Matthew Burke

Title: Elements of the Theory of Quasi-categories
Abstract: We outline some of the theory of quasi-categories that is required to set up the functor calculus. First we review the definition of left, right and inner factorisation systems and describe an alternative characterisation of these factorisation systems that makes certain computations more straightforward. Then we define quasi-categories and prove that the internal hom of quasi-categories is a quasi-category. In order to define a tangent bundle functor we first need to define the (large) quasi-category of quasi-categories and recall how the representable (infinity) functors are defined in this setting. If we have time we describe how to use this work to define the functor of excisive functors that conjecturally constitutes the tangent bundle functor.