Universes for category theory

TL;DR

This paper investigates the dependence of universal constructions on Grothendieck universes in category theory, proving that certain constructions like adjoints are independent of the universe choice.

Contribution

It proves that bounded constructions such as adjoints of accessible functors are independent of the Grothendieck universe parameter U.

Findings

Adjoints of accessible functors do not depend on the universe U

Limits and Kan extensions are also independent of U

Universal constructions are stable under changes of the universe

Abstract

The Grothendieck universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-sets or even the category of all locally U-small categories, where U is an "arbitrary but fixed" universe, all without worrying about which set-theoretic operations one may legitimately apply to these entities. Unfortunately, as soon as one allows the possibility of changing U, one also has to face the fact that universal constructions such as limits or adjoints or Kan extensions could, in principle, depend on the parameter U. We will prove this is not the case for adjoints of accessible functors between locally presentable categories (and hence, limits and Kan extensions), making explicit the idea that "bounded" constructions do not depend on the choice of U.