Structure and classification of monoidal groupoids

Topology and its Applications, 2020

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.

2005

By regarding the classical non abelian cohomology of groups from a 2dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids and, cocycles are lax 2-functors and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand group extensions are fibrations of categories. Furthermore, n-simplices in the nerve of a 2category are lax 2-functors.

Applied general topology, 2021

The aim of this paper is to obtain a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of an internal category in the category of group-groupoids. Corresponding group2-groupoids, we obtain some categorical structures related to crossed modules and group-groupoids and prove categorical equivalences between them. These results enable us to obtain 2-dimensional notions of group-groupoids. 2010 MSC: 20L05; 18D05; 18D35; 20J15.

Operads: Proceedings of Renaissance Conferences, 1997

Basic examples of monoidal categories are the following: Example 1.1. Let C be any category with nite products. Then these products may be used to give it a monoidal structure C = (C; ; 1; a; l; r), where is the binary product, 1 is the terminal object (which exists as the empty product), and a, l, r are uniquely determined by the universal property of the products. A monoid in this monoidal category is what is usually called an internal monoid in a category with products. Also in this \cartesian" situation one may de ne what it means for a monoid G = (G; : G G ! G; : 1 ! G) to be an internal group object: there must exist an endomorphism : G ! G satisfying (G)d = p = (G)d where d : X ! X X and p : X ! 1 are the canonical morphisms (which are only available in the cartesian case). In particular, taking C to be the category Ens of sets and functions, one obtains just monoids and groups in the ordinary sense; or, taking the categories of spaces, simplicial sets, etc., one obtains topological or simplicial monoids and groups. Example 1.2. The category R-mod of modules over a commutative ring R may be given a monoidal structure using the tensor product over R. We shall denote this monoidal category

Journal of the Australian Mathematical Society, 1989

AbsractIn this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

2011

Semigroup Forum, 2015

We interpret Grillet's symmetric third cohomology classes of commutative monoids in terms of strictly symmetric monoidal abelian groupoids. We state and prove a classification result that generalizes the well-known one for strictly commutative Picard categories by Deligne, Fröhlich and Wall, and Sinh.

Maltepe Üniversitesi, 2019

In this extended abstract we state the topological version of categorical equivalence between internal groupoids and crossed modules in the category of groups with operations; and explain how to develop crossed module aspect of monodromy groupoids for topological internal groupoids.

In this extended abstract considering the topological version of categorical equivalence of crossed modules and group-groupoids we develop crossed module aspects of monodromy group-groupoids for topological group-groupoids and give some examples for monodromy groupoids.

1998

Categories are known to be useful for organizing structural aspects of mathematics. However, they are also useful in finding out what structure can be dismissed (coherence theorems) and hence in aiding calculations. We want to illustrate this for finite set theory, linear algebra, and group representation theory. We begin with some combinatorial set theory. Let N denote the set of natural numbers. We identify each n⁄⁄Œ⁄⁄N with the finite set n = { j⁄⁄Œ⁄⁄N: 0 £ j < n}. However, we must be careful to distinguish the cartesian product m ⁄⁄ ¥ ⁄⁄n = { (i⁄⁄,⁄⁄j) : 0 £ i < m, 0 £ j < n} from the isomorphic set mn. Let S denote the skeletal category of finite sets; explicitly, the objects are the n⁄⁄Œ⁄⁄N and the morphisms are the functions between these sets. We need to discuss the explicit construction of finite products in S⁄⁄. Let p 0 m n, : mn aAm and p1 m n, : mn aAn be the functions given by p 0 m n, (k) = i and p1 m n, (k) = j where k = i ⁄⁄n + j. That p 0 m n, and p1 m n, a...