalent to a complete lattice). So Set C opSet^{C^{op}} is a preorder, which (by considering constant functors) implies that CC is the empty category 00. But Set 0 opSet^{0^{op}} isn’t empty.

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

Re: 2-toposes

It’s also easy to extend this result to locally small categories. If CSet C opC\simeq \mathrm{Set}^{C^\mathrm{op}} and CC is locally small, then Set C op\mathrm{Set}^{C^\mathrm{op}} is also locally small, and therefore CC must be small. Then apply your argument.

(By the way, it seems that an alternate argument when CC is small is: if CSet C opC\simeq \mathrm{Set}^{C^\mathrm{op}}, then Set C op\mathrm{Set}^{C^\mathrm{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 CC is not locally small, it has no Yoneda embedding CSet C opC\to \mathrm{Set}^{C^\mathrm{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

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 SS, no function S2 SS \to 2^S is surjective. Can it be shown that for categories CC, no functor CSet C opC \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 HH (real, say), we have a bilinear map ,:H×H \langle -, - \rangle: H \times H \to \mathbb{R} and Riesz tells us that every map HH \to \mathbb{R} is of the form x,\langle x, - \rangle for some xHx \in H. Cantor’s theorem says that if we take a set SS and a map ,:S×S2 \langle -, - \rangle: S \times S \to 2 then not every map S2S \to 2 is of the form x,\langle x, - \rangle for some xSx \in S. My new question can be phrased as: given a category CC and a functor ,:C op×CSet, \langle -, - \rangle: C^{op} \times C \to Set, can it be shown that not every functor CSetC \to \Set is isomorphic to X,\langle X, - \rangle for some XCX \in C? (A functor C op×CSetC^{op} \times C \to Set is usually called a (C,C)(C, C)-module, or (C,C)(C, C)-bimodule, or a profunctor or distributor from CC to CC.)

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

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 ,\langle -,-\rangle, 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 S2 SS\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×S2S\times S\to 2 or functor C op×CSetC^{\mathrm{op}} \times C\to \mathrm{Set} whose slices can represent any function S2S\to 2 or functor C opSetC^{\mathrm{op}} \to \mathrm{Set}. Thus the use of the same notation ,\langle-,-\rangle 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

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 XX be a vector space. Then any bilinear form ,\langle -, - \rangle on XX that makes XX into a Hilbert space also has the ‘Riesz property’.
  • Let SS be a set. Then there is no ,\langle -, - \rangle whatsoever on SS 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

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\delta_X is only one of two equally necessary pillars on which the argument stands, the second being a fixed-point-free self-map τ\tau (such as logical negation in the case of the set 2).

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

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

Re: 2-toposes

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}\{\mathrm{true},\mathrm{false}\} where false is less than true. Let me summarize the relevant points here.

Of course, if an nn-topos is a kind of nn-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, nn-categories are supposed to be categories enriched over (n1)(n-1)-categories, but if (1)(-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)(n,k)-category’ terminology to the case k=n+1k=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)(-1)-categories or (1)(-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 HH has a 1-topos of sheaves, from which you can recover HH as the poset of subobjects of the subobject classifier. If you take HH 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)(-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 A2^A (or power object Ω A\Omega^A in any topos) of a set AA, while larger than AA, is completely determined by AA; 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\kappa for any strong limit cardinal* κ\kappa, and all of these are elementary toposes. It is customary, however, to fix some κ\kappa, which is not only a strong limit but inaccessible**, and call the category of sets smaller than κ\kappa `the’ category of sets Set\mathrm{Set}. (If we then redefine `set’ to mean `set smaller than κ\kappa’ and define `proper class’ to mean `set not smaller than κ\kappa’, then we obtain the usual way of thinking about Set\mathrm{Set} as the category of `all’ sets.)

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

In general, to make a nice 2-category into a 2-topos, you pick an object Ω\Omega like Set\mathrm{Set} together with a `classifying discrete opfibration’ over it, and think of Ω A op\Omega^{A^\mathrm{op}} as the `presheaf object’ of AA. 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\mathrm{CAT} has a Yoneda embedding. The only categories that do are those whose hom-sets are smaller than κ\kappa. 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\mathrm{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\mathrm{Set}. For fixed λ\lambda, different choices of κ\kappa will give different 2-toposes.

For example, the 2-category of finite categories corresponds to taking λ=ω\lambda=\omega. In order to make this a 2-topos, we need to choose a natural number n=κn=\kappa so that Set\mathrm{Set} can be the category of sets smaller than nn. 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\Omega is a 1-topos. There seems to be nothing in the definition which would force κ\kappa 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 AA is smaller than κ\kappa, so is 2 A2^A.

** Being inaccessible means essentially that the sets smaller than κ\kappa 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

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\Omega 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

Street and Walters observed that a short abstract definition of “Grothendieck topos” is that it’s a locally small category AA for which the yoneda embedding APSh(A)A\to \mathrm{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

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

I’m aware of the Day embedding construction for enlarging the enriching category VV, 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\mathrm{SET} is really the only sensible candidate for an enlargement of Set\mathrm{Set}.

On the other hand, while [V op,SET][V^{\mathrm{op}},\mathrm{SET}] is one possible enlargement of VV, in any particular case there are usually other `natural’ candidates. For example, if V=TopV=\mathrm{Top}, then maybe the enlargement should be TOP\mathrm{TOP}. If VV 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\mathrm{SET} is not the same as [Set op,SET][\mathrm{Set}^{\mathrm{op}},\mathrm{SET}]! What I don’t like is having to choose some non-canonical enlargement of VV just in order to be able to do ordinary VV-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

David wrote:

Is ‘higher order’ the best choice here? That’s already used for the logic of a 1-topos. Why not nn-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 nn. So, you could be ahead of him when it comes to pondering ‘nn-logic’, and its relation to Klein nn-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

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 nn.

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

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

There is of course the topos of GG-sets. One would hope that for a 2-group 𝒢\mathcal{G} there is a 2-topos of 𝒢\mathcal{G}-categories (or groupoids). Since this is nothing but Hom(Σ𝒢,Cat)Hom(\Sigma\mathcal{G},\mathrm{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

Re: 2-toposes

“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).”

I guess Isn’t true: Let C the Presheaves category on a finite or small category A, C is the small-colimit completion of A, then the Presheaves category of C is a “double completion” of A and by Universal propriety is isomorphic to C.

Posted by: Sergio Buschi (a secondary school teacher) on August 17, 2008 11:59 AM | Permalink | Reply to this

Re: 2-toposes

Sorry, what I said aove is wrong..(mistake abut similar propriety of Ind-categories and about reflexions proriety true for full subcategories but no in this situation)..

Posted by: Buschi Sergio on August 18, 2008 1:25 PM | Permalink | Reply to this

Re: 2-toposes

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 (pointonthesurfaceoftheearth,pointintime) (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×S^2 \times \mathbb{R}. So in this context, you can view the truth values as the subsets of S 2×S^2 \times \mathbb{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

Re: 2-toposes

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 (pointonthesurfaceofearth,pointintime) (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× 2S^2 \times \mathbb{R}^2. So in this context, you can view the truth values as the subsets of S 2×S^2 \times \mathbb{R}.

I do follow this reasoning, but I am not sure how this applies to truth values with values in SetSet. 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 CatCat, 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

Re: 2-toposes

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 {}Set \{\bullet\} \to Set while apparently it is instead a functor pointedSetSet. \mathrm{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

Re: 2-toposes

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//GX // 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

Re: 2-toposes

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

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

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

All objects xx get mapped to the single object in GG, and morphism (x,g)(x, g) is mapped to gg.

Now horizontally, i.e., the functor from X//GX // G to Pointed set. xx is sent to pointed set (X,x)(X, x). Morphism (x,g)(x, g) is sent to the obvious map from (X,x)(X, x) to (X,gx)(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×GGm: G \times G \to G, would be arrows such as (g,h)gh(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

Re: 2-toposes

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 XX and a subobject i:AXi: A \hookrightarrow X. This means we have a set XX together with a fibering by (1)(-1)-categories (i.e., by elements 0, 1: 0 if the fiber over xx is empty, 1 if the fiber is occupied). To each such fibering we have, tautologically, a map

χ i:X2=(1)Cat\chi_i: X \to \mathbf{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\mathbf{2} would correspond to the identity map 22\mathbf{2} \to \mathbf{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}\{1\} \hookrightarrow \{0, 1\} as the universal subobject.

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

F:BSetF: B \to Set

classifies “something”, a functor p:EBp: E \to B, so that for an object bb the set F(b)F(b) is precisely the fiber p 1(b)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 SetSet corresponds to the identity SetSetSet \to Set. This means the fiber over a set SS is SS itself: the objects of the fiber are points sSs \in S. Thus, objects of the total category of the universal discrete fibration are ordered pairs (S,sS)(S, s \in S), that is to say, pointed sets. We arrive at the underlying functor

(Pointedsets)Set(Pointed sets) \to Set

as the universal discrete fibration.

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

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

Ain’t it cool?

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

Re: 2-toposes

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=1CatCat = 1-Cat. (Or should it be Grpd?)

Then what fibring over Cat corresponds to the identity CatCatCat \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

David wrote:

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

Given a CatCat-valued functor X:ACatX: A \to Cat, the corresponding (op)fibred category EE over AA is defined as follows. An object of EE is a pair (a,x)(a, x) with aAa \in A and xX(a)x \in X(a). A map (a,x)(b,y)(a, x) \to (b, y) in EE is a map f:abf: a \to b in AA together with a map (Xf)(x)y(X f)(x) \to y in X(b)X(b).

In the special case A=CatA = Cat, X=idX = id, an object of EE is indeed a pointed category (C,c)(C, c), but a map (C,c)(D,d)(C, c) \to (D, d) is a functor F:CDF: C \to D together with a map F(c)dF(c) \to d in DD. Hence the fibre over CCatC \in Cat is indeed CC 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

Re: 2-toposes

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, CC, would be CC, just as fibred above set XX was XX.

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

Re: 2-toposes

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)E(Cat) over CatCat has pairs (C,c)(C, c) for 00-cells, pairs (F:CD,ϕ:Fcd)(F: C \to D, \phi: F c \to d) for 11-cells (F,ϕ):(C,c)(D,d)(F, \phi): (C, c) \to (D, d), and pairs (η:FG,α:ϕψηc)(\eta: F \to G, \alpha: \phi \to \psi \circ \eta c) for 22-cells (η,α):(F,ϕ)(G,ψ)(\eta, \alpha): (F, \phi) \to (G, \psi).

Here, if you look at the fiber of E(Cat)E(Cat) over a category CC (whose 1-cells map to 1 C1_C and whose 2-cells map to 1 1 C1_{1_C}), you get CC back. Also, the homomorphism down to E(Cat)CatE(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:CD,ϕ:dFc)(F: C \to D, \phi: d \to F c) for 11-cells instead. There are variations based on reversing directions in 22-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

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

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

Re: 2-toposes

When I first saw Mark Weber’s definition of a 2-topos I also wondered why the classifier in Cat\mathrm{Cat} was given by Set *Set\mathrm{Set}^*\to\mathrm{Set} and not by 1Set1\to\mathrm{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 *Set\mathrm{Set}^*\to\mathrm{Set} : in Cat, any discrete opfibration EBE\to B is the pullback of an arrow BSetB\to\mathrm{Set} along Set *Set\mathrm{Set}^*\to\mathrm{Set}.

The second possibility is to classify with a comma object. Then you use 1Set1\to\mathrm{Set} : any discrete opfibration EBE\to B is the comma object of an arrow F:BSetF:B\to\mathrm{Set} along 1Set1\to\mathrm{Set} (i.e. the forgetful functor from the category of elements of FF to BB).

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.

E Set * 1 B F Set = Set\array{ E &\to &\mathrm{Set}^* &\to &1\\ \downarrow &&\downarrow &\Downarrow &\downarrow \\ B &\overset{F}{\to} &\mathrm{Set}&=&\mathrm{Set} }

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

Re: 2-toposes

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

Re: 2-toposes

Urs expressed some confusion about “variable truth values” and “set-valued truth values”.

I understand the confusion. My explanation was only of variable truth values. I didn’t try to say anything about “set-valued truth values”.

There are two orthogonal processes: categorification and nonclassicalization. By the latter I mean, for instance:

  • taking the notion “classical truth value” (true or false) and replacing it by “element of a Heyting algebra”
  • taking the notion “set” and replacing it by “object of a topos”.

“Variable truth value” is a suggestive way of saying “element of a Heyting algebra”. “Variable set” is a suggestive way of saying “object of a topos”.

Categorification, on the other hand, takes a notion such as “0-topos” and replaces it by “1-topos”. Since according to Mike, a 0-topos is a (complete) Heyting algebra, the concepts arrange themselves into a grid:

classicaltruthvalue | variabletruthvalue =elementof{true,false} | =elementofaHeytingalgebra | =elementofa0topos + set | variableset =objectofSet | =objectofatopos | =objectofa1topos + category | variablecategory =objectofCAT | =objectofa2topos \begin{aligned} classical truth value& | & variable truth value\\ = element of \{ true, false \} & | & = element of \mathrm{a} Heyting algebra \\ & | & = element of \mathrm{a}\, 0-topos\\ & + & \\ set& | & variable set\\ = object of\, \mathbf{Set}& | & = object of \mathrm{a} topos\\ & | & = object of \mathrm{a}\, 1-topos\\ & + & \\ category& | & variable category\\ = object of\, \mathbf{CAT}& | & = object of \mathrm{a}\, 2-topos\\ \end{aligned}

Moving right is nonclassicalization. Moving down is categorification.

In particular, we have a progression truth value / set / category. I wouldn’t talk about “set-valued truth values”, any more than I’d talk about “category-valued sets” (shifting everything up one dimension.) But you can say that sets are to 2-toposes as truth values are to 1-toposes.

Posted by: Tom Leinster on January 15, 2008 1:22 PM | Permalink | Reply to this

Re: 2-toposes

Good way of thinking about it.

I think your left-to-right axis, labelled “nonclassicalization”, is the same thing as the parameter I’ve labelled k in my fantasy below.

Posted by: James Cranch on January 15, 2008 10:11 PM | Permalink | Reply to this

Re: 2-toposes

I urge everyone interested in nn-topoi to read what Mike Shulman writes in pages 40–46 of the paper Lectures on nn-Categories and Cohomology. He says a lot about what nn-topoi should be like — and more generally, (n,m)(n,m)-topoi!

Posted by: John Baez on January 13, 2008 1:55 AM | Permalink | Reply to this

Re: 2-toposes

One could dream of a further generalisation, coming from using the idea of categorification in a different place.

Mike’s appendix says that an (n,m)-topos is some (n,m)-category which looks like the (n,m)-category of all (n-1,m-1)-categories.

Of course, we care about toposes in their own right rather than merely as universes; we might certainly get interested in what the collection of all toposes is like.

So we’re entitled to say that an (n,m)-2-topos is some (n,m)-category which looks like the (n,m)-category of all (n-1,m-1)-toposes.

This notation is defined so that an (n,m)-0-topos is an (n,m)-category, and an (n,m)-1-topos is what Mike calls an (n,m)-topos. Clearly enough, we can iterate this to get a description of an (n,m)-k-topos: it’s an (n,m)-category which looks like the (n,m)-category of all (n-1,m-1)-(k-1)-toposes.

What should they look like?

Of course, our motivating example is supposed to be the usual 2-category of toposes (and geometric morphisms and natural transformations between them). It’s also reasonable to expect the category of complete Heyting algebras to be an example.

I don’t understand the usual 2-categorical properties of the category of toposes well enough to make a sensible guess. But I think it’s a natural question.

Posted by: James Cranch on January 14, 2008 11:28 AM | Permalink | Reply to this
Read the post 101 things to do with a 2-classifier
Weblog: The n-Category Café
Excerpt: What to do with a 2-classifier.
Tracked: January 17, 2008 10:26 AM
Read the post 2-Structure Types
Weblog: The n-Category Café
Excerpt: 2-structure types
Tracked: April 2, 2008 10:00 AM

Re: 2-Toposes

I would have expected that the canonical example of a 2-topos would have been Gpd, not Cat. There is a school of thought that says that 0-groupoids are sets and 0-categories are posets, after all. This is reinforced by the fact that the categorification of a commutative C*-algebra is a symmetric 2-H*-algebra — the former has spectrum given by a set, the second by a groupoid (okay, a supergroupoid).

Does this point of view fit at all with the notion of 2-topos discussed here?

Posted by: Jamie Vicary on August 19, 2008 11:53 PM | Permalink | Reply to this

Re: 2-Toposes

Is

1FinSetFinSet, 1 \to FinSet \to FinSet,

the first map taking ** to \empty, the second forming the disjoint union with 11, the categorified natural number object for the 2-topos of categories?

John has some characterisations of FinSet here.

Posted by: David Corfield on November 4, 2008 12:26 PM | Permalink | Reply to this

Re: 2-Toposes

Perhaps does it depend on the precise definition of categorified natural number object, but if you take the obvious one, i.e. (pseudo-)initial algebra for the endo-2-functor +1:CatCat-+ 1: \mathrm{Cat}\to\mathrm{Cat} (mapping a category CC to the coproduct of CC with the category 11), then the natural number object is the set of natural numbers (seen as a discrete category).
Posted by: Mathieu Dupont on November 5, 2008 11:18 AM | Permalink | Reply to this

Re: 2-Toposes

Yes, that looks reasonable. Does this mean ‘recursive functor’ theory is unlikely to be much more interesting than recursive function theory?

Posted by: David Corfield on November 5, 2008 11:50 AM | Permalink | Reply to this

Re: 2-Toposes

[…] the categorified natural number object […]

I need some reminder on what game we are playing here. What is an ordinary natural number object precisely?

What is it in a presheaf topos, in particular. (I suppose I can guess what it is in SetSet.)

Another issue which I am becoming interested in: suppose we have a notion of nn-stacks on some site (0-stacks being sheaves). Supposedly these should form an (n+1)(n+1)-topos. What would we expect the subobject classifier object to be like?

(By the way, while googling around I came across “topos-physics.org” (which likely was mentioned on my own blog before without me remembering it) and which provides among other things a presheaf-topos dictionary: for instance here is subobject classifier).

Posted by: Urs Schreiber on November 5, 2008 11:06 PM | Permalink | Reply to this

Re: 2-Toposes

A natural numbers object in a cartesian closed category CC is an initial object

10NSN 1 \overset{0}{\rightarrow} N \overset{S}{\rightarrow} N

in the category of all diagrams

1aAfA 1 \overset{a}{\rightarrow} A \overset{f}{\rightarrow} A

in CC. In a presheaf topos it’s the constant presheaf.

Posted by: David Corfield on November 6, 2008 8:46 AM | Permalink | Reply to this

Re: 2-Toposes

It’s hazardous not to visit the Café for even a few days. Too much goes on!

But, to answer the question: in a presheaf topos Set CSet^C, the natural numbers object is just the constant presheaf at the ordinary natural numbers (in Set). So, the value at every object cc is \mathbb{N}; the value at every morphism cdc \to d is the identity on \mathbb{N}.

Posted by: Todd Trimble on November 7, 2008 4:27 AM | Permalink | Reply to this
Read the post Equipments
Weblog: The n-Category Café
Excerpt: An enhanced structure on a 2-category, called a "proarrow equipment," lets us define weighted limits and develop a good deal of "formal category theory."
Tracked: November 23, 2009 5:23 AM

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