Lex colimits

Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of “exactness” conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We introduce a general notion of exactness, of which each of the structures listed above, and others besides, are particular instances. The notion can be understood as a form of cocompleteness “in the lex world” – more precisely, in the 2-category of finitely complete categories and finite-limit preserving functors.

💡 Research Summary

The paper “Lex colimits” develops a unified framework for a wide variety of categorical exactness notions—regular, Barr‑exact, lextensive, coherent, adhesive, and many others—by interpreting them as instances of a single concept: Φ‑exactness. The setting is the 2‑category LEX of finitely complete categories and finite‑limit preserving functors (the “lex world”).

The authors begin by recalling the free‑cocompletion pseudomonad P on CAT, whose pseudo‑algebras are precisely cocomplete categories. By restricting and corestricting P to LEX they obtain a new pseudomonad Pℓ; its pseudo‑algebras are called small‑exact categories, which turn out to be exactly the ∞‑pretoposes.

For any class Φ of weights (each weight is a small presheaf ϕ:Kᵒᵖ→Set with K finitely complete), they define Φ‑lex‑cocompleteness: a finitely complete category C is Φ‑lex‑cocomplete if every ϕ‑weighted colimit of a finite‑limit preserving diagram D:K→C exists. Typical choices of Φ recover familiar colimit types: coequalisers of kernel pairs (regularity), coequalisers of equivalence relations (Barr‑exactness), finite coproducts, unions of subobjects, pushouts along monos, etc.

Φ‑exactness is then the conjunction of Φ‑lex‑cocompleteness with the “limit‑colimit compatibilities” that hold in Set (and more generally in any Grothendieck topos or ∞‑pretopos). Two equivalent characterisations are given:

1. A small finitely complete C is Φ‑exact iff it admits a full embedding into a V‑topos (a V‑enriched presheaf category) that is reflective via a finite‑limit preserving left adjoint. In other words, C is a reflective subcategory of its free Φ‑lex‑cocompletion Φℓ C, and the reflector preserves finite limits.

2. Equivalently, the restricted Yoneda embedding C→Φℓ C has a finite‑limit preserving left adjoint; thus C is a lex‑reflective subcategory of Φℓ C.

These conditions capture precisely the idea of “cocompleteness in the lex world”: the pseudomonad Pℓ on LEX encodes small‑exactness, and taking a full sub‑pseudomonad determined by Φ yields the notion of Φ‑exactness. Consequently, the free Φ‑exact completion of any finitely complete C exists and is given by the full subcategory Φℓ C of the presheaf category