If $C$ has all geometric realizations of simplicial objects, what other colimits does it have?

Question

Let $C$ be an $\infty$-category. Suppose that every diagram $\Delta^{\mathit{op}} \to C$ has a colimit. Is there any characterization of small categories $I$ such that every diagram $I \to C$ has a colimit?

Answer 1

This is not a definitive answer, but more an attempt to provide some background.

You're essentially asking for the saturation, in the $\infty$-categorical setting, of (the terminal weight on) $\Delta^{\mathrm{op}}$. The saturation has been studied in the abstract by Albert and Kelly in the enriched case. In this setting, it's natural to work with weighted colimits. Albert and Kelly observe that it's natural to consider a slightly stronger property: you should look at diagrams $D$ such that every $\infty$-category with geometric realizations has $D$-colimits and every geometric-realization-preserving functor between categories with geometric realizations preserves $D$-colimits. Such a $D$ is said to lie in the saturation of $\{\Delta^{\mathrm{op}}\}$. Albert and Kelly actually show that it doesn't make a difference which question you consider in the case of weighted colimits, but they leave this question open for conical colimits.

Albert and Kelly's main result (in the case of enriched categories) is that a weight $\phi$ lies in the saturation of $\Phi$ if and only if $\phi$ lies in the free cocompletion of (the opposite of) its domain under $\Phi$-colimits. Presumably something similar should hold in the $\infty$-categorical setting. (I don't know whether weighted limits appear in HTT, but they shouldn't be hard to define, and at any rate the definition should be derivable in a standard way from Riehl and Verity's modules.

Some well-known saturated classes include filtered colimits, sifted colimits, simply-connected limits, the L-finite limits, which are the saturation of finite limits in ordinary category theory, and absolute colimits, which are the saturation of the empty class of colimits (for any fixed enriching category).

In 1-category theory, it is "almost" the case that the saturation of filtered colimits + reflexive coequalizers is all sifted colimits. See, for example here (though better results are now available). And my understanding is that this "near equation" persists to $\infty$-category theory. In any event, the nearest 1-categorical analog of $\Delta^{\mathrm{op}}$ is the weight for reflexive coequalizers, so we can look at the saturation of this class in 1-category theory.

By Albert and Kelly's result, to compute the saturation in enriched category theory, it suffices to compute the free cocompletion under this class of colimits. An elementary description of the free completion under reflexive coequalizers is given, e.g. in Adámek et al's book Algebraic Theories, Ch. 17. I don't know whether the free completion under geometric realization is similarly complicated in $\infty$-category theory.