tailieunhanh - Braided regular crossed modules bifibered over regular groupoids

We show that the forgetful functor from the category of braided regular crossed modules to the category of regular (or whiskered) groupoids is a fibration and also a cofibration. | Turk J Math (2017) 41: 1385 – 1403 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Braided regular crossed modules bifibered over regular groupoids 1 Alper ODABAS ¸ 1 , Erdal ULUALAN2,∗ Eski¸sehir Osmangazi University, Faculty of Science and Art, Department of Mathematics and Computer Science Eski¸sehir, Turkey 2 Dumlupınar University, Faculty of Science and Art, Department of Mathematics, K¨ utahya, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: We show that the forgetful functor from the category of braided regular crossed modules to the category of regular (or whiskered) groupoids is a fibration and also a cofibration. Key words: Crossed modules, groupoids, fibration, cofibration of categories 1. Introduction Crossed modules of groups were introduced by Whitehead in [14] to study relative homotopy groups as models for homotopy (connected) 2-types. If f : G → H is a homomorphism of groups, then there is a pullback or reindexing functor f ∗ : CM/H → CM/G , where CM/G is the category of crossed G -modules. The left adjoint f∗ to this functor f ∗ was constructed in [5]. Thus, Whitehead’s crossed modules fibered and cofibered over groups. Analogous constructions in the crossed modules category in commutative algebras and Lie algebras were given in [13] and [8], respectively. For further accounts of fibered and cofibered categories and an introduction to their literature, see [10, 11] and the references therein. Brown and Sivera [6] showed that the forgetful functor Φ1 : XMod → Gpd from the category of crossed modules of groupoids to the category of groupoids that sends a crossed module M → P to its base groupoid P is a fibration and a cofibration of categories. That is, crossed modules of groupoids bifibered over groupoids. If we consider the category of braided regular crossed modules (cf. [4]) as BRCM instead of XMod