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:
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 18.104.22.168 of Higher Topos Theory.
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:
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 $\infty$".
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 22.214.171.124(a) of Higher Topos Theory, but I don't see a way to generalize that proof.