Butterflies in a Semi-Abelian Context

TL;DR

This paper introduces internal butterflies as a new concept to internalize monoidal functors between internal groupoids in semi-abelian categories, establishing a bicategory structure that generalizes existing frameworks.

Contribution

It defines internal butterflies in semi-abelian categories and proves their bicategory is the bicategory of fractions of internal groupoids under certain conditions.

Findings

Internal butterflies form a bicategory B(C).

B(C) is the bicategory of fractions of Grpd(C).

Main result applies when Huq and Smith commutators coincide.

Abstract

It is known that monoidal functors between internal groupoids in the category Grp of groups constitute the bicategory of fractions of the 2-category Grpd(Grp) of internal groupoids, internal functors and internal natural transformations in Grp with respect to weak equivalences. Monoidal functors can be described equivalently by a kind of weak morphisms introduced by B. Noohi under the name of "butter ies". In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category C, and we show that they are morphisms of a bicategory B(C): Our main result states that, when in C the notions of Huq commutator and Smith commutator coincide, then the bicategory B(C) of internal butterflies is the bicategory of fractions of Grpd(C) with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects).