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February 16, 2011

Concrete ∞-Categories

Posted by Mike Shulman

Here’s a somewhat frivolous question which just occurred to me. A concrete category is usually defined to mean a category equipped with a functor CSetC\to Set which is faithful, which is to say, surjective on equations (= 2-morphisms) between 1-morphisms. (One sometimes adds additional conditions as well.) In the language of stuff, structure, property, it is a category whose objects can be regarded as sets equipped with structure.

It makes sense to define a concrete 2-category to be one equipped with a functor CCatC \to Cat which is locally faithful, i.e. surjective on equations (= 3-morphisms) between 2-morphisms. For instance, the 2-category of monoidal categories is concrete in this sense: two monoidal transformations are equal if their underlying natural transformations are, but we cannot say the same for monoidal functors. Thus a concrete 2-category is a 2-category whose objects are regarded as categories equipped with extra stuff.

But what about ∞-categories? When we get all the way up to the top, it seems like there’s no condition left to impose. Is a concrete ∞-category just an ∞-category equipped with a completely arbitrary functor to ∞Cat (or ∞Gpd, if we have an (∞,1)-category)?

Posted at February 16, 2011 3:19 AM UTC

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23 Comments & 1 Trackback

Re: Concrete ∞-Categories

Not too dissimilar a conclusion to yours about infinitely categorified Klein geometry.

Posted by: David Corfield on February 16, 2011 1:51 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

It does look a bit similar, doesn’t it? But intuitively, I feel like surely some ∞-categories are “concrete” and others aren’t. For instance, the forgetful functor from monoidal ∞-groupoids to ∞-groupoids really seems like it makes the former into a concrete ∞-category. But the functor CGpdC\to \infty Gpd which is constant at the terminal object doesn’t really seem as though it should be considered as making CC a “concrete” ∞-category. Although I suppose one could argue that it’s our intuition that needs retraining, I’d like to hear the argument.

Posted by: Mike Shulman on February 16, 2011 4:55 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

It is maybe remarkable that this phenomenon has more in common with the notion of concrete objects in a local topos than just the terminology. I find myself in the middle trying to understand this (have been talking to Dave Carchedi about related things for half the day).

I’ll restrict your setup to goupoids and \infty-groupoids.

For instance let Core(Set)Core(Set) be the maximal groupoid in SetSet and say a functor HCore(Set)H \to Core(Set) makes the groupoid HH concrete if it is faithful. Equivalently that means that the functor is 0-truncated (full and faithful would be (1)(-1)-truncated and an equivalence would be (2)(-2)-truncated).

Next let Core(Grpd)Core(Grpd) be the maximal 2-groupoid in GrpdGrpd. A concrete 2-groupoid in your sense above would be given by a 1-truncated functor HCore(Grpd)H \to Core(Grpd), I suppose.

Generally, if we’d continue in this way we would say that a concrete nn-groupoid is given by an \infty-functor HCore((n1)Grpd)H \to Core((n-1)Grpd) which is (n1)(n-1)-truncated.

In each case this would really be saying something only about the highest degree nontrivial morphisms.

Now the analogous setup: in a cohesive topos, an object XX is concrete if XcoDiscΓXX\to coDisc \Gamma X is a monomorphism. In other words: a cohesive set is concrete if the cohesive 0-functor XcoDiscΓXX \to coDisc \Gamma X is (1)(-1)-truncated.

Given then a cohesive (,1)(\infty,1)-topos, let XX be an nn-truncated obect in it. We might be tempted to say that it is concrete if the canonical morphism XcoDiscΓXX \to coDisc \Gamma X is (n1)(n-1)-truncated.

For instance in SmoothGrpdSmooth\infty Grpd =Sh (,1)(CartSp smooth) = Sh_{(\infty,1)}(CartSp_{smooth}) for GG a Lie group, the object BG\mathbf{B}G is concrete in this sense, due to the fact that it has a diffeological space of morphisms (and diffeological spaces are precisely the concrete 0-truncated objects).

However, there seems to be a problem with this way of speaking: also BG conn\mathbf{B}G_{conn}, the coefficient object for GG-bundles with connection, is concrete in this sense, because the conditon as stated above only pays attention to morphisms in degree 1 and does not notice the non-concrete cohesive structure in the objects (sheaves of differential forms in positive degree are as far from concrete as possible). In other words, BG conncoDiscΓBG conn\mathbf{B}G_{conn} \to coDisc \Gamma \mathbf{B}G_{conn} is 0-truncated but far from (1)(-1)-truncated.

This is a low degree indication of the analogue of the problem that you mention in the above entry: as we keep raising nn, the condition that XcoDiscΓXX \to coDisc \Gamma X be (n1)(n-1)-truncated misses an increasing amount of information that intuitively we’d think should matter for concreteness, until it vanishes entirely for untruncated objects.

So what’s the right way to think about such conditons? For concrete objects in a cohesive \infty-topos, I am thinking that a good way to say that an nn-truncated object XX is concrete is to say that

  • XcoDiscΓXX \to coDisc \Gamma X is (n1)(n-1)-truncated;

  • and τ n1XcoDiscΓτ n1X\tau_{\leq n-1} X \to coDisc \Gamma \tau_{\leq n-1} X is (n2)(n-2)-truncated;

  • and so on

  • and finally τ 0XcoDiscΓτ 0X\tau_{\leq 0} X \to coDisc \Gamma \tau_{\leq 0} X is (1)(-1)-truncated.

In other words, that’s close to saying that all its categorical homotopy groups – its homotopy sheaves – are concrete sheaves.

With that definition for instance BG\mathbf{B}G would be concrete, but BG conn\mathbf{B}G_{conn} would not, as it should be.

Now, this tower of conditions going all the way from nn to 0 is too strong for the analog of the notion of “concrete category”. But maybe some variant of this would be good to consider. Maybe such a tower for an interval of values?

Posted by: Urs Schreiber on February 16, 2011 7:44 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

It is maybe remarkable that this phenomenon has more in common with the notion of concrete objects in a local topos than just the terminology.

I don’t think it’s particularly surprising. The relationship that pops into my mind would be:

Theorem: Let U:CDU\colon C\to D be a functor with a right adjoint RR. For an object cCc\in C, the unit cRUcc \to R U c is monic if and only if UU is faithful on morphisms with target cc.

Proof: The composite C(c,c)D(Uc,Uc)C(c,RUc)C(c',c) \to D(U c', U c) \xrightarrow{\cong} C(c', R U c) is given by composing with the unit, so both conditions say that this function is injective. \square

Thus, in a local topos, or any category whose forgetful functor has a right adjoint, the concrete objects are roughly “the largest possible subcategory on which Γ\Gamma is faithful.” In particular, the category of concrete objects is in fact concrete.

All your talk of nn-truncated objects started to make me uneasy because I’d expected an object to simply “be concrete” or not, without needing to fix an nn relative to which to ask the question. But then I realized that we have the same problem for categories!

Namely, a non-faithful functor U:CSetU\colon C\to Set does not allow us to regard CC as a “concrete 1-category,” but at least according to the definition I proposed above, the composite CUSetCatC \xrightarrow{U} Set \hookrightarrow Cat does allow us to regard CC as a “concrete 2-category”!

Of course this is just a manifestation of the same issue, that when we head off to infinity, “top-dimensional” phenomena get pushed off the end of the natural numbers into nothingness. But for some reason it hadn’t clicked in my mind before just now that something odd already happens at the boundary between 11 and 22.

I suppose it even already happens between 00 and 11. A poset PP, equipped with a functor U:P2U\colon P\to 2, is “concrete” if UU is full, i.e. reflects \le. But the composite P2SetP \to 2 \hookrightarrow Set is always faithful, so any poset is a concrete category.

Maybe this is the real question: is it right that an arbitrary nn-category can be regarded as a “concrete” (n+1)(n+1)-category? If so, then the situation with \infty is obvious, just because +1=\infty + 1 = \infty.

Posted by: Mike Shulman on February 17, 2011 5:12 AM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

Theorem:

Yes, that’s the observation called lemma 2.3 in Johnstone’s latest.

I don’t think it’s particularly surprising.

Let me put it more positively: that is a nice relationship you have pointed out there! Simple as it may be. Thanks for making me aware of this. I have added the remark to Quasitoposes of concrete cohesive objects.

I’d expected an object to simply “be concrete” or not, without needing to fix an n relative to which to ask the question.

Yes, exactly, so this means there is an open question to be solved: what is the right notion of concrete objects in the untruncated context? I had proposed that one answer might be: an object whose nn-truncation for all nn is (n+1)(n+1)-concrete (or whichever way you count).

For categories this would presumeably mean something like, for instance: a 2-functor Γ:CGrpd\Gamma : C \to Grpd exhibits a concrete 2-category if

  1. Γ\Gamma is locally faithful;

  2. τ 0Γ:τ 0Cτ 0GrpdSet\tau_{\leq 0}\Gamma : \tau_{\leq 0} C \to \tau_{\leq 0}Grpd \simeq Set is faithful.

For instance the 2-category 2Grp2Grp of 2-groups would then be just plain concrete in this sense.

Also for objects in examples of cohesive \infty-toposes that I know and care about, this notion reproduces roughly what one would expext intuitively. But I am lacking for this definition any nice abstract characterization similar to the nice characterizations of 1-concrete 0-truncated objects that we have been talking about. That makes me nervous. I feel I haven’t gotten to the bottom of this yet.

What also makes me nervous is that the duality between pieces have points and discrete objects are concrete breaks in higher categorical dimension. The condition pieces have points holds over any \infty-cohesive site, but demanding that the morphims

DiscScoDiscS Disc S \to coDisc S

be (1)(-1)-truncated is in typical nontrivial examples of cohesive \infty-toposes equivalent to demanding that SS is a 0-truncated \infty-groupoid.

However, it is true in these examples that the morphism of homotopy sheaves

π nDiscSπ ncoDiscS \pi_n Disc S \to \pi_n coDisc S

is a monomorphism for all nn \in \mathbb{N}, which should mean that τ nDiscS\tau_{\leq n} Disc S is (n+1)(n+1)-concrete for all nn.

What’s going on here?

Posted by: Urs Schreiber on February 17, 2011 10:53 AM | Permalink | Reply to this

Re: Concrete ∞-Categories

Sorry, maybe I misunderstood what you meant by “remarkable.” (-: And I guess I didn’t really understand what you were proposing, either.

So your definition would mean that a 2-category with forgetful 2-functor U:CCatU: C\to Cat is concrete if not only is UU locally faithful, but two 1-morphisms in CC which become isomorphic in CatCat are already isomorphic in CC. (I’m pretty sure that τ 0Cat\tau_{\le 0} Cat and τ 0Gpd\tau_{\le 0} Gpd are not the same as SetSet, although they include it as a full subcategory.)

I fail to see why 2Grp2Grp would be concrete in this sense. I’m pretty sure that MonCatMonCat is not concrete in this sense: one and the same functor between monoidal categories can have more than one inequivalent structure of monoidal functor.

Posted by: Mike Shulman on February 17, 2011 6:42 PM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

So your definition would mean […]

No, you are misreading the notation “τ n\tau_{\leq n}” as forming the homotopy (n+1)(n+1)-category. But I mean it as in HTT def. 5.5.6.1: as forming the full subcategory on nn-truncated objects.

I think I am using the notation consistently, but inherent in prop. 5.5.6.18 is the following notational problem: the nn-truncation reflection is denoted

τ nCτ nC. \tau_{\leq n} C \stackrel{\overset{\tau_{\leq_n}}{\leftarrow}}{\hookrightarrow} C \,.

That means τ K\tau_{\leq} K means something very different for KK regarded as an ambient \infty-category than it means for KK regarded as an object in an ambient \infty-category.

Posted by: Urs Schreiber on February 17, 2011 7:40 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

Augh! That’s terrible notation! (In particular, Joyal has used τ 0\tau_0 and τ 1\tau_1 of a quasicategory to mean its homotopy 0- and 1-categories, respectively.) Can’t we write something like Trunc n(C)Trunc_n(C) for the full subcategory of nn-truncated objects?

Posted by: Mike Shulman on February 17, 2011 8:10 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

Can’t we write something like Trunc n(C)Trunc_n(C)

Sure. So to recapitulate, what I am wondering about is this:

  1. Given a local \infty-topos

    (p p p !):HGrpd (p^{*} \dashv p_{*} \dashv p^!) : \mathbf{H} \to \infty Grpd

    does it make sense to say that the concrete objects XX are those for which

    τ nXp !p *τ nX \tau_n X \to p^! p_* \tau_n X

    is (n1)(n-1)-truncated for all nn?

  2. Given an (,1)(\infty,1)-functor

    CGrpd, C \to \infty Grpd \,,

    does it make sense to say that it exhibits a concrete (,1)(\infty,1)-category if

    Trunc n(C)Trunc nGrpdnGrpd Trunc_n(C) \to Trunc_n \infty Grpd \simeq n Grpd

    is faithful on (n+1)(n+1)-morphisms for all nn?

  3. Would the restriction

    p *:Conc(H)Grpd p_* : Conc(\mathbf{H}) \to \infty Grpd

    of p *p_* to the full subcategory Conc(H)HConc(\mathbf{H}) \hookrightarrow \mathbf{H} on the concrete objects exhibit a concrete (,1)(\infty,1)-category?

Or something like that?

There is a concrete test-case (not a truncated one! ;-) that I want to understand, and which should serve to gauge or fine-tune or correct the definitions here. But maybe I’ll post that to the thread on punctual connectness.

Posted by: Urs Schreiber on February 17, 2011 9:26 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

Okay, now I think I finally understand what you’re saying. (-: It feels odd to me to give truncated objects a special role. For instance, with your proposed definition, it could happen that a full subcategory of a non-concrete nn-category is concrete (throw away the truncated objects). Also the opposite of a concrete category might not be concrete, since the notion of “truncated object” is not self-dual. But those are aesthetic objections, and might be overruled if the notion turns out to be of mathematical value.

More mathematically, I think the original notion of concreteness for 1-categories does not satisfy your proposed definition, which would say that (in addition to being faithful on 1-morphisms), the functor to Set should be full on morphisms between subterminal objects. For instance, consider the slice category Top/XTop/X, for some non-discrete topological space XX. The obvious forgetful functor to Set is faithful. And if UXU\to X and VXV\to X are injective continuous maps (hence subterminal in Top/XTop/X) with UVU\subseteq V set-theoretically, but not as monomorphisms into XX (for instance, maybe UU has the subspace topology and VV the discrete topology), then the forgetful functor is not full on morphisms from UU to VV.

However, the Isbell-Freyd characterization mentioned by Richard suggests that really we should be looking at regular subterminals here, and in that case the functor is, indeed, full on morphisms between them. It also suggests that we should be looking at (regular) subobjects of any object, not just the terminal object.

So we come back to the question of what is a “regular subobject” in an nn-category that I raised here in response to Richard, only now thinking not of well-poweredness but of faithfulness on the subcategory. And now a certain sort of converse of the characterization is easy:

  • If CC and DD have equalizers, then a functor U:CDU: C\to D which preserves equalizers is faithful if and only if it is full on containments between regular subobjects. For the converse, given f,gf,g parallel with equalizer ee, if U(f)=U(g)U(f)=U(g), then U(e)U(e) must be an isomorphism (being the equalizer of a parallel pair of morphisms), hence so must ee be.

With this perspective, there’s no objection to inserters for 2-categories, since an inserter is always 0-truncated. This suggests that maybe a 2-category (with finite limits) should be called concrete if it has a functor U:CCatU\colon C\to Cat (which preserves finite limits, to make things easy for now) such that

  1. UU is locally faithful (which is equivalent, by an argument like that above, to saying that UU is full on containments between equifier subobjects); and
  2. for each xCx\in C, the restriction of U:C/xCat/U(x)U\colon C/x \to Cat/U(x) to the full subcategory of C/xC/x spanned by the inserters (which is a 1-category, since inserters are 0-truncated) is faithful.

Do the natural examples of concrete 2-categories satisfy that? Is it equivalent to any more “basic” property more akin to local-faithfulness?

Posted by: Mike Shulman on February 18, 2011 7:56 AM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

I wrote:

Do the natural examples of concrete 2-categories satisfy that? Is it equivalent to any more “basic” property more akin to local-faithfulness?

Actually, I think that second property is also implied by local faithfulness. Which is perhaps not surprising. For if m:axm\colon a\to x is the inserter of f,g:xyf,g\colon x\rightrightarrows y, with universal 2-cell α:fmgm\alpha\colon f m \to g m, and n:bxn\colon b\to x is the inserter of h,k:xzh,k\colon x\rightrightarrows z with universal 2-cell β:hnkn\beta\colon h n \to k n, then two morphisms r,s:abr,s\colon a\to b over xx are isomorphic, by the universal property of inserters, just when βr=βs\beta r = \beta s, which is an equality of 2-cells and hence reflected by a locally faithful 2-functor.

So what I wanted to do in the previous comment isn’t actually getting us anything new.

Posted by: Mike Shulman on February 18, 2011 8:08 PM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

Freyd’s JPAA paper on concreteness gives some alternative characterisations of the notion. Perhaps these alternative characterisations generalise more readily.

Posted by: Richard Garner on February 17, 2011 12:49 AM | Permalink | Reply to this

Re: Concrete ∞-Categories

Thanks for the reference! I think the most interesting characterization is that a finitely complete category is concretizable iff it is regularly well-powered, i.e. each object has only a set of regular subobjects (subobjects that are equalizers). I added a brief version of Isbell’s proof of necessity to the nLab; Freyd’s proof of sufficiency is rather more involved.

For a 2-category, Isbell’s proof seems to generalize directly to say that if a 2-category admits a locally faithful functor to CatCat, then it is well-powered with respect to subobjects that are equifiers. I have no idea whether Freyd’s proof of sufficiency could be generalized.

So is there a reasonable replacement for equifiers here? In 2-categories we could consider inverters, and I think Isbell’s proof would show that if a 2-category admits a locally conservative functor to CatCat, then it is well-powered with respect to subobjects that are inverters. But it’s not clear to me whether local conservativity is a property to expect of a concrete 2-category; certainly concrete 1-categories need not be conservative over SetSet (consider TopTop). What else is there? CatCat itself is not well-powered with respect to “subobjects” that are inserters or descent objects.

And I can’t think of a good analogue of any of these for ∞-categories.

Posted by: Mike Shulman on February 18, 2011 7:28 AM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

And I can’t think of a good analogue of any of these for ∞-categories.

Do you mean analogues of the characterization statements or analogues of the notions of subobjects?

There is the evident notion of a regular monomorphism in an (,1)(\infty,1)-category, as you know. But I gather that’s what you mean by descent objects.

Is GrpdGrpd well-powered with respect to regular (2,1)(2,1)-monomorphisms?

Posted by: Urs Schreiber on February 18, 2011 10:00 AM | Permalink | Reply to this

Re: Concrete ∞-Categories

Yes, “regular monics” in an (∞,1)-category are the ∞-analogue of descent objects, and analogues of the notion of subobject are what I was looking for here.

And no, I’m pretty sure that Gpd is not “well-powered” for descent objects. For instance, for any groupoid X, and any group G, we can form the truncated simplicial object XX×BG* X \;\underoverset{\to}{\to}{\leftarrow}\; X\times B G \;\underoverset{\to}{\to}{\to}\; \ast where the single codegeneracy is the projection, and the cofaces above and below it are 1 X1_X crossed with *BG\ast \to B G. The descent object of this is the groupoid whose objects are pairs (x,g)(x,g) with xXx\in X and gGg\in G, and whose morphisms (x,g)(y,h)(x,g) \to (y,h) are morphisms xyx\to y in XX together with the assertion that g=hg=h. In other words, the descent object is X×GX\times G where GG is regarded as a discrete groupoid. Since there are a proper class of GG, there are a proper class of descent object “subobjects” of XX.

Posted by: Mike Shulman on February 20, 2011 9:12 AM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

Yes, “regular monics” in an (∞,1)-category are the ∞-analogue of descent objects,

Just a remark on tastes: I’d prefer to call descent objects only those \infty-limits/homotopy limits of cosimplicial diagrams that actually do arise in descent problems, the result of homming a codescent objects/hypercovers into a coefficient object. For instance any equalizer in a 1-category is a regular \infty-monic, but we’d hardly want to call every equalizer a “descent object in a 1-category”. (?)

For instance, […]

Ah, right, of course. every loop space object

G BGG \simeq {*} \prod_{\mathbf{B}G} {*}

being the \infty-limit over

BG{*} \to \mathbf{B}G \leftarrow {*}

is equivalently the \infty-limit over

BG{\bullet} \stackrel{\to}{\to} \mathbf{B}G \stackrel{\to}{\stackrel{\to}{\to}} {\bullet}.

Thanks.

(And sorry for the whitespacy typesetting, but the blog’s parser get’s confused with “*{*}“s in the formulas. Newlines help.)

I have another question concerning your remark above that my attempt at defining concrete \infty-categories was not invariant under passing to opposites: why would we want to require that?

I understand that there are faithful functors Set opSetSet^{op} \to Set and accordingly for every concrete category also its opposite can be regarded as concrete. But somehow this feels more like a “coincidence”. Is it not? I am ready to abandon my attempt (it’s just an attempt), but I am not sure that I understand why its non-invariance under passing to opposites is good reason to do so.

Posted by: Urs Schreiber on February 21, 2011 11:12 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

I’d prefer to call descent objects only those ∞-limits/homotopy limits of cosimplicial diagrams that actually do arise in descent problems

Well, the terminology as I used it is pretty standard in 2-category theory. In strict 2-category theory it refers to a specific sort of strict 2-limit, which is more specific than just a non-strict 2-limit of a cosimplicial diagram. Also there are “lax” descent objects, which are a different type of 2-limit (strict or non-strict) that are “genuinely weighted” and can’t be described just as a non-strict conical 2-limit of a cosimplicial diagram.

I don’t quite understand the reason for your preference; is there a danger of confusion somewhere?

the blog’s parser get’s confused with “*\ast“s in the formulas.

I’ve found that using \ast instead of * in formulas avoids all the confusion.

not invariant under passing to opposites: why would we want to require that?

Probably you’re right that we don’t. I’m not sure why I thought that would be a thing to expect; it certainly does feel like something of an accident that Set opSet^{op} is concretizable. (The faithful functor Set opSetSet^{op}\to Set that I know is the contravariant powerset functor—I wonder whether Set opSet^{op} is concretizable in predicative mathematics?)

Posted by: Mike Shulman on February 21, 2011 11:37 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

it certainly does feel like something of an accident that Set opSet^op is concretizable.

Does concreteness continue up the ladder? There are three duals to CatCat: Cat opCat^{op}, Cat coCat^{co} and Cat coopCat^{co op}. Is any of them a concrete 2-category? Are there nice characterisations, just as Set opSet^{op} is equivalent to the category of complete atomic Boolean algebras?

Posted by: David Corfield on February 22, 2011 8:57 AM | Permalink | Reply to this

Re: Concrete ∞-Categories

There are three duals to CatCat: Cat opCat^{op}, Cat coCat^{co} and Cat coopCat^{coop}.

By the way, this reminds me of something interesting that I had meant to mention:

Chris Schommer-Pries and collaborators have a proof that the (,1)(\infty,1)-category of (,n)(\infty,n)-categories

  1. is unique , up to equivalence;

  2. has as nontrivial automorphism precisely the nn different op-s

if only we impose four or five mild, simple and entirely natural conditions (notably saying that posets and strict nn-categories embed in the expected way).

The argument is a slight variant of the proof by Carlos Simpson and Jacob Lurie that (,1)Cat(\infty,1)Cat has precisely one nontrivial automorphism (up to equivalence) given by the single op-operation.

Posted by: Urs Schreiber on February 22, 2011 11:00 AM | Permalink | Reply to this

Re: Concrete ∞-Categories

I look forward to seeing it!

Posted by: Mike Shulman on February 22, 2011 9:38 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

Does concreteness continue up the ladder?

Well, Cat coCat^{co} is isomorphic to CatCat; the isomorphism is () op(-)^{op}, which Urs just referred to. (-: Hence Cat coopCat^{co op} is isomorphic to Cat opCat^{op}, so there’s really only one question there (and so on up the ladder).

IF you’re willing to ignore size issues, then the analogue of the contravariant powerset functor is [,Set]:Cat opCat[-,Set] \colon Cat^{op}\to Cat. I think that is actually locally full and faithful, not just locally faithful. I’m not sure whether there’s some way to finesse the size issues here, though. And Freyd’s characterization of concreteness suggests that size issues are, in some sense, the essence of concreteness (reminds me also of this post), so ignoring them is probably not a good strategy.

This leads back to my comments about predicativismSet opSet^{op} is concrete because we have a 0-category of all (-1)-categories, but that impredicative assumption is an anomaly, since we don’t ever have a 1-category of all 0-categories, a 2-category of all 1-categories, etc.

Posted by: Mike Shulman on February 22, 2011 9:23 PM | Permalink | PGP Sig | Reply to this

Re: Concrete ∞-Categories

Oh, and the characterization of Cat opCat^{op} corresponding to [,Set][-,Set] would be something like “cocomplete categories equipped with a strongly-generating set of tiny objects, and functors having both left and right adjoints (equivalently, by the adjoint functor theorem, preserving small limits and colimits) whose left adjoint preserves the chosen generators.” That’s obviously similar to “complete atomic Boolean algebras” except that we have to consider the “atoms” as structure, rather than a property. If you make it into a property and consider instead “cocomplete categories for which there exists a strongly-generating set of tiny objects, and functors having both left and right adjoints” then you get the subcategory Cat cc opCat^{op}_{cc} of Cauchy-complete categories.

Posted by: Mike Shulman on February 22, 2011 9:36 PM | Permalink | Reply to this

Re: Concrete ∞-Categories

Fascinating. So that’s the categorified set-CABA duality. Presumably there’s a whole range of such dualities, e.g., an analogue to Stone space-BA duality. SetSet like 22 must be equipable with a long list of structures, so can play the ambimorphic object in many ways. In fact we saw one in Forssell’s work, mentioned a little way down here, where SetSet is both finite product category and also a category with all limits and colimits, and an object in the category, of such categories with functors which preserve limits, filtered colimits, and regular epimorphisms.

Hmm, perhaps I ought to go back to the thought in that post. I’m sure there was something to that idea that if modal logic theories and their semantics could be represented by algebras and coalgebras, respectively, of endofunctors acting on each side of Stone dual categories, then first-order modal logic could be represented by similarly using Forssell’s dual 2-categories.

Posted by: David Corfield on February 23, 2011 8:47 AM | Permalink | Reply to this
Read the post Doctrinal and Tannakian Reconstruction
Weblog: The n-Category Café
Excerpt: Seeking what is common between Gabriel-Ulmer style dualities and Tannaka duality
Tracked: July 11, 2011 11:37 AM

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