We study and give examples of braided groupoids, and a fortiori, non-degenerate solutions of the quiver-theoretical braid equation.

The Pi family of reversible programming languages for boolean circuits is presented as a syntax combinators witnessing type isomorphisms algebraic data types. In this paper, we give denotational semantics language, using weak groupoids à la Homotopy Type Theory, and show how to derive an equational theory it, by 2-combinators equivalences isomorphisms. We establish correspondence between the sy...

In this work we deal with actions of vector groupoid which is a new concept in the literature‎. ‎After we give the definition of the action of a vector groupoid on a vector space‎, ‎we obtain some results related to actions of vector groupoids‎. ‎We also apply some characterizations of the category and groupoid theory to vector groupoids‎. ‎As the second part of the work‎, ‎we define the notion...

We show how to recover a discrete twist over an ample Hausdorff groupoid from pair consisting of algebra and what we call quasi-Cartan subalgebra. identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), prove assignment twisted Steinberg algebras such our construction are mutually inverse. algebraic pairs correspond effective groupoids prin...

Motivated by properties of higher tangent lifts geometric structures, we introduce concepts weighted structures for various objects on a manifold F equipped with homogeneity structure. The latter is smooth action the monoid $$(\mathbb {R},\cdot )$$ multiplicative reals. Vector bundles are particular cases and them call $$\mathrm{V\!B}$$ -structures. In case Lie algebroids groupoids, include -al...

The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical ω-groupoids with connections where the initial definition of the tensor product li...

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product...

Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory forétale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homolog...