Let M be a torus bundle, i.e. an (n + 1)-dimensional manifold, which fibers over the circle with the fiber an n-dimensional torus. A covariant functor from M to the Cuntz-Krieger algebra (a C∗-algebra) is constructed. The functor maps homeomorphic torus bundles to the stably isomorphic Cuntz-Krieger algebras. It is shown that in general the K-theory of the Cuntz-Krieger algebra counts the torsi...

For an arbitrary-type functor F, the notion of split coalgebras, that is, coalgebras for which the canonical projections onto the simple factor split, generalizes the well-known notion of simple coalgebras. In case F weakly preserves kernels, the passage from a coalgebra to its simple factor is functorial. This is the simplification functor. It is left adjoint to the inclusion of the subcategor...

The purpose of these notes is to define an equivalence between the natural homology theories associated to operads and the homology of functors over certain categories of operators (PROPs) related to operads.

In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched.

We give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [H. Krause, Approximations and adjoints in homotopy categories, Math. Ann. 353 (2012), 765–781]. Then we provide a criterion for the existence of r...

We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra α : X → FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states. A first crucial step is the lifting of the functor F on Set to a functor F in the category PMet of pseudometric spaces. We present two differen...

Let k be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian p-adic Lie extensions E/F , where F is a local field with residue field k, and a category whose objects are pairs (K,A), where K ∼= k((T )) and A is an abelian p-adic Lie subgroup of Autk(K). In this paper we extend this equivalence to allow Gal(E/F ...

This text is a preliminary version of material used for a course at the University of Copenhagen, part of ”Workshop and Masterclass on Homological stability” August 26-30, 2013. This lecture concerns the method developped, in collaboration with Aurélien Djament, in [DV10] and used in [Dja12] and [DVa] to compute the stable homology of a family of groups with coefficients given by a polynomial f...