The n-Category Café

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 $C\to \infty Gpd$ which is constant at the terminal object doesn’t really seem as though it should be considered as making $C$ a “concrete” ∞-category. Although I suppose one could argue that it’s our intuition that needs retraining, I’d like to hear the argument.

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 $U\colon C\to D$ be a functor with a right adjoint $R$. For an object $c\in C$, the unit $c \to R U c$ is monic if and only if $U$ is faithful on morphisms with target $c$.

Proof: The composite $C(c',c) \to D(U c', U c) \xrightarrow{\cong} C(c', R U c)$ 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 $\Gamma$ is faithful.” In particular, the category of concrete objects is in fact concrete.

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

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

I suppose it even already happens between $0$ and $1$. A poset $P$, equipped with a functor $U\colon P\to 2$, is “concrete” if $U$ is full, i.e. reflects $\le$. But the composite $P \to 2 \hookrightarrow Set$ is always faithful, so any poset is a concrete category.

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

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 $n$-truncation for all $n$ is $(n+1)$-concrete (or whichever way you count).

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

1. $\Gamma$ is locally faithful;

2. $\tau_{\leq 0}\Gamma : \tau_{\leq 0} C \to \tau_{\leq 0}Grpd \simeq Set$ is faithful.

For instance the 2-category $2Grp$ 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

$$Disc S \to coDisc S$$

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

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

$$\pi_n Disc S \to \pi_n coDisc S$$

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

What’s going on here?

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 $U: C\to Cat$ is concrete if not only is $U$ locally faithful, but two 1-morphisms in $C$ which become isomorphic in $Cat$ are already isomorphic in $C$. (I’m pretty sure that $\tau_{\le 0} Cat$ and $\tau_{\le 0} Gpd$ are not the same as $Set$, although they include it as a full subcategory.)

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

So your definition would mean […]

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

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

$$\tau_{\leq n} C \stackrel{\overset{\tau_{\leq_n}}{\leftarrow}}{\hookrightarrow} C \,.$$

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

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

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 $n$-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 $Top/X$, for some non-discrete topological space $X$. The obvious forgetful functor to Set is faithful. And if $U\to X$ and $V\to X$ are injective continuous maps (hence subterminal in $Top/X$) with $U\subseteq V$ set-theoretically, but not as monomorphisms into $X$ (for instance, maybe $U$ has the subspace topology and $V$ the discrete topology), then the forgetful functor is not full on morphisms from $U$ to $V$.

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 $n$-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 $C$ and $D$ have equalizers, then a functor $U: C\to D$ which preserves equalizers is faithful if and only if it is full on containments between regular subobjects. For the converse, given $f,g$ parallel with equalizer $e$, if $U(f)=U(g)$, then $U(e)$ must be an isomorphism (being the equalizer of a parallel pair of morphisms), hence so must $e$ 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 $U\colon C\to Cat$ (which preserves finite limits, to make things easy for now) such that

1. $U$ is locally faithful (which is equivalent, by an argument like that above, to saying that $U$ is full on containments between equifier subobjects); and
2. for each $x\in C$, the restriction of $U\colon C/x \to Cat/U(x)$ to the full subcategory of $C/x$ 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?

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 $m\colon a\to x$ is the inserter of $f,g\colon x\rightrightarrows y$, with universal 2-cell $\alpha\colon f m \to g m$, and $n\colon b\to x$ is the inserter of $h,k\colon x\rightrightarrows z$ with universal 2-cell $\beta\colon h n \to k n$, then two morphisms $r,s\colon a\to b$ over $x$ are isomorphic, by the universal property of inserters, just when $\beta r = \beta s$, 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.

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 $Cat$, 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 $Cat$, 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 $Set$ (consider $Top$). What else is there? $Cat$ 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.

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 $(\infty,1)$-category, as you know. But I gather that’s what you mean by descent objects.

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

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 $$X \;\underoverset{\to}{\to}{\leftarrow}\; X\times B G \;\underoverset{\to}{\to}{\to}\; \ast$$ where the single codegeneracy is the projection, and the cofaces above and below it are $1_X$ crossed with $\ast \to B G$. The descent object of this is the groupoid whose objects are pairs $(x,g)$ with $x\in X$ and $g\in G$, and whose morphisms $(x,g) \to (y,h)$ are morphisms $x\to y$ in $X$ together with the assertion that $g=h$. In other words, the descent object is $X\times G$ where $G$ is regarded as a discrete groupoid. Since there are a proper class of $G$, there are a proper class of descent object “subobjects” of $X$.

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 $Set^{op}$ is concretizable. (The faithful functor $Set^{op}\to Set$ that I know is the contravariant powerset functor—I wonder whether $Set^{op}$ is concretizable in predicative mathematics?)