2 - Homotopical categories: from model categories to (∞, 1)-categories

Summary

A homotopical category is a category equipped with some collection of morphisms traditionally called “weak equivalences” that somewhat resemble isomorphisms but fail to be invertible in any reasonable sense, and might in fact not even be reversible: that is, the presence of a weak equivalence X → Y need not imply the presence of a weak equivalence Y → X. Frequently, the weak equivalences are defined as the class of morphisms in a category K that are “inverted by a functor” F : K → L, in the sense of being precisely those morphisms in K that are sent to isomorphisms in L. For instance:

• – Weak homotopy equivalences of spaces or spectra are those maps inverted by the homotopy group functors π_∗ : Top → GrSet or π_∗ : Spectra → GrAb.

• – Quasi-isomorphisms of chain complexes are those maps inverted by the homology functor H_∗ : Ch → GrAb.

• – Equivariant weak homotopy equivalences of G-spaces are those maps inverted by the homotopy functors on the fixed point subspaces for each compact subgroup of G.

The term used to describe the equivalence class represented by a topological space up to weak homotopy equivalence is a homotopy type. Since the weak homotopy equivalence relation is created by the functor π_∗ , a homotopy type can loosely be thought of as a collection of algebraic invariants of the space X, as encoded by the homotopy groups π_∗X. Homotopy types live in a category called the homotopy category of spaces, which is related to the classical category of spaces as follows: a genuine continuous function X → Y certainly represents a map (graded homomorphism) between homotopy types. But a weak homotopy equivalence of spaces, defining an isomorphism of homotopy types, should now be regarded as formally invertible.

In their 1967 manuscript Calculus of fractions and homotopy theory, Gabriel and Zisman [100] formalized the construction of what they call the category of fractions associated to any class of morphisms in any category together with an associated localization functor π : K → K[𝑊⁻¹] that is universal among functors with domain K that invert the class 𝑊 of weak equivalences. This construction and its universal property are presented in §2.2.