Aspherical abelian groupoids and their directions (Q5960401)

Let \(\mathcal E\) be a left exact category. The category of internal groupoids in \(\mathcal E\) is denoted by \({\mathcal G}rd {\mathcal E}\) and the object of objects functor by \(()_0 : {\mathcal G}rd{\mathcal E}\to{\mathcal E}\). It is a fibration whose fibres \({\mathcal G}rd_X{\mathcal E}\) are protomodular categories. The notion of protomodular categories had been introduced by the author in a previous paper. The author investigates what are the normal subobjects and the abelian objects in these protomodular fibres.NEWLINENEWLINENEWLINEA groupoid is proved to be abelian when the locally defined Mal'cev operation on triples of parallel arrows defined by \(p(f,g,h) = hg^{-1}f\) is functorial, or when its groups of endomorphisms are abelian. Let \({\mathcal E}\) be, moreover, Barr exact. The category of autonomous Mal'cev operations on objects of global support in \(\mathcal E\) is denoted by \({\mathcal A}ut{\mathcal M}{\mathcal E}g\) and the category of abelian internal groups in \(\mathcal E\) by \({\mathcal A}b{\mathcal E}\). NEWLINENEWLINENEWLINEA direction functor \(d: {\mathcal A}ut{\mathcal M}{\mathcal E}g\to {\mathcal A}b{\mathcal E}\) is defined and proved to be a cofibration whose fibres \(d^{-1}A\) have closed monoidal structures, and are equivalents to the category of \(A\)-torsors. Then the first cohomology group \(H^1({\mathcal E},A)\) is the set of connected components of \(d^{-1}A\). A groupoid in \(\mathcal E\) is said to be aspherical if it is connected with a global support object of objects. Whence a direction functor \(d_1 : {\mathcal A}b{\mathcal G}rd{\mathcal E}g\to {\mathcal A}b{\mathcal E}\) from the category of aspherical abelian groupoids in \(\mathcal E\) to \({\mathcal A}b{\mathcal E}\) exists. The set of connected components of the fibre \(d^{-1}_1A\) is claimed to be the second cohomology group \(H^2({\mathcal E},A)\).