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 (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. 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.

Butterflies in a semi-abelian context / O. Abbad, S. Mantovani, G. Metere, E.M. Vitale. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 238(2013 May 01), pp. 140-183. [10.1016/j.aim.2013.01.008]

Butterflies in a semi-abelian context

S. Mantovani
Secondo
;
2013

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 (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. 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.
Bicategory of fractions; Butterfly; Internal groupoid; Semi-abelian category; Weak equivalence
Settore MAT/02 - Algebra
1-mag-2013
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/217767
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