We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting “nicely” on a tree gives rise to a self-covering of a topological groupoid, and how the group can be reconstructed from the groupoid and its covering. The connection is via finite-state automata. These define decomposition rules, or self-similar tilin...

Abstract. In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of squa...

Every AF-algebra A arises as the C *-algebra of a locally finite pointed directed graph in the sense of Kumjian, Pask, Raeburn, and Renault. For AF-algebras, the diagonal subalgebra defined by Strˇatilˇa and Voiculescu is consistent with Kumjian's notion of diagonal, and the groupoid arising from a well-chosen Bratteli diagram for A coincides with Kumjian's twist groupoid constructed from a dia...

The groupoid interpretation of dependent type theory given by Hofmann and Streicher associates to each closed type a category whose objects represent the elements of that type and whose maps represent proofs of equality of elements. The categorial structure ensures that equality is reflexive (identity maps) and transitive (closure under composition); the groupoid structure, which demands that e...

We describe the point and contact equivalence groupoids of an important class two-dimensional quasilinear hyperbolic equations. In particular, we prove that this is normalized in usual sense with respect to transformations, its groupoid generated by first-order prolongation groupoid, vertex group wave equation a family admissible transformations between trivially Darboux-integrable

The notion of recognition of a language by a finite semigroup can be generalized to recognition by finite groupoids, i.e. sets equipped with a binary operation ‘ · ’ which is not necessarily associative. It is well known that L can be recognized by a groupoid iff L is context-free. But it is also known that some subclasses of groupoids can only recognize regular languages. For example, loops re...

We will show that deformation quantization in skein theory of oriented 3-manifolds is induced from a topological deformation quantization of the fundamental 2-groupoid of the space of immersions of circles in M . The structure of skein module and its relations with string topology homomorphisms appear through representations of the groupoid structure into the set the objects. The deformation of...

The main objective of this work is to study mathematical properties of computational paths. Originally proposed by de Queiroz & Gabbay (1994) as ‘sequences of rewrites’, computational paths can be seen as the grounds on which the propositional equality between two computational objects stand. Using computational paths and categorical semantics, we take any type A of type theory and construct a ...