Lex $\infty$-colimits

Question

In the paper lex colimits, Garner and Lack gave a general characterization of "exactness properties" for categories (and enriched categories). A "lex-weight" is a weight for colimits whose domain category has finite limits. If $\Phi$ is a class of lex-weights, then a finitely complete category is "$\Phi$-lex-cocomplete" if it admits all $\Phi$-weighted colimits of left-exact diagrams, and "$\Phi$-exact" if moreover the functor assigning these colimits itself preserves finite limits. Intuitively, $\Phi$-exactness of $C$ means that "all exactness relations between $\Phi$-lex-colimits and finite limits which hold in Grothendieck topoi hold in $C$", which is justified by a theorem that a small $\Phi$-lex-cocomplete category is $\Phi$-exact just when it embeds in a Grothendieck topos by a functor preserving finite limits and $\Phi$-lex-colimits.

They also gave many examples of "exactness properties" which can be characterized as $\Phi$-exactness for some $\Phi$, such as regularity, Barr-exactness, lextensivity, coherency, adhesivity, etc. In basically all examples, the proof that the given explicitly defined property is sufficient for $\Phi$-exactness proceeds by constructing a subcanonical Grothendieck topology out of the colimits in question and showing that the colimits are preserved by the resulting embedding into the topos of sheaves for this topology.

My question is about whether there is a version of this theory for $(\infty,1)$-categories. The first part of it seems extremely formal, and I would expect probably works just the same, although obviously there are a lot of details to work out. At present I am more interested in the construction of examples. This is interesting for two reasons:

1. In the $(\infty,1)$-categorical context, for any kind of colimit, there is an obvious notion of "exactness", namely "descent", as in for example section 6.5 of Rezk's note "Toposes and homotopy toposes" or Lemma 6.1.3.5 of Higher Topos Theory.

2. However, there is not (yet) a fully general notion of "Grothendieck topology" for $(\infty,1)$-categories which suffices to present all Grothendieck $(\infty,1)$-topoi.

Here then, at last, is a concrete question.

Suppose $C$ is a small $(\infty,1)$-category with finite limits, which also admits colimits of some specified sort, and that these colimits satisfy descent. Does $C$ necessarily embed in a Grothendieck $(\infty,1)$-topos by a functor preserving finite limits and the specified colimits?

There are two obvious candidates for such an $(\infty,1)$-topos:

1. The topos of sheaves on $C$ for the topology whose covering sieves consist of coprojections for the specified colimits. This almost certainly does not work. It doesn't even work in general in the 1-categorical case, although in many particular cases it does. In general, Garner and Lack showed that some additional covering families have to be added to ensure "postulatedness" of the colimits. It's unclear to me whether this will remain possible when we go "off to $\infty$".

2. The subcategory of presheaves on $C$ which preserve the specified colimits (i.e. take them to limits of $\infty$-groupoids). This satisfies the embedding property almost by definition, but I don't immediately see why it should be an $(\infty,1)$-topos. If this worked, it would be a generalization of Remark 6.3.5.17(a) of Higher Topos Theory, but I don't see a way to generalize that proof.

Answer 1

Probably not.

Claim: Let $C$ be a small $(\infty,1)$-category with finite limits and colimits which admits an embedding $V:C\to E$ into a Grothendieck $(\infty,1)$-topos preserving finite limits and colimits. Then $C$ has a countable coproduct of copies of the terminal object $1$.

Proof: Let $S^1\in C$ be the pushout of $1 \leftarrow 1+1 \to 1$, and let $Z\in C$ be the pullback of $1\to S^1 \leftarrow 1$. Since $V$ preserves these limits and colimits, $V(Z)$ is the analogous object of $E$, which is the countable coproduct of copies of $1$ (this is true in presheaf $(\infty,1)$-toposes and preserved by left exact localizations). Thus, since $V$ is fully faithful, $Z$ is also the countable coproduct of copies of $1$. $\quad\Box$

It seems unlikely to me that assuming descent for finite colimits, which is (at least intuitively) a "finitary" property, would be sufficient to guarantee an infinitary property like the existence of some countable coproduct. It's reasonable to expect that the object $Z$ would be an "internal object of integers", but not (it seems to me) that it would be such a coproduct in an externally infinitary sense.

On the other hand, it does seem likely to me that Proposition 7.3 of the Garner-Lack paper (any small $\Phi$-exact category $C$ embeds $\Phi$-exactly in a topos) should work in the $(\infty,1)$-case, at least when the generating class of lex weights $\Phi$ is a small set. The topos in question is the category of presheaves on $C$ that preserve $\Phi^*$-lex-colimits, where $\Phi^*$ is the "lex saturation" of $\Phi$. The proof (in the case of small $\Phi$) shows this category to be the intersection of two lex-reflective subcategories of the topos of presheaves on $\Phi_l C$, the closure of the representables in the presheaves on $C$ under finite limits and $\Phi$-lex-colimits; thus it is also lex-reflective, hence a topos.

I haven't checked the details, but on a quick scan I don't see anything in this proof that would obviously fail in the $\infty$-world. One difference in the $\infty$-world is that apparently not every lex-reflective subcategory of a topos is accessible (hence also a topos), but in this case the two reflections are both abstractly equivalent to presheaves on $C$, hence accessible. (The case of proper-class $\Phi$ does seem suspicious, as in that case they use a proper-class intersection of lex-reflective subcategories, appealing to a fact that the lex-reflective subcategories of a topos form a small complete lattice, which seems less likely to be true for $\infty$-toposes. But for something like $\Phi = \{ \text{pushouts}\}$ it should be fine.)

I think the disconnect is that even in the $\infty$-world, descent for a certain class $\Phi$ of colimits is (probably) not strong enough to ensure $\Phi$-exactness, so the analogue of Prop. 7.3 can't be applied. A $\Phi$-exact category is one where the inclusion $C \to \Phi_l C$ has a left exact left adjoint, and so in particular it has a left adjoint, which means that $C$ must be not just $\Phi$-lex-cocomplete but $\Phi^*$-lex-cocomplete. The latter is a stronger condition in general even for 1-categories, e.g. there is a $\Phi_{rc}$ for which $\Phi_{rc}$-lex-cocompleteness means the existence of reflexive coequalizers, whereas $\Phi_{rc}^*$-lex-cocompleteness requires also the existence of "equivalence relations freely generated by reflexive relations". Similarly in the $\infty$-case, for $\Phi = \{ \text{pushouts}\}$, $\Phi^*$-lex-cocompleteness includes also countable coproducts of copies of $1$ (and lots of other stuff like $\Omega^k S^n$!).

Note also that even in the 1-categorical case, "equivalence relations freely generated by reflexive relations" are an infinite colimit, even though $\Phi_{rc}$ itself consists only of finite limits. In particular, therefore, there are colimits that an elementary 1-topos admits but is not necessarily exact for (in contrast to a Grothendieck 1-topos, which is exact for all lex colimits), hence cannot be preserved by any embedding into a Grothendieck 1-topos. So really nothing new is happening in the $\infty$-case: once again there may be colimits (such as pushouts) that an "elementary $(\infty,1)$-topos" admits but is not necessarily exact for, hence cannot be preserved by any embedding into a Grothendieck $(\infty,1)$-topos. The only difference is that perhaps this is more surprising in the $\infty$-world, where we might have acquired an intuition that "everything that's wrong with colimits in 1-categories has been fixed by descent". (-: