Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until M. Stronkowski and D. Stanovský’s recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes t...

Dependent type theory associates to any elements x, y of a type A an identity type x =A y, the type of proofs of equality of these elements. In PER Martin-Löf’s extensional type theory, the identity type is a subsingleton inhabited precisely when x and y are judgmentally equal. Semantically, types are thus sets equipped with equivalence relations given by their identity types. Groupoid Martin-L...

We give a simple combinatorial model for plethysm. Precisely, the bialgebra dual to plethystic substitution is realised as homotopy cardinality of incidence an explicit simplicial groupoid, obtained from surjections by construction reminiscent Waldhausen S and Quillen Q-construction.

This unpublished note contains some materials taken from my old study note on groupoids and small categories ([1]). It contains a proof for the fact that a groupoid is any group bundle over an equivalence relation. Moreover, the action of a category G on a category H as well as the resulting semi-direct product category H ×α G will be defined (when either G is a groupoid or H (0) = G). If both ...

We refine the concept of the symmetry group of a geometric object through its symmetry groupoid, which incorporates both global and local symmetries in a common framework. The symmetry groupoid is related to the weighted differential invariant signature of a submanifold, that is introduced to capture its fine grain equivalence and symmetry properties. Applications to the recognition and symmetr...

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [19] of P. Resende. As a motivating example, we describe the connection between the quantale and the C∗-algebra that both classify Penrose tilings, which was left as an o...

We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlister’s P–theorem for incompressible ordered groupoids. AMS 2000 Mathematics sub...

It is well known that a measured groupoid G defines a von Neu-mann algebra W * (G), and that a Lie groupoid G canonically defines both a C *-algebra C * (G) and a Poisson manifold A * (G). We show that the maps (G) are functorial with respect to suitable categories. In these categories Morita equivalence is isomorphism of objects, so that these maps preserve Morita equivalence.

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of CuntzKrieger type. Here we develop a theory of the fundamental groupoid of a kgraph, and relate it to the fundamental groupoid of an associated graph called the 1-skeleton. We also explore the failure, in general, of k-graphs to faithfully embed into their fu...