Biproducts without pointedness

TL;DR

This paper introduces a generalized notion of biproducts in arbitrary categories without enrichment, providing new characterizations and examples, thereby broadening the understanding of biproducts beyond traditional settings.

Contribution

It defines biproducts up to isomorphism without enrichment assumptions, unifies existing definitions, and offers new examples and characterizations in category theory.

Findings

Generalized biproducts up to isomorphism in arbitrary categories

Characterization of categories with all binary biproducts via ambidextrous adjunctions

New examples of biproducts recognized by the new definition

Abstract

We show how to define biproducts up to isomorphism in an arbitrary category without assuming any enrichment. The resulting notion coincides with the usual definitions whenever all binary biproducts exist or the category is suitably enriched, resulting in a modest yet strict generalization otherwise. We also characterize when a category has all binary biproducts in terms of an ambidextrous adjunction. Finally, we give some new examples of biproducts that our definition recognizes.