Notes on commutation of limits and colimits

TL;DR

This paper explores the structure of classes of colimits in Set, revealing infinitely many intermediate classes between pseudo-filtered colimits and all colimits, and characterizing classes that contain pullbacks or equalizers.

Contribution

It identifies infinitely many distinct classes of colimits between pseudo-filtered and all colimits, and characterizes classes containing pullbacks or equalizers.

Findings

Existence of infinitely many distinct closed classes of colimits.

Classes containing pullbacks or equalizers are contained within pseudo-filtered colimits.

Provides a detailed analysis of the Galois connection between limits and colimits.

Abstract

We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all (small) colimits. On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is contained in that of pseudo-filtered colimits.