Butterflies in a semi-abelian context

Abstract

It is known that monoidal functors between internal groupoids in the\ud category Grp of groups constitute the bicategory of fractions of the\ud 2-category Grpd(Grp) of internal groupoids, internal functors and\ud internal natural transformations in Grp, with respect to weak\ud equivalences (that is, internal functors which are internally fully\ud faithful and essentially surjective on objects). Monoidal functors can\ud be equivalently described by a kind of weak morphisms introduced by B.\ud Noohi under the name of butterflies. In order to internalize monoidal\ud functors in a wide context, we introduce the notion of internal\ud butterflies between internal crossed modules in a semi-abelian category\ud C, and we show that they are morphisms of a bicategory B(C). Our main\ud result states that, when in C the notions of Huq commutator and Smith\ud commutator coincide, then the bicategory B(C) of internal butterflies is\ud the bicategory of fractions of Grpd(C) with respect to weak\ud equivalences. (C) 2013 Elsevier Inc. All rights reserved

Peiffer product and Peiffer commutator for internal pre-crossed modules

“…if f and g in PXMod B (C) are such that f · σ = g · σ, then f = g. Proof. This is a pre-crossed module version of a standard fact about internal reflexive graphs and fully faithful morphisms between them (see [1] for the crossed module case). We just recall here the construction of the induced action: it is the (unique) arrow ξ making the diagram below commute.…”

Fibred-categorical obstruction theory

“…Under this biequivalence, 2-cells of crossed modules correspond to internal natural transformations (in fact, natural isomorphisms). The biequivalence holds true for crossed modules internal to many other algebraic settings (see [1] for the semi-abelian case). The construction of the groupoid associated with a given crossed module (and vice versa) can be easily found in the literature (see [24] for the original source, and also Proposition 2.5 in [1]).…”

Bipullbacks of fractions and the snail lemma

“…where the coherence axioms can be deduced from the definition of δ and the commutativity of (1). It remains to prove the uniqueness of such a 2-cell ϕ. Suppose…”

“…The class of weak equivalences in Grpd(A) has a right calculus of fractions (in the bicategorical sense) [19]. The bicategory of fractions of Grpd(A) with respect to this class of weak equivalences has been described in [15] (see also [1] if A is semi-abelian, [12] if A is monadic, [2] if A has enough regular projective objects, and [18] for a description in terms of anafunctors). The objects are internal groupoids, and the arrows are particular profunctors called fractors: a fractor E : A B is a diagram of the form σ is a regular epimorphism, and R[σ] is its kernel pair;…”