When does $\operatorname{Ind}_\kappa(C^I) \simeq \operatorname{Ind}_\kappa(C)^I$?

Abstract

We investigate under which condition the $\kappa$-ind completion of a functor category $C^I$ is equivalent to the category of functors from $I$ to the $\kappa$-ind completion of $C$. A published theorem implies this is true for any Cauchy complete category $C$ and $\kappa$-small category $I$, but we show this is not the case in general. We prove two results that seem to cover all applications of this incorrect theorem we could find in the literature: The result holds if $C$ has $\kappa$-small colimits and $I$ is $\kappa$-small, or if $C$ is an arbitrary category and $I$ is well-founded and $\kappa$-small. In both cases, we show that the conditions are optimal in the sense that the result holds for all $C$ if and only if $I$ satisfies the given assumption.