In mathematics, particularly in homotopy theory, a **model category** is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by Daniel G.Quillen ( 1967 ).

- Motivation
- Formal definition
- Definition via weak factorization systems
- First consequences of the definition
- Examples
- Topological spaces
- Chain complexes
- Further examples
- Some constructions
- Characterizations of fibrations and cofibrations by lifting properties
- Homotopy and the homotopy category
- Quillen adjunctions
- See also
- Notes
- References
- Further reading
- External links

In recent decades, the language of model categories has been used in some parts of algebraic *K*-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.

Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets.

Another model category is the category of chain complexes of *R*-modules for a commutative ring *R*. Homotopy theory in this context is homological algebra. Homology can then be viewed as a type of homotopy, allowing generalizations of homology to other objects, such as groups and *R*-algebras, one of the first major applications of the theory. Because of the above example regarding homology, the study of closed model categories is sometimes thought of as homotopical algebra.

The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating others to weaken some of the assumptions to define a model category. In practice the distinction has not proven significant and most recent authors (e.g., Mark Hovey and Philip Hirschhorn) work with closed model categories and simply drop the adjective 'closed'.

The definition has been separated to that of a model structure on a category and then further categorical conditions on that category, the necessity of which may seem unmotivated at first but becomes important later. The following definition follows that given by Hovey.

A **model structure** on a category *C* consists of three distinguished classes of morphisms (equivalently subcategories): weak equivalences, fibrations, and cofibrations, and two functorial factorizations and subject to the following axioms. A fibration that is also a weak equivalence is called an **acyclic** (or **trivial**) **fibration**^{ [1] } and a cofibration that is also a weak equivalence is called an **acyclic** (or **trivial**) **cofibration** (or sometimes called an **anodyne morphism**).

- Axioms

*Retracts*: if*g*is a morphism belonging to one of the distinguished classes, and*f*is a retract of*g*(as objects in the arrow category , where 2 is the 2-element ordered set), then*f*belongs to the same distinguished class. Explicitly, the requirement that*f*is a retract of*g*means that there exist*i*,*j*,*r*, and*s*, such that the following diagram commutes:*2 of 3*: if*f*and*g*are maps in*C*such that*gf*is defined and any two of these are weak equivalences then so is the third.*Lifting*: acyclic cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to acyclic fibrations. Explicitly, if the outer square of the following diagram commutes, where*i*is a cofibration and*p*is a fibration, and*i*or*p*is acyclic, then there exists*h*completing the diagram.*Factorization*:- every morphism
*f*in*C*can be written as for a fibration*p*and an acyclic cofibration*i*; - every morphism
*f*in*C*can be written as for an acyclic fibration*p*and a cofibration*i*.

- every morphism

A **model category** is a category that has a model structure and all (small) limits and colimits, i.e., a complete and cocomplete category with a model structure.

The above definition can be succinctly phrased by the following equivalent definition: a model category is a category **C** and three classes of (so-called) weak equivalences *W*, fibrations *F* and cofibrations *C* so that

**C**has all limits and colimits,

- is a weak factorization system
- satisfies the 2 of 3 property.
^{ [2] }

The axioms imply that any two of the three classes of maps determine the third (e.g., cofibrations and weak equivalences determine fibrations).

Also, the definition is self-dual: if *C* is a model category, then its opposite category also admits a model structure so that weak equivalences correspond to their opposites, fibrations opposites of cofibrations and cofibrations opposites of fibrations.

The category of topological spaces, **Top**, admits a standard model category structure with the usual (Serre) fibrations and with weak equivalences as weak homotopy equivalences. The cofibrations are not the usual notion found here, but rather the narrower class of maps that have the left lifting property with respect to the acyclic Serre fibrations. Equivalently, they are the retracts of the relative cell complexes, as explained for example in Hovey's *Model Categories*. This structure is not unique; in general there can be many model category structures on a given category. For the category of topological spaces, another such structure is given by Hurewicz fibrations and standard cofibrations, and the weak equivalences are the (strong) homotopy equivalences.

The category of (nonnegatively graded) chain complexes of *R*-modules carries at least two model structures, which both feature prominently in homological algebra:

- weak equivalences are maps that induce isomorphisms in homology;
- cofibrations are maps that are monomorphisms in each degree with projective cokernel; and
- fibrations are maps that are epimorphisms in each nonzero degree

or

- weak equivalences are maps that induce isomorphisms in homology;
- fibrations are maps that are epimorphisms in each degree with injective kernel; and
- cofibrations are maps that are monomorphisms in each nonzero degree.

This explains why Ext-groups of *R*-modules can be computed by either resolving the source projectively or the target injectively. These are cofibrant or fibrant replacements in the respective model structures.

The category of arbitrary chain-complexes of *R*-modules has a model structure that is defined by

- weak equivalences are chain homotopy equivalences of chain-complexes;
- cofibrations are monomorphisms that are split as morphisms of underlying
*R*-modules; and - fibrations are epimorphisms that are split as morphisms of underlying
*R*-modules.

Other examples of categories admitting model structures include the category of all small categories, the category of simplicial sets or simplicial presheaves on any small Grothendieck site, the category of topological spectra, and the categories of simplicial spectra or presheaves of simplicial spectra on a small Grothendieck site.

Simplicial objects in a category are a frequent source of model categories; for instance, simplicial commutative rings or simplicial *R*-modules admit natural model structures. This follows because there is an adjunction between simplicial sets and simplicial commutative rings (given by the forgetful and free functors), and in nice cases one can lift model structures under an adjunction.

A *simplicial model category* is a simplicial category with a model structure that is compatible with the simplicial structure.^{ [3] }

Given any category *C* and a model category *M*, under certain extra hypothesis the category of functors Fun (*C*, *M*) (also called *C*-diagrams in *M*) is also a model category. In fact, there are always *two* candidates for distinct model structures: in one, the so-called projective model structure, fibrations and weak equivalences are those maps of functors which are fibrations and weak equivalences when evaluated at each object of *C*. Dually, the injective model structure is similar with cofibrations and weak equivalences instead. In both cases the third class of morphisms is given by a lifting condition (see below). In some cases, when the category *C* is a Reedy category, there is a third model structure lying in between the projective and injective.

The process of forcing certain maps to become weak equivalences in a new model category structure on the same underlying category is known as Bousfield localization. For example, the category of simplicial sheaves can be obtained as a Bousfield localization of the model category of simplicial presheaves.

Denis-Charles Cisinski has developed^{ [4] } a general theory of model structures on presheaf categories (generalizing simplicial sets, which are presheaves on the simplex category).

If *C* is a model category, then so is the category Pro(*C*) of pro-objects in *C*. However, a model structure on Pro(*C*) can also be constructed by imposing a weaker set of axioms to *C*.^{ [5] }

Every closed model category has a terminal object by completeness and an initial object by cocompleteness, since these objects are the limit and colimit, respectively, of the empty diagram. Given an object *X* in the model category, if the unique map from the initial object to *X* is a cofibration, then *X* is said to be **cofibrant**. Analogously, if the unique map from *X* to the terminal object is a fibration then *X* is said to be **fibrant**.

If *Z* and *X* are objects of a model category such that *Z* is cofibrant and there is a weak equivalence from *Z* to *X* then *Z* is said to be a **cofibrant replacement** for *X*. Similarly, if *Z* is fibrant and there is a weak equivalence from *X* to *Z* then *Z* is said to be a **fibrant replacement** for *X*. In general, not all objects are fibrant or cofibrant, though this is sometimes the case. For example, all objects are cofibrant in the standard model category of simplicial sets and all objects are fibrant for the standard model category structure given above for topological spaces.

Left homotopy is defined with respect to cylinder objects and right homotopy is defined with respect to path space objects. These notions coincide when the domain is cofibrant and the codomain is fibrant. In that case, homotopy defines an equivalence relation on the hom sets in the model category giving rise to homotopy classes.

Cofibrations can be characterized as the maps which have the left lifting property with respect to acyclic fibrations, and acyclic cofibrations are characterized as the maps which have the left lifting property with respect to fibrations. Similarly, fibrations can be characterized as the maps which have the right lifting property with respect to acyclic cofibrations, and acyclic fibrations are characterized as the maps which have the right lifting property with respect to cofibrations.

The *homotopy category* of a model category *C* is the localization of *C* with respect to the class of weak equivalences. This definition of homotopy category does not depend on the choice of fibrations and cofibrations. However, the classes of fibrations and cofibrations are useful in describing the homotopy category in a different way and in particular avoiding set-theoretic issues arising in general localizations of categories. More precisely, the "fundamental theorem of model categories" states that the homotopy category of *C* is equivalent to the category whose objects are the objects of *C* which are both fibrant and cofibrant, and whose morphisms are left homotopy classes of maps (equivalently, right homotopy classes of maps) as defined above. (See for instance Model Categories by Hovey, Thm 1.2.10)

Applying this to the category of topological spaces with the model structure given above, the resulting homotopy category is equivalent to the category of CW complexes and homotopy classes of continuous maps, whence the name.

A pair of adjoint functors

between two model categories *C* and *D* is called a Quillen adjunction if *F* preserves cofibrations and acyclic cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and acyclic fibrations. In this case *F* and *G* induce an adjunction

between the homotopy categories. There is also an explicit criterion for the latter to be an equivalence (*F* and *G* are called a *Quillen equivalence* then).

A typical example is the standard adjunction between simplicial sets and topological spaces:

involving the geometric realization of a simplicial set and the singular chains in some topological space. The categories **sSet** and **Top** are not equivalent, but their homotopy categories are. Therefore, simplicial sets are often used as models for topological spaces because of this equivalence of homotopy categories.

- ↑ Some readers find the term "trivial" ambiguous and so prefer to use "acyclic".
- ↑ Riehl (2014 , §11.3)
- ↑ Definition 2.1. of .
- ↑ Cisinski, Denis-Charles. Les préfaisceaux comme modèles des types d'homotopie. (French) [Presheaves as models for homotopy types] Astérisque No. 308 (2006), xxiv+390 pp. ISBN 978-2-85629-225-9 MR 2294028
- ↑ Barnea, Ilan; Schlank, Tomer M. (2016), "A projective model structure on pro-simplicial sheaves, and the relative étale homotopy type.",
*Adv. Math.*,**291**: 784–858, arXiv: 1109.5477 , Bibcode:2011arXiv1109.5477B, MR 3459031

In topology, a branch of mathematics, a **fibration** is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space. A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.

In mathematics, certain functors may be *derived* to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

In mathematics, **localization of a category** consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. **Calculus of fractions** is another name for working in a localized category.

In mathematics, a **simplicial set** is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.

In mathematics, in particular homotopy theory, a continuous mapping

In mathematics, specifically in homotopy theory in the context of a model category *M*, a **fibrant object***A* of *M* is an object that has a fibration to the terminal object of the category.

In mathematics, the **homotopy category** is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

In homotopy theory, a branch of mathematics, a **Quillen adjunction** between two closed model categories **C** and **D** is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(**C**) and Ho(**D**) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.

In mathematics, a **Waldhausen category** is a category *C* equipped with some additional data, which makes it possible to construct the K-theory spectrum of *C* using a so-called **S-construction**. It's named after Friedhelm Waldhausen, who introduced this notion to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces.

In algebraic geometry and algebraic topology, branches of mathematics, **A**^{1}**homotopy theory** is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line **A**^{1}, which is. The theory has seen spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.

In mathematics, **directed algebraic topology** is a refinement of algebraic topology for **directed spaces**, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in Concurrency, Network traffic control, General Relativity, Noncommutative Geometry, Rewriting Theory, and Biological systems.

In mathematics, more specifically category theory, a **quasi-category** is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.

In mathematics, a **weak equivalence** is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.

In mathematics, more specifically in homotopy theory, a **simplicial presheaf** is a presheaf on a site taking values in simplicial sets. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a **simplicial sheaf** on a site is a simplicial object in the category of sheaves on the site.

In mathematics, especially in algebraic topology, the **homotopy limit and colimit**^{pg 52} are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we have a diagram

In category theory, a branch of mathematics, a (left) **Bousfield localization** of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences.

In algebraic topology, a **presheaf of spectra** on a topological space *X* is a contravariant functor from the category of open subsets of *X*, where morphisms are inclusions, to the good category of commutative ring spectra. A theorem of Jardine says that such presheaves form a simplicial model category, where *F* →*G* is a weak equivalence if the induced map of homotopy sheaves is an isomorphism. A **sheaf of spectra** is then a fibrant/cofibrant object in that category.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, in particular in category theory, the **lifting property** is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

In mathematics, **homotopy theory** is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry and category theory.

- Denis-Charles Cisinski:
*Les préfaisceaux commes modèles des types d'homotopie*, Astérisque, (308) 2006, xxiv+392 pp. - Dwyer, William G.; Spaliński, Jan (1995), "Homotopy theories and model categories" (PDF),
*Handbook of algebraic topology*, Amsterdam: North-Holland, pp. 73–126, doi:10.1016/B978-044481779-2/50003-1, MR 1361887 - Philip S. Hirschhorn:
*Model Categories and Their Localizations*, 2003, ISBN 0-8218-3279-4. - Mark Hovey:
*Model Categories*, 1999, ISBN 0-8218-1359-5. - Klaus Heiner Kamps and Timothy Porter:
*Abstract homotopy and simple homotopy theory*, 1997, World Scientific, ISBN 981-02-1602-5. - Georges Maltsiniotis:
*La théorie de l'homotopie de Grothendieck*. Astérisque, (301) 2005, vi+140 pp.

- Riehl, Emily (2014),
*Categorical homotopy theory*, Cambridge University Press, doi:10.1017/CBO9781107261457, ISBN 978-1-107-04845-4, MR 3221774

- Quillen, Daniel G. (1967),
*Homotopical algebra*, Lecture Notes in Mathematics, No. 43,**43**, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0097438, MR 0223432 CS1 maint: discouraged parameter (link)

- "Do we still need model categories?"
- "(infinity,1)-categories directly from model categories"
- Paul Goerss and Kristen Schemmerhorn, Model Categories and Simplicial Methods

- Model category in
*nLab* - Model category in Joyal's catlab

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