This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view. In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories wit...

This is the first of a short series of papers intended to provide one common semantics for several different types of specification language, in order to allow comparison and translations. The underlying idea is that a specification describes the behaviour of a system, depending on parameters. We can represent this behaviour as a functor which acts on structures representing the parameters, and...

We survey some uses of adjoint functors in graph theory pertaining to colourings, complexity reductions, multiplicativity, circular colourings and tree duality. The exposition of these applications through adjoint functors unifies the presentation to some extent, and also raises interesting questions.

In [CY98] Crane and Yetter introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter in [Yet98] is a proper generalization of Gerstenhaber’s deformation theory for associative algebras [Ger63, Ger64, GS88]. In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that u...

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A∞-functor Y : A op → A∞(A,C) is a full embedding for an arbitrary unital A∞-category A. Since A∞-algebras were introduced by Stasheff [Sta63, II] there existed a possibility to consider A∞-generalizations of categories. It did not happen until A∞-categories were encountered in studies of mirror...

This is a second paper on quotients of Hom-functors and their applications to the representation theory of nite general linear groups in non-describing characteristic. After some general result on quotients of Hom-functors and their connection to Harish-Chandra theory these contructions are used to obtain a full classiication of thè-modular irreducible representations of GL n (q) for some prime...

of “Rational Homotopy Calculus of Functors” by Ben C. Walter, Ph.D., Brown University, July 2005. We construct a homotopy calculus of functors in the sense of Goodwillie for the categories of rational homotopy theory. More precisely, given a homotopy functor between any of the categories of differential graded vector spaces (DG), reduced differential graded vector spaces, differential graded Li...