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...

We prove in this paper that the genus zero data of a modular functor determines the modular functor. We do this by establishing that the S-matrix in genus onewith one point labeled arbitrarily can be expressed in terms of the genus zero information and we give an explicit formula. We do not assume the modular functor in question has duality or is unitary, in order to establish this. CONTENTS

We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our treatment of the problem is v...

Brian Day Received November 7, 19~9 The purpose of the present paper is to develop in further detail the remarks, concerning the relationship of Kan functor extensions to closed structures on functor categories, made in "Enriched functor categories" | 1] §9. It is assumed that the reader is familiar with the basic results of closed category theory, including the representation theorem. Apart fr...

Looking at the operation of forming neat α-reducts as a functor, with α an infinite ordinal, we investigate when such a functor obtained by truncating ω dimensions, has a right adjoint. We show that the neat reduct functor for representable cylindric algebras does not have a right adjoint, while that of polyadic algebras is an equivalence. We relate this categorical result to several amalgamati...

In this research announcement, we propose a new interpretation of the Engle Pereira Rovelli (EPR) quantization of the Barrett-Crane (BC) model using a functor we call the time functor, which is the first example of a co-lax, amply renormalizable (claren) functor. Under the hypothesis that the universe is in the Kodama state, we construct a holographic version of the model. Generalisations to ot...

A completion functor is constructed on the category of completely normal Cauchy groups and Cauchy-continuous homomorphlsms. A competlon functor is also obtained for a corresponding category of convergence groups.

We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined ...

We construct the polynomial induction functor, which is right adjoint to restriction functor from category of representations a general linear group its Weyl group. This construction leads representation-theoretic proof Littlewood’s plethystic formula for multiplicity an irreducible representation symmetric in such restriction. The unimodality certain bipartite partition functions follows.

We prove that the united K-theory functor is a surjective functor from the category of real simple purely infinite C*-algebras to the category of countable acyclic CRT-modules.