Ind-completion and accessibility of functor categories

Abstract

We investigate under which conditions the κ-Ind completion of a functor category $C^I$ is equivalent to the category of functors from I to the κ-Ind completion of C. A published result implies this is true for any Cauchy complete category C and κ-small category I, but we show that 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 κ-small colimits and I is κ-small, or if C is an arbitrary category and I is well-founded and κ-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.