The talk began with some philosophical remarks on the need to more publicly present unified views of higher dimensional categorical notions, i.e., to "present the mathematics of definitions" rather than just the definitions. By way of example the talk provided a unified treatment of, inter alia, functors, natural transformations, 2-natural transformations, lax natural transformations, modifications, and lax modifications.
We explored the slogan "View category theory as directed, globular, homologiical algebra plus functoriality". It turns out that using just three concepts – homotopy of maps of complexes (of arbitrary degree), functoriality of chain maps, and the mapping cone of chain maps – we recover/discover the details of all of the notions mentioned above, and obtain proposals for the definitions of their higher dimensional analogues.
(This is joint work with Richard Wood.)