The n-Category Café

An object classifier in an $(n,1)$-category $\mathbf{H}$ is an object $U$ such that for any $X$, the hom-groupoid $\mathbf{H}(X,U)$ is naturally equivalent to (a small full subgroupoid of) the groupoid of objects over $X$, $Core(\mathbf{H}/X)$. However, since $\mathbf{H}$ is an $(n,1)$-category, its hom-category $\mathbf{H}(X,U)$ is an $(n-1,1)$-category, and so the collection of objects over $X$ classified by maps into $U$ must be a sub-$(n-1,1)$-category of the $(n,1)$-category $\mathbf{H}/X$. The obvious way to make this work is to take it to consist of $(n-2)$-truncated objects. For instance, in the classical case when $\mathbf{H}$ is a 1-category, we can only expect to classify a sub-poset of $\mathbf{H}/X$, and the standard poset to classify is that of subobjects of $X$ (i.e. $(-1)$-truncated objects of $\mathbf{H}/X$) — we then get the standard topos-theoretic notion of a subobject classifier.

However, because $\infty-1=\infty$, when $\mathbf{H}$ is an $(\infty,1)$-category it can contain classifiers for objects with no “truncation” restriction being necessary. To a topologist, object classifiers are familiar under the name of classifying spaces: for any $F$, the classifying space $\mathbf{B}Aut(F)$ is an object classifier for the subgroupoid of $\mathbf{H}/X$ consisting of the spaces over $X$ all of whose fibers are $F$. On the other hand, any $(\infty,1)$-topos contains an object classifier for all “$\kappa$-compact objects”, for any sufficiently large cardinal $\kappa$. (The size restriction is necessary to avoid Cantorian paradoxes.)

This latter property basically characterizes $(\infty,1)$-topos among locally presentable $(\infty,1)$-categories. However, one can ask whether $(\infty,1)$-categories that aren’t $(\infty,1)$-toposes can contain classifiers for other classes of objects. Gepner and Kock give a positive answer to this question, using an elegant categorification of the Garner-Lack theory of Grothendieck quasitoposes.

A Grothendieck (1-)topos is the category of sheaves on some site. A Grothendieck quasitopos is the category of separated presheaves on some site. Grothendieck quasitoposes have many of the properties of toposes: they are locally cartesian closed, complete and cocomplete, and locally presentable, but instead of a subobject classifier they have only a “strong-subobject classifier”: this is like a subobject classifier but it classifies only strong monomorphisms.

Thus, when in search of more exotic object classifiers, it is natural to look for a notion of $(\infty,1)$-quasitopos. This, in turn, requires categorifying the notion of “separated presheaf”. Gepner and Kock start from the characterization that a presheaf $X\in Set^{C^{op}}$ is separated if and only if $X \to L X$ is a monomorphism in $Set^{C^{op}}$, where $L$ is the (left exact) reflection of $Set^{C^{op}}$ into the subcategory of sheaves. This definition is convenient because every $(\infty,1)$-topos is a left-exact reflective subcategory of an $(\infty,1)$-presheaf category, but we don’t yet have a sufficiently general notion of “site” to be able to characterize all $(\infty,1)$-toposes as consisting of “sheaves”.

However, categorifying this definition does require choosing a way to categorify the notion of “monomorphism”, for which there are multiple choices. Rather than fixing a particular choice, Gepner and Kock show that it doesn’t matter: you can use any stable factorization system which is preserved by the reflector. Thus we arrive at their

Definition: Let $\mathbf{P}$ be an $(\infty,1)$-topos, $(\mathcal{E},\mathcal{M})$ a stable factorization system on $\mathbf{P}$, and $L:\mathbf{P}\to \mathbf{P}$ a left exact reflector which preserves $(\mathcal{E},\mathcal{M})$. An object $X\in \mathbf{P}$ is $\mathcal{M}$-separated if $X\to L X$ is in $\mathcal{M}$. An $(\infty,1)$-quasitopos is an $(\infty,1)$-category consisting of the $\mathcal{M}$-separated objects in such a situation.

In particular, if we were to interpret this definition in 1-category theory, we would get something apparently more general than the usual notion of (Grothendieck) quasitopos. (We would get the usual notion if we fixed $(\mathcal{E},\mathcal{M})$ to be (epi, mono).) It would be interesting to see how much of the theory of 1-quasitoposes carries over to these more general “quasitoposes relative to a factorization system”, and also how much of it extends to the $\infty$-case.

In particular, any category of concrete (∞,1)-sheaves is an $(\infty,1)$-quasitopos. Motivated by these, there is an nLab page called (∞,1)-quasitoposes; this gives a definition which is clearly closely related to Gepner and Kock’s definition, but is not identical. I think that the nLab definition is precisely the special case of Gepner and Kock’s definition where

1. $\mathbf{P}$ is the $(\infty,1)$-topos of $(\infty,1)$-sheaves on some $(\infty,1)$-site $(C,J)$,
2. $(\mathcal{E},\mathcal{M})$ is the ($n$-connected, $n$-truncated) factorization system for some $n$, and
3. $L$ is the reflector corresponding to a different, larger topology $K$ on the same category $C$.

On reflection, I think Gepner and Kock’s choice to use an arbitrary left-exact reflector $L$ is preferable, since not every $(\infty,1)$-topos consists of sheaves. In particular, their definition includes all $(\infty,1)$-toposes as $(\infty,1)$-quasitoposes, which seems like a desirable feature. It’s less obvious to me whether one wants to allow any stable factorization system (rather than just the (connected, truncated) ones). Quite possibly one does (and I’m very fond of stable factorization systems myself), but the theory of $(\infty,1)$-quasitoposes is currently so embryonic that I’d like to see more theory to justify one choice or the other.

One bit of theory justifying Gepner and Kock’s choice is that they get object classifiers at their level of generality. How? Well, in 1-category theory, every quasitopos contains a topos. In a presentation as the separated presheaves, this topos consists of the sheaves. Its objects can also be characterized internally in the quasitopos as the “coarse” ones (those that are orthogonal to all monic epis). The first description works just as well $(\infty,1)$-categorically: every object of the reflective subcategory corresponding to $L$ is $\mathcal{M}$-separated, since $\mathcal{M}$ contains the equivalences, and these objects form an $(\infty,1)$-topos. I don’t know whether they have an internal characterization in this case.

Anyway, the upshot is that any object classifier in the $(\infty,1)$-topos of coarse objects is also an object classifier in the whole $(\infty,1)$-quasitopos. The morphisms it classifies will always be “fiberwise coarse”; in particular, plenty of objects will not be classified by any of the object classifiers obtained in this way. Gepner and Kock write

Presumably there should be a direct characterization of this … class in terms of some notion of strong map, depending on the factorization system… just as in the classical case [it] consists of strong monos.

Sounds probably related to the question of finding an internal characterization of the coarse objects.

In addition to posing good questions like these for future research, there’s more good stuff in Gepner and Kock’s paper. They construct an object classifier in an $n$-topos which classifies some $(n-1)$-truncated objects (as opposed to the $(n-2)$-truncated ones that one would naively expect to have to require). They also construct one in motivic homotopy theory (which I know absolutely nothing about). And they give another proof (the first proof by Denis-Charles Cisinski was here on this blog) that every locally presentable, locally cartesian closed $(\infty,1)$-category can be presented by a right proper Cisinski model category.

This last fact is, with current technology, central to our ability to actually model type theory in $(\infty,1)$-categories (i.e. to deal with the strictness/coherence issues). We still don’t have a general solution for this which includes (univalent) universes, although I’ve recently heard some rumors that there may be something along those lines forthcoming soon. (Of course, like all rumors, they could be exaggerated…)