Commuting homotopy colimits and arbitrary products in spaces

Question

Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, but a general $\infty$-category is fine as well). I am interested in the following question:

What is a practical criterion for checking that the canonical comparison map $$ \text{colim}_D (X^I) \rightarrow (\text{colim}_D X)^I $$ is an equivalence?

The question is inspired by the following Proposition due to Adamek, Koubek, Velebil (see below) for the case of a (1-categorical) diagram with values in Sets:

Proposition 4.5.: A small diagram $X : D \rightarrow Set$ commutes with products of cardinality smaller than $\lambda$ in the above sense iff

1. given less than $\lambda$ many elements $(d_i,x_i)$ in the category of elements $\text{Elts}(X)$ of $X$, there exists an object $d$ in $D$ such that each $(d_i,x_i)$ lies in the same component of $\text{Elts}(X)$ as some element of $X(d)$;
2. given less than $\lambda$ many pairs $(d, x_i)$ and $(d', x'_i)$ of elements of $X$ such that for each $i$ the pair lies in one component of $\text{Elts}(X)$, there exists a zig-zag $Z$ in $D$ connecting $d$ and $d'$ such that each of the pairs above can be connected by a zig-zag in $\text{Elts}(X)$ whose underlying zig-zag is $Z$.

To give some intuition for this statement: Condition $1$ is precisely surjectivity of the comparison map, and $2$ is precisely injectivity.

My thoughts on the matter: The reason why I assume there should be a generalization to colimits in spaces, is that the usage of the word "zigzag" can be literally thought of as meaning path in the realization of the category of elements $\text{Elts}(X)$, which is a model for the (homotopy) colimit. Also, since $\pi_0$ commutes with both colimits and arbitrary products, a necessary condition is that $\pi_0(X)$ satisfies conditions (1) and (2). I suspect a strengthened requirement of (2) to be needed.

Adámek, Jiří; Koubek, Václav; Velebil, J., A duality between infinitary varieties and algebraic theories., Commentat. Math. Univ. Carol. 41, No. 3, 529-541 (2000). ZBL1035.08004.

Answer 1

I will answer my own question, in hope that it is helpful to someone.

Given a functor $X:D \rightarrow Spc$ of $\infty$-categories, we can take the unstraightening of $X$ (the appropriate generalization of the category of elements of X), as a left fibration $Un(X) \rightarrow D$. It is a standard fact that the colimit over X can be computed as the realization $|Un(X)| = Un(X)[Un(X)^{-1}]$, see e.g. Lurie HTT, Cor 3.3.4.6.

It is also clear that $Un( X^I ) \simeq Un( X )^I$. Hence we have commutativity with products iff

$$(\prod Un( X ) )[ \prod Un( X )^{-1} ] \rightarrow \prod Un( X )[ Un( X )^{-1} ]$$

is an equivalence (of $\infty$-categories!). This is a classical problem in homotopy theory. The major obstruction is that (higher dimensional) hammock-type diagrams must be bounded in size, i.e. equivalent modulo higher dimensional hammocks to hammocks of a given size constraint. This is for example the case if $Un( X )$ satisfies a functorial 3-arrow calculus.

Alternatively, one can give concrete descriptions of the homotopy groups of a realization of an $\infty$-category via infinite simplicial or cubical subdivisions. The challenge then remains to find control over the size of representing elements.