nLab localization of model categories



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


(,1)(\infty,1)-Category theory



In the way that model categories serve as presentations for their homotopy categories or more generally of their associated \infty -categories so the general notion of localization of a model category [Hirschhorn (2002), Def. 3.1.1] serves to invert morphisms in the homotopy category such as to present localization of that \infty-category.

To quote Hirschhorn (2002), p. xi:

‘Localizing a model category with respect to a class of maps does not mean making the maps into isomorphisms; instead, it means making the images of those maps in the homotopy category into isomorphisms. Since the image of a map in the homotopy category is an isomorphism if and only if the map is a weak equivalence, localizing a model category with respect to a class of maps means making maps into weak equivalences.’

In practice, the most prominent notion of localization of model categories is Bousfield localization of model categories (see there for more), which is a special case, by Hirschhorn (2002), Thm. 3.3.19.



The main account on the general notion of localization of model categories is:

[p. 47:] In Section 3.1 we define left and right localizations of model categories, and explain the connection between left localizations, local objects, and local equivalences […] in Section 3.3 we discuss (left and right) Bousfield localizations of categories, a special case of left and right localizations (see Theorem 3.3.19).

For more references dedicated to the special case of Bousfield localization of model categories see there.

Last revised on April 22, 2023 at 11:01:33. See the history of this page for a list of all contributions to it.