Flatness, weakly lex colimits, and free exact completions
| Authors | |
|---|---|
| Year of publication | 2024 |
| Type | Article in Periodical |
| Magazine / Source | Annali di Matematica Pura ed Applicata |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.1007/s10231-023-01383-2 |
| Doi | http://dx.doi.org/10.1007/s10231-023-01383-2 |
| Keywords | Enriched categories; Flatness; Free completions; Lex colimits; Regular/exact categories |
| Attached files | |
| Description | 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 $$F:{{\mathcal {C}}}\rightarrow {{\mathcal {E}}}$$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. |
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