Lifting PIE limits with strict projections

TL;DR

This paper provides a unified, direct proof for lifting PIE limits to 2-categories of algebras, clarifying which projections are strict and detecting strictness, and extends to weak algebra morphisms including lax morphisms.

Contribution

It offers a unified proof for lifting PIE limits in 2-categories, explicitly characterizing strict projections and extending to weak morphisms, unifying previous separate results.

Findings

Unified proof for lifting PIE limits

Explicit characterization of strict projections

Extension to weak and lax morphisms

Abstract

We give a unified direct proof of the lifting of PIE limits to the 2-category of algebras and (pseudo) morphisms, which specifies precisely which of the projections of the lifted limit are strict and detect strictness. In the literature, these limits were lifted one by one, so as to keep track of these projections in each case. We work in the more general context of weak algebra morphisms, so as to include lax morphisms as well. PIE limits are also all simultaneously lifted in this case, provided some specified arrows of the diagram are pseudo morphisms. Again, this unifies the previously known lifting of many particular PIE limits, which were also treated separately.