We study when the stable category A of an abelian category A modulo a full additive subcategory T is balanced and, in case T is functorially finite in A, we study a weak version of balance for A . Precise necessary and sufficient conditions are given in case T is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (gen...

Suppose given a Frobenius category E , i.e. an exact category with a big enough subcategory B of bijectives. Let E := E/B denote its classical stable category. For example, we may take E to be the category of complexes C(A) with entries in an additive category A, in which case E is the homotopy category of complexes K(A). Suppose given a finite poset D that satisfies the combinatorial condition...

Suppose given a Frobenius category E , i.e. an exact category with a big enough subcategory B of bijectives. Let E := E/B denote its classical homotopy category. For example, we may take E to be the category of complexes C(A) with entries in an additive category A, in which case E is the homotopy category of complexes K(A). Suppose given a finite poset D that satisfies the combinatorial conditi...

respectively. We refer to one of these as the parent category, with R or C as the indexing gadget. From a categorical perspective the different kinds of parent categories have a lot in common. We could say that Mod -R is just an additive version of Ĉ . That is a bit too glib, but the analysis of the two kinds of categories is quite similar. For each parent category there is a notion of a locali...

In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a Q-linear additive Jacobian functor N 7→ J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd ...

We prove a full completeness theorem for multiplicative-additive linear logic (i.e. MALL) using a double gluing construction applied to Ehrhard’s ∗-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free MALL...

We give a classification of substructures (= closed subbifunctors) given skeletally small extriangulated category by using the defects, in similar way to author’s exact structures additive category. More precisely, for an category, possible are bijection with Serre subcategories abelian consisting defects conflations. As byproduct, we prove that poset on it is isomorphic some

The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Ou...

We de…ne combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey prope...

We study a variation of Turaev’s homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space X and define an SX -structure to be a monoidal 2-functor from this to the 2-category of idempotent-complete additive k-linear categories. We initiate the study of the algebraic structure arising from these functors. In particular we show that, und...