We prove that the folk model structure on strict ∞-categories transfers to the category of strict ∞-groupoids (and more generally to the category of strict (∞, n)categories), and that the resulting model structure on strict ∞-groupoids coincides with the one defined by Brown and Golasiński via crossed complexes.

We prove that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively.

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.

Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is...

We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of principal groupoids such a notion coincide with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids ...

This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2-functor Z : nCob2→2Vect, by analogy with the description of a TQFT as a functor Z : nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the Dijkgraaf-Witten model. To give this definition rigorously, we first define a bic...

A Lagrangian subspace L of a weak symplectic vector space is called split if it has an isotropic (hence Lagrangian) complement. When the structure strong, sufficient for to have closed complement, which can then be moved become isotropic. The purpose this note develop theory compositions and reductions canonical relations spaces. We give conditions on coisotropic C V imply that induced relation...

We deeply analyse the structural organisation of the fibration of points and of the monad of internal groupoids. From that we derive: 1) a new characterization of internal groupoids among reflexive graphs in the Mal’cev context; 2) a setting in which a Mal’cev category is necessarily a protomodular category.