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...

One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K : A → B between categories is Frobenius if there exists a functor G : B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the...