Cloud computing platforms have created the possibility for computationally limited users to delegate demanding tasks strong but untrusted servers. Verifiable algorithms help build trust in such interactions by enabling server provide a proof of correctness his results which user can check very efficiently. In this article, we present doubly-efficient interactive algorithm verifiable polynomial ...

A cosymplectic groupoid is a Lie with multiplicative structure. We provide several structural results for groupoids and we discuss the relationship between groupoids, Poisson of corank 1, oversymplectic 1.

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual sense, but with respect to a discounted distance measure that contracts arbitrarily long sequences of edges within the same sub-groupoid (coset) and only co...

X iv :0 80 3. 04 72 v1 [ m at h. R A ] 4 M ar 2 00 8 On decomposition of commutative Moufang groupoids B.V.Novikov It is well-known that the multiplicative groupoid of an alternative/Jordan algebra satisfies Moufang identities [1, 4]. Therefore it seems interesting to study the structure of such groupoids. In this note we apply to Moufang groupoids an approach which is widespread in Semigroup T...

Morphisms from H to G in the Burnside category of groups are provided by sets with commuting right and left actions of H and G, which are free and finite over G. We extend this construction to groupoids by considering a G-action on a set X as a homomorphism G → Aut(X). This has an alternative expression in terms of “bi-actions,” a variant of the discrete version of a topological construction wh...

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We extend the construction of the tangent pro-longation of an abstract Lie algebroid to show that the Lie algebroid structure of any LA-groupoid may be prolonged to the Lie algebroid of its groupoid structure. In the case of a double groupoid, this prolonged structure for ei...

Double categories were introduced by Ehresmann [Ehr65] as an example of structured category. Examples of double groupoids were shown to arise from crossed modules in [BS76b, BS76a] and this result was applied in [BH78] to give a 2-dimensional version of the Van Kampen Theorem for the fundamental group, namely a colimit theorem for the fundamental crossed module of a based pair. These results we...

Building on Hrushovski’s work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski’s notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uni...

In this work we give a generalization of matched pairs of (finite) groups to describe a general class of depth two inclusions of factor von Neumann algebras and the C*-quantum groupoids associated with, using double groupoids. Date: Preliminary version of the 03/27/07. 1991 Mathematics Subject Classification. (2000) 46L37, 20L05, 20G42, 20 99.

We outline the main features of the definitions and applications of crossed complexes and cubical ω-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types in non simply connected situations.