Free extensivity via distributivity

TL;DR

This paper explores the relationship between pseudodistributive laws and extensive categories, showing how certain free completions lead to cartesian closed structures and exponentiable objects.

Contribution

It introduces the notion of doubly-infinitary lextensive categories and demonstrates their free generation results, connecting pseudomonads with categorical properties.

Findings

Extensive categories with pullbacks are pseudoalgebras for certain pseudomonads.

Freely generated doubly-infinitary lextensive categories are cartesian closed.

Objects with finitely many connected components are exponentiable in these categories.

Abstract

We consider the canonical pseudodistributive law between various free limit completion pseudomonads and the free coproduct completion pseudomonad. When the class of limits includes pullbacks, we show that this consideration leads to notions of extensive categories. More precisely, we show that extensive categories with pullbacks and infinitary lextensive categories are the pseudoalgebras for the pseudomonads resulting from the pseudodistributive laws. Moreover, we introduce the notion of doubly-infinitary lextensive category, and we establish that the freely generated such categories are cartesian closed. From this result, we further deduce that, in freely generated infinitary lextensive categories, the objects with a finite number of connected components are exponentiable. We conclude our work with remarks on descent theoretical aspects of this work, along with results concerning non-canonical isomorphisms, and relationship with other work.