Eilenberg and Mac Lane developed or discovered category theory in the context of algebraic topology; the very utility of topological invariants such as homology, co-homology, and the homotopy groups is contingent on their functoriality. These particular invariants are moreover homotopy invariants. This property can be rephrased as those invariants factoring through the homotopy category, where homotopic maps are identified. We try and reverse this observation. Can we, starting with a functor, interpret it as a homotopy invariant by finding the right notion of homotopy? What data do we need to do so, and how universal are these constructions?
In part one we will motivate and define Quillen's concept of a model category, and in part two we will outline how such a structure begets a homotopy category.
In lieu of pizza, it is suggested that one attend the department picnic.
From Invariants, to functors, to homotopy, to model categories, and back to homotopy: part 1