Let I be a small indexing category, G : I → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (re...

Let I be a small indexing category, G : I → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (re...

Given groupoids G and H as well as an isomorphism Ψ : Sd G∼= Sd Hbetween subdivisions, we construct an isomorphism P : G∼= H . If Ψ equals SdF forsome functor F , then the constructed isomorphism P is equal to F . It follows thatthe restriction of Sd to the category of groupoids is conservative. These results donot hold for arbitrary categories.

A very simple proof of the finite embeddability property for residuated distributivelattice-ordered groupoids and some related classes of algebras is presented. In particular, this gives an answer to the question posed in [3, Problem 4.2]. The presented construction allows for improvement of the upper bound on the complexity of the decision procedure for the universal theory of residuated distr...

Mimicking a recent article of Stefaan Vaes, in which was proved that every locally compact quantum group can act outerly, we prove that we get the same result for measured quantum groupoids, with an appropriate definition of outer actions of measured quantum groupoids. This result is used to show that every measured quantum groupoid can be found from some depth 2 inclusion of von Neumann algebras.

The notion of geometric nerve of a 2-category (Street, [18]) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding sim-plicial maps. These facts allow us to prove a representation theorem of the general non abelian cohomolo...

We study isomorphism classes of symplectic dual pairs P ← S → P , where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P , these Morita self-equivalences of P form a group Pic(P ) under a natural “tensor product” operation. Variants of this construction are also studied, fo...

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...