On the local cartesian closure of exact completions

TL;DR

This paper establishes a precise condition under which the exact completion of a category with weak finite limits is (locally) cartesian closed, clarifying previous claims and providing necessary and sufficient criteria.

Contribution

It provides a new necessary and sufficient condition for the local cartesian closure of exact completions, correcting and extending prior characterisations.

Findings

Weak finite limits are insufficient for previous characterisation proofs.

The paper offers a strengthened hypothesis for the proof to hold.

The characterisation aligns with existing results for ex/lex completions.

Abstract

This paper presents a necessary and sufficient condition on a category with weak finite limits for its exact completion to be (locally) cartesian closed. A paper by Carboni and Rosolini already claimed such a characterisation using a different property on the base category, but we shall show that weak finite limits are not enough for their proof to go through. We shall also indicate how to strengthen the hypothesis for that proof to work. It will become clear that, in the case of ex/lex completions, their characterisation is still valid and it coincides with the one presented here.