Quantum groupoids are a joint generalization of groupoids and quantum groups. We propose a definition of a compact quantum groupoid that is based on the theory of C *-algebras and Hilbert bimodules. The essential point is that whenever one has a tensor product over C in the theory of quantum groups, one now uses a certain tensor product over the base algebra of the quantum groupoid.

In this paper we use Quillen's model structure given by Dwyer-Kan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicialidentities and give a sim...

We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered R-categories, thus answering questions of what might be ‘commutative versions’ of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applic...

Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere’s functorial notion of algebraic theory and Ehresmann’s functorial notion of theory called sketc...

We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In [7] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegene...

Hofmann and Streicher showed that there is a model of the intensional form of Martin-Löf’s type theory obtained by interpreting closed types as groupoids. We show that there is also a model when closed types are interpreted as strict ω-groupoids. The nonderivability of various truncation and uniqueness principles in intensional type theory is then an immediate consequence. In the process of con...

In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if [Formula: see text] is a groupoid and if [Formula: see text] is a function [Formula: see text], then [Formula: see text] is a left-twisted semigroup with respect to [Formula: see text] if for all [Formula: see text...

Any semi-abelian category A appears, via the discrete functor, as a full replete reflective subcategory of the semi-abelian category of internal groupoids in A. This allows one to study the homology of n-fold internal groupoids with coefficients in a semi-abelian category A, and to compute explicit higher Hopf formulae. The crucial concept making such computations possible is the notion of prot...

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the of integrable Lie algebroids and complete Lie algebroid comorphisms to the of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of com...

We present Hausdorff versions for Lie Integration Theorems 1 and 2 apply them to study symplectic groupoids arising from Poisson manifolds. To prepare these results we include a discussion on equivalences propose an algebraic approach holonomy. also subsidiary results, such as generalization of the integration subalgebroids non-wide case, explore in detail case foliation groupoids.