On semiflexible, flexible and pie algebras

TL;DR

This paper introduces pie algebras for 2-monads, establishing their relationship with flexible and semiflexible algebras, and characterizing them through weighted limit functors and 'free at the object' conditions.

Contribution

It defines pie algebras for 2-monads and explores their properties and relationships with existing algebra classes, providing new characterizations and insights.

Findings

Pie algebras are 'free at the object' in many cases.

Characterizations of pie, flexible, and semiflexible weights via weighted limit functors.

Strict monoidal categories are pie when their underlying monoid is free.

Abstract

We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones. We see that in many cases, the pie algebras are precisely those "free at the level of objects" in a suitable sense; so that, for instance, a strict monoidal category is pie just when its underlying monoid of objects is free. Pie algebras are contrasted with flexible and semiflexible algebras via a series of characterisations of each class; particular attention is paid to the case of pie, flexible and semiflexible weights, these being characterised in terms of the behaviour of the corresponding weighted limit functors.