Let C be a category with finite colimits, and let (E ,M) be a factorisation system on C with M stable under pushouts. Writing C;M for the symmetric monoidal category with morphisms cospans of the form c → m ←, where c ∈ C and m ∈ M, we give method for constructing a category from a symmetric lax monoidal functor F : (C;M,+) → (Set,×). A morphism in this category, termed a decorated corelation, ...

We define A∞-bimodules similarly to Tradler and show that this notion is equivalent to an A∞-functor with two arguments which takes values in the differential graded category of complexes of k-modules, where k is a ground commutative ring. Serre A∞-functors are defined via A∞-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A∞-category A over a field k adm...

We define A∞-bimodules similarly to Tradler and show that this notion is equivalent to an A∞-functor with two arguments which takes values in the differential graded category of complexes of k-modules, where k is a ground commutative ring. Serre A∞-functors are defined via A∞-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A∞-category A over a field k adm...

The category is introduced as an ordered 5-tuple of the form 〈O,M,dom,cod, ·, id〉 where O (objects) and M (morphisms) are arbitrary nonempty sets, dom and cod map M onto O and assign to a morphism domain and codomain, · is a partial binary map from M×M to M (composition of morphisms), id applied to an object yields the identity morphism. We define the basic notions of the category theory such a...

This thesis presents several module systems, in particular Mei and DMei, designed for mechanized mathematics systems. Mei is a λ-calculus style module system that supports higher-order functors in a natural way. The semantics of functor application is based on substitution. A novel coercion mechanism integrates a parameter passing mechanism based on theory interpretations with simple λ-calculus...

We study modular subspaces corresponding to two deformation functors associated to an isolated singularity X0: the functor DefX0 of deformations of X0 and the functor Def X0 of deformations with section of X0. After recalling some standard facts on the cotangent cohomology of analytic algebras and the general theory of deformations with section, we give several criteria for modularity in terms ...

We study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a comparison theorem for determinant functors on a triangulated category with a non-degenerate bounded t-structure and determinant functors on its heart. For a small t...

We define a functor which gives the “global rank of a quiver representation” and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other “rank functors” for a quiver Q, which induce ring homomorphisms (called “rank functions”) from the representation ring of Q to Z. These rank functions give discrete numerical invariant...

Coverings in the representation theory of algebras were introduced for the Auslander-Reiten quiver of a representation finite algebra in [15] and later for finite dimensional algebras in [2, 7, 11]. The best understood class of covering functors is that of Galois covering functors F : A → B determined by the action of a group of automorphisms of A. In this work we introduce the balanced coverin...