2011/04/21 by Omar Abbad, Abbad, Omar, Sandra Mantovani +6
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.CT #msc:18A40 #msc:18B40 #msc:18D05 #msc:18E10
paper · pdf · doi:10.48550/arxiv.1104.4275
arxiv created 2011/04/21 · arxiv updated 2011/04/22
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).