Flatness, weakly lex colimits, and free exact completions

Abstract

We capture in the context of lex colimits, introduced by Garner and Lack, the universal property of the free regular and Barr-exact completions of a weakly lex category. This is done by introducing a notion of flatness for functors $F\colon\mathcal C\to\mathcal E$ with lex codomain, and using this to describe the universal property of free $\Phi$-exact completions in the absence of finite limits, for any given class $\Phi$ of lex weights. In particular, we shall give necessary and sufficient conditions for the existence of free lextensive and free pretopos completions in the non-lex world, and prove that the ultraproducts, in the categories of models of such completions, satisfy an universal property.