Skew structures in 2-category theory and homotopy theory

Homotopy Coherent Category Theory

Homotopy Theory and TDA with a View Towards Category Theory

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...

Hom complexes and homotopy theory in the category of graphs

We investigate a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph ×homotopy is characterized by the topological properties of the Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; Hom complexes were introduced by Lovász and further studied ...

Abstract Homotopy Theory: The Interaction of Category Theory and Homotopy theory A revised version of the 2001 article

Homotopy Theory: The Interaction of Category Theory and Homotopy theory A revised version of the 2001 article

Structures in Higher-Dimensional Category Theory

This is an exposition of some of the constructions which have arisen in higherdimensional category theory. We start with a review of the general theory of operads and multicategories (cf. [Bur], [Her], [Lei1]). Using this we give a new definition of n-category (a variation on Batanin’s); we also give an informal definition in pictures. Next we discuss Gray-categories and their place in coherenc...