This paper establishes a relation between the notion of a codifferential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad T , which is furthermore an algebra modality, i.e. a natural assignment of an associative algebra structure to each object of the form T (C). Finally, a codi...

We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finite-dimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Ta...

In this paper we prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in X are equivalent to chain complexes in X ; (b) the observation of...

BACKGROUND Regression models for survival data have traditionally been based on the Cox regression model. However, its validity relies heavily on assumption of proportional hazards. Another restriction of the Cox model is insufficiency in dealing with time-varying covariate effects, since the regression coefficients are assumed constant. These weaknesses have generated interest in alternative a...

We review some properties of finitely accessible categories related to approximations, and we analyze the relationship between approximations in a finitely accessible additive category and in its associated generalized module category.

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental additive functor holds for all additive functors, like K-theory, cyclic homology, topological Hochschild homology, etc. Finally, we reduce this fundamental isomorphism conjecture to K-theoretic ones.

For any finite group G, we define a bifunctor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

The article is devoted to extension groups in the category of functors from a small category to an additive category with an Abelian structure in the sense of Heller. It is proved that under additional assumptions there exists a spectral sequence which converges to the extension groups.

A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from hom-sets to homsemimodules) are functorial, and use them to make precise the intuition that “compact objects a...