Fibred-categorical obstruction theory

Obstruction Theory for Extensions of Categorical Groups

For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H) → H → Aut(H)→ Out(H)→ 1 is raised to a categorical group level by using a suitable notion of exactness. Breen’s Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane’s obstructi...

Categorical homotopy theory

Categorical Homotopy Theory

This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves wi...

Ergodic Theory for Markov Fibred Systems and Parabolic Rational Maps

A parabolic rational map of the Riemann sphere admits a nonatomic /¡-conformai measure on its Julia set where h = the HausdorfT dimension of the Julia set and satisfies 1/2 < h < 2 . With respect to this measure the rational map is conservative, exact and there is an equivalent cr-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case...

Fibred and Indexed Categories for Abstract Model Theory

Indexed and Fibred category theory have a long tradition in computer science as a language to formalize different presentations of the notion of a logic, as for instance, in the theory of institutions and general logics, and as unifying models of (categorical) logic and type theory as well. Here we introduce the notions of indexed and fibred frames and construct a rich mathematical workspace wh...