# Model categories and their localizations

@inproceedings{Hirschhorn2003ModelCA, title={Model categories and their localizations}, author={Philip S. Hirschhorn}, year={2003} }

Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left Bousfield localizations Existence of right Bousfield localizations Fiberwise localization Homotopy theory in model categories: Summary of part 2 Model categories Fibrant and cofibrant approximations Simplicial model categories Ordinals, cardinals, and transfinite composition Cofibrantly generated model categories… Expand

#### 1,365 Citations

Calculus of functors and model categories

- Mathematics
- 2006

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [5]. In this paper we construct various localizations of the projective model… Expand

On equivariant homotopy theory for model categories

- Mathematics
- 2016

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's… Expand

Simplicial model category structures on the category of chain functors

- Mathematics
- 2007

The model structure on the category of chain functors Ch, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in… Expand

Homotopy Limit Functors on Model Categories and Homotopical Categories

- Mathematics
- 2005

Model categories: An overview Model categories and their homotopy categories Quillen functors Homotopical cocompleteness and completeness of model categories Homotopical categories: Summary of part… Expand

Maps and localizations in the category of Segal spaces

- Mathematics
- 2005

The category of Segal spaces was proposed by Charles Rezk in 2000 as a suitable candidate for a model category for homotopy theories. We show that Quillen functors induce morphisms in this category… Expand

A characterization of simplicial localization functors and a discussion of DK equivalences

- Mathematics
- 2012

Abstract In a previous paper, we lifted Charles Rezk’s complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of “relative categories”. Here,… Expand

THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES

- Mathematics
- 2005

Abstract Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of… Expand

Covariant Model Structures and Simplicial Localization

- Mathematics
- 2015

In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model… Expand

Simplicial model structures on pro-categories

- Mathematics
- 2020

We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our… Expand

Calculus of functors and model categories, II

- Mathematics
- 2007

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large… Expand

#### References

SHOWING 1-7 OF 7 REFERENCES

Acyclic Models and Triples

- Mathematics
- 1966

We shall prove two theorems on the “triple” cohomology of algebras [1] using a method of acyclic models suggested by H. Appelgate. Specifically, we show that the triple cohomology coincides with… Expand

Locally presentable and accessible categories, London Math

- Soc. Lecture Note Series, vol. 189, Cambridge University Press, Cambridge,
- 1994

Rosicky, Locally presentable and accessible categories

- London Math. Soc. Lecture Note Series,
- 1994

Idempotent functors in homotopy theory, Manifolds—Tokyo 1973 (Proc

- Internat. Conf., Tokyo, 1973), Univ. Tokyo Press, Tokyo,
- 1975

Idempotent functors in homotopy theory, Manifolds—Tokyo

- (Proc. Intern a l Conf.,
- 1973

Dugundji, Categorical homotopy and fibrations

- Trans. Amer. Math. Soc
- 1969

Acyclic models and triples, Proceedings of the Conference on Categorical Algebra, La Jolla, 1965 (S

- Eilenberg, D. K. Harrison, S. MacLane, and H. Röhrl, eds.), Springer-Verlag,
- 1966