TAILIEUCHUNG - Braiding for internal categories in the category of whiskered groupoids and simplicial groups

In this work, we define the notion of ‘braiding’ for an internal groupoid in the category of whiskered groupoids and we give a relation between this structure and simplicial groups by using higher order Peiffer elements in the Moore complex of a simplicial group. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 145 – 164 ¨ ITAK ˙ c TUB doi: Braiding for internal categories in the category of whiskered groupoids and simplicial groups Erdal ULUALAN∗, Sedat PAK Dumlupınar University, Science Faculty, Department of Mathematics, K¨ utahya, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: In this work, we define the notion of ‘braiding’ for an internal groupoid in the category of whiskered groupoids and we give a relation between this structure and simplicial groups by using higher order Peiffer elements in the Moore complex of a simplicial group. Key words: Simplicial groups, crossed modules, groupoids 1. Introduction Brown and Gilbert [12] have defined a braiding map for a regular crossed module over groupoids. They have proved that the category of braided regular crossed modules is equivalent to that of simplicial groups with Moore complex of length 2. Braided monoidal categories were defined by Joyal and Street in [21]. They have also defined crossed semi-modules for monoids with a bracket operation and given an equivalence between the category of braided monoidal categories and the category of crossed semi-modules with bracket operations. For further work about braided monoidal categories see also [8] and [22]. Categorical groups are monoidal groupoids in which every object is invertible, up to isomorphism, with respect to the tensor product (cf. Breen [10] and Joyal-Street [21, 20]). These structures sometimes are equipped with a braiding or a symmetry (cf. [9, 19, 21, 20]). Garzon and Miranda, [19], gave the relation between the category of categorical groups equipped with a braiding and the category of reduced 2-crossed modules by using Brown-Spencer theorem given in [14]. For these categorical notions see also [5, 6, 13]. In order to define the notion of commutativity for a

• Toán học
• Internal categories
• Simplicial groups
• Crossed modules
• Whiskered groupoids
• Peiffer elements