nLab model category

Idea

A model category (sometimes called a Quillen model category or a closed model category, but not related to “closed category”) is a context for doing homotopy theory. Quillen developed the definition of a model category to formalize the similarities between homotopy theory and homological algebra: the key examples which motivated his definition were the category of topological spaces, the category of simplicial sets, and the category of chain complexes.

So, what is a model category? For starters, it is a category equipped with three classes of morphisms, each closed under composition and called weak equivalences, fibrations and cofibrations:

• The weak equivalences play the role of ‘homotopy equivalences’ or something a bit more general (such as weak homotopy equivalences). Already in the case of topological spaces, it is useful to say that two spaces have the same homotopy type if there is a map from one to the other that induces isomorphisms on homotopy groups for any choice of basepoint in the first space. These maps are more general than homotopy equivalences, so they are called ‘weak equivalences’.

• The fibrations play the role of ‘nice surjections’. For example, in the category Top of topological spaces with its usual Quillen model structure on topological spaces, a locally trivial fiber bundle is a fibration. More generally the fibrations here are the Serre fibrations.

• The cofibrations play the role of ‘nice inclusions’. For example, in the category Top of topological spaces with its usual model structure on topological spaces, an NDR pair is typically a cofibration.

A bit more technically: we can define an (∞,1)-category starting from any category with weak equivalences. The idea is that this (∞,1)-category keeps track of objects in our original category, morphisms between objects, homotopies between morphisms, homotopies between homotopies, and so on, ad infinitum. However, the extra structure of a model category makes it easier to work with this (∞,1)-category. We can obtain this (∞,1)-category in various ways, such as simplicial localization of the underlying category with weak equivalences, or (if the model category is simplical) the homotopy coherent nerve of the simplicial subcategory $M_{cf}\subset M$ of cofibrant-fibrant objects. We say this (∞,1)-category is presented (or modeled) by the model category, and that the objects of the model category are models for the objects of this $(\infty,1)$-category. Not every (∞,1)-category is obtained in this way (otherwise it would necessarily have all small homotopy limits and homotopy colimits).

In this sense model categories are ‘models for homotopy theory’ or ‘categories of models for homotopy theory’. (The latter sense was the one intended by Quillen, but the former is also a useful way to think.)

Recall that the idea of categories with weak equivalences is to work just with 1-morphisms instead of with n-morphisms for all $n$, but to carry around extra information to remember which 1-morphisms are really equivalences in the full (∞,1)-category, i.e. isomorphisms in the corresponding homotopy category.

In a model category the data of weak equivalences is accompanied by further auxiliary data that helps to compute the (∞,1)-categorical hom-space, the homotopy category and derived functors. See homotopy category of a model category for more on that.

If the model category happens to be a combinatorial simplicial model category $\mathbf{A}$ it presents the category $\mathbf{A}^\circ$ in the form of a simplicially enriched category given by the full SSet-enriched subcategory on objects that are both fibrant and cofibrant.

Definition

The following is a somewhat terse account. For a more detailed exposition see at Introduction to Homotopy Theory the section Abstract homotopy theory.

Definition

A model structure on a category $\mathcal{C}$ is a choice of three distinguished classes of morphisms

• cofibrations $Cof \subset Mor(\mathcal{C})$,

• fibrations $Fib \subset Mor(\mathcal{C})$,

• weak equivalences $W \subset Mor(\mathcal{C})$

satisfying the following conditions:

1. $W$ makes $\mathcal{C}$ into a category with weak equivalences,

   (meaning that it contains all isomorphisms and is closed under two-out-of-three: given a composable pair of morphisms $f,g$, if two out of the three morphisms $f, g, g f$ are in $W$, so is the third);

2. $(Cof, Fib \cap W)$ and $(Cof \cap W, Fib)$ are two weak factorization systems on $\mathcal{C}$.

Definition

A model category is a complete and cocomplete category $\mathcal{C}$ equipped with a model structure according to def. .

This equivalent version of the definition was observed in (Joyal, def. E.1.2), highlighted in (Riehl 09). This definition already implies all the closure conditions on classes of morphisms which other definitions in the literature explicitly ask for, see below.

Definition

(terminology)

• The morphisms in $W \cap Fib$ (the fibrations that are also weak equivalences) are called trivial fibrations or acyclic fibrations

• The morphisms in $W \cap Cof$ (the cofibrations that are also weak equivalences) are called trivial cofibrations or acyclic cofibrations.

• An object is called cofibrant if the unique morphism $\emptyset \to X$ from the initial object is a cofibration

• An object is called fibrant if the unique morphism $X\to *$ to the terminal object is a fibration.

Remark

Often, the fibrant and cofibrant objects are the ones one is “really” interested in, but the category consisting only of these is not well-behaved (as a 1-category). The factorizations supply fibrant and cofibrant replacement functors which allow us to treat any object of the model category as a ‘model’ for its fibrant-cofibrant replacement.

Variants

Slight variations on the axioms

Quillen’s original definition required only finite limits and finite colimits, which are enough for the basic constructions. Colimits of larger cardinality are sometimes required for the small object argument, however. This change was popularized by Dwyer, Hirschhorn & Kan 1997, published as Dwyer, Hirschhorn, Kan & Smith 2004.

Robert W. Thomason proposed to require that the factorizations given by (ii) are actually functorial factorization systems,

resulting in the notion of a Thomason model category. Mark Hovey later included the data of such a functorial factorization (and not just its existence) into his definition of a model category. In practice, Quillen’s small object argument means that many model categories can be made to have functorial factorizations. (But not all: an example of a model category with non-functorial factorizations can be found in Isaksen 2001.)

Enhancements of the axioms

There are several extra conditions that strengthen the notion of a model category:

• A monoidal model category is monoidal category that is also a model category in a compatible way.

• An enriched model category is an enriched category over a monoidal category, that is also a model category in a compatible way.

• An algebraic model category is one where the two defining weak factorization systems are refined to algebraic weak factorization systems.

• A cofibrantly generated model category is one with a good compatible notion of cell complexes.

• A combinatorial model category is a cofibrantly generated one that in addition is a locally presentable category.

• An accessible model category is one on a locally presentable category that admits accessible factorizations, which can therefore be enhanced to algebraic weak factorization systems.

• A left/right proper model category is one where the weak equivalences are stable under pushforward along cofibrations / pullback along fibrations

Weaker axiom systems

There are several notions of category with weak equivalences with similar but less structure than a full model category.

• A category of fibrant objects has a notion of just weak equivalences and fibrations, none of cofibrations. As the name implies, all of its objects are fibrant; the canonical example is the subcategory of fibrant objects in a model category.

• A Waldhausen category dually has a notion of weak equivalences and cofibrations, and all of its objects are cofibrant.

• There is also a slight variant of the full notion of model category by Thomason that is designed to make the global model structure on functors more naturally accessible: this is the notion of Thomason model category.

• A category with path objects is similar to a category of fibrant objects but has path space objects.

• Semimodel categories relax some of the conditions on lifting properties.

• Weak model categories relax these conditions even further.

• Premodel categories are an even weaker notion that allows for a nice 2-categorical treatment.

Properties

Closure of morphism classes under retracts

As a consequence of the definition, the classes $Cof, Fib$, and $W$ each

• are closed under retracts in the arrow category $Arr C$,

• are closed under composition,

• contain all the isomorphisms of $C$.

For $Cof$ and $Fib$ and $W \cap Cof$ and $W \cap Fib$ this and further closure properties are discussed in detail at weak factorization system in the section Closure properties.

In the presence of functorial factorizations, it follows immediately that also $W$ is closed under retracts: for any retract diagram may then be funtorially factored with the middle morphism factored through $W \cap Cof$ followed by $W \cap Fib$, and so the statement follows from the above closure of these classes under retracts.

Without assuming functorial factorization the statement still holds:

Proposition

Given a model category in the sense of def. , then its class of weak equivalences is closed under forming retracts (in the arrow category).

(Joyal, prop. E.1.3), highlighted in (Riehl 09)

Proof

Let

$$\array{ id \colon & A &\longrightarrow& X &\longrightarrow& A \\ & {}^{\mathllap{f}} \downarrow && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{f}} \\ id \colon & B &\longrightarrow& Y &\longrightarrow& B }$$

be a commuting diagram with $w \in W$ a weak equivalence. We need to show that then also $f \in W$.

First consider the case that $f \in Fib$.

In this case, factor $w$ as a cofibration followed by an acyclic fibration. Since $w \in W$ and by two-out-of-three this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain the commuting diagram

$$\array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{id}} \\ id \colon & A' &\overset{s}{\longrightarrow}& X' &\overset{\phantom{AA}t\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{f}}_{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,,$$

where $s$ is uniquely defined and where $t$ is any lift of the top middle vertical acyclic cofibration against $f$. This now exhibits $f$ as a retract of an acyclic fibration. These are closed under retract by this prop..

Now consider the general case. Factor $f$ as an acyclic cofibration followed by a fibration and form the pushout in the top left square of the following diagram

$$\array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{\in W \cap Cof}}\downarrow &(po)& \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{\in W \cap Cof}} \\ id \colon & A' &\overset{}{\longrightarrow}& X' &\overset{\phantom{AA}\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W }} && \downarrow^{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,,$$

where the other three squares are induced by the universal property of the pushout, as is the identification of the middle horizontal composite as the identity on $A'$. Since acyclic cofibrations are closed under forming pushouts by this prop., the top middle vertical morphism is now an acyclic fibration, and hence by assumption and by two-out-of-three so is the middle bottom vertical morphism.

Thus the previous case now gives that the bottom left vertical morphism is a weak equivalence, and hence the total left vertical composite is.

Redundancy in the defining factorization systems

It is clear that:

Remark

Given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:

• given $W$ and $C$, we have $F = RLP(W \cap C)$;

• given $W$ and $F$, we have $C = LLP(W \cap F)$;

• given $C$ and $F$, we find $W$ as the class of morphisms which factor into a morphism in $LLP(F)$ followed by a morphism in $RLP(C)$.

(Here $RLP(S)$ denotes the class of morphisms with the right lifting property against $S$ and $LLP(S)$ denotes the class of morphisms with the left lifting property against $S$.)

But, in fact, already the cofibrations and the fibrant objects determine the model structure.

Proposition

A model structure $(C,W,F)$ on a category $\mathcal{C}$ is determined by its class of cofibrations and its class of fibrant objects.

This statement appears for instance as (Joyal, prop. E.1.10)

Proof

Let $\mathcal{E}$ with $C,F,W \subset Mor(\mathcal{E})$ be a model category.

By remark it is sufficient to show that the cofibrations and the fibrant objects determine the class of weak equivalences. Moreover, these are already determined by the weak equivalences between cofibrant objects, because for $u : A \to B$ any morphism, functorial cofibrant replacement $\emptyset \hookrightarrow \hat A \stackrel{\simeq}{\to} A$ and $\emptyset \hookrightarrow \hat B \stackrel{\simeq}{\to} B$ with 2-out-of-3 implies that $u$ is a weak equivalence precisely if $\hat u : \hat A \to \hat B$ is.

By the nature of the homotopy category $Ho$ of $\mathcal{E}$ and by the Yoneda lemma, a morphism $\hat u : \hat A \to \hat B$ between cofibrant objects is a weak equivalence precisely if for every fibrant object $X$ the map

$$Ho(\hat u, X) : Ho(\hat B, X) \to Ho(\hat A, X)$$

is an isomorphism, namely a bijection of sets. The equivalence relation that defines $Ho(\hat A,X)$ may be taken to be given by left homotopy induced by cylinder objects, which in turn are obtained by factoring codiagonals into cofibrations followed by acyclic fibrations. So all this is determined already by the class of cofibrations, and hence weak equivalences are determined by the cofibrations and the fibrant objects.

Basic closure properties of class of model categories

Proposition

(product model structure)
For $\mathcal{C}$, $\mathcal{D}$ a pair of model categories, the product category of their underlying categories carries the product model structure $\mathcal{C} \times \mathcal{D}$ whose weak equivalences/fibrations/cofibrations are the pairs of the corresponding such morphisms in $\mathcal{C}$ and $\mathcal{D}$.

More generally, for an indexed set $\big(\mathcal{C}_s\big)_{s \in S}$ of model categories, there is the analogous product model structure on their $S$-indexed product category $\underset{s \in S}{\prod} \mathcal{C}$, whose objects and morphisms are dependent $S$-tuples of objects and morphisms in the $\mathcal{C}_i$.

(e.g. Hovey (1999), Exp. 1.1.6)

Proposition

(opposite model structure)
If a category $C$ carries a model category-structure, then the opposite category $C^{op}$ carries the opposite model structure:

its weak equivalences are those morphisms whose dual was a weak equivalence in $C$, its fibrations are those morphisms that were cofibrations in $C$ and similarly for its cofibrations.

(e.g. Hovey (1999), Rem. 1.1.7)

Proposition

(transferred model structure along adjoint equivalence)
Given an right adjoint equivalence $R \colon \mathcal{D} \overset{\sim}{\longrightarrow} \mathcal{C}$ between categories and a model category-structure on $C$, then $D$ carries the transferred model structure whose weak equivalences/fibrations/cofibrations are the preimages under $R$ of the corresponding morphisms in $\mathcal{C}$. With respect to this transferred structure, the original adjunction $L \dashv R$ is a Quillen equivalence.

(cf. this Example at transferred model structure)

Homotopy and Homotopy category

See at

• homotopy in a model category

• homotopy category of a model category

Examples

Trivial model structure

Example

(trivial model structure)
Every bicomplete category carries its trivial model structure whose weak equivalences are the isomorphisms and all morphisms are both cofibrations and fibrations.

Remark

Trivial as this example may be, to the extent that model categories are presentations of certain $\infty$-categories it shows that/how ordinary (bicomplete) 1-categories are subsumed as a special case.

Classical model structures

The archetypical model structures are the

• classical model structure on topological spaces

and the

• classical model structure on simplicial sets.

These model categories are Quillen equivalent and encapsulate much of “classical” homotopy theory. From a higher-categorical viewpoint, they can be regarded as models for ∞-groupoids (in terms of CW complexes or Kan complexes, respectively).

The passage to stable homotopy theory is given by model structures on spectra built out of either of these two classical model structures. See at Model categories of diagram spectra for a unified treatment.

Accordingly homological algebra with its derived categories and derived functors (which may be thought of as a sub-topic of stable homotopy theory via the stable Dold-Kan correspondence) is reflected by

• model structures on chain complexes

• model structures on dg-algebras

In fact, the original definition of model categories in Quillen67 was motivated by the analogy between constructions in homotopy theory and homological algebra.

Categorical model structures

Of interest to category theorists is that many notions of higher categories come equipped with model structures, witnessing the fact that when retaining only invertible transfors between $n$-categories they should form an $(\infty,1)$-category. Many of these are called

• canonical model structures, including “categorical” model structures for
  • categories
  • (strict or weak) 2-categories
  • strict ∞-categories, and
  • strict ∞-groupoids.

Model categories have successfully been used to compare many different notions of (∞,1)-category. The following definitions of $(\infty,1)$-category all form Quillen equivalent model categories:

• simplicially enriched categories
• quasi-categories (via the Joyal model structure on simplicial sets)
• Segal categories
• complete Segal spaces

There are related model structures for enriched higher categories:

• model structure on enriched categories
• model structure on dg-categories

Other “higher categorical structures” can also be expected to form model categories, such as the

• model structure on dendroidal sets

which generalizes the Joyal model structure from (∞,1)-categories to (∞,1)-operads.

There is also another class of model structures on categorical structures, often called Thomason model structures (not to be confused with the notion of “Thomason model category”). In the “categorical” or “canonical” model structures, the weak equivalences are the categorical equivalences, but in the Thomason model structures, the weak equivalences are those that induce weak homotopy equivalences of nerves. Thomason model structures are known to exist on 1-categories and 2-categories, at least, and are generally Quillen equivalent to the Quillen model structure on topological spaces and thus (via the singular simplicial complex and geometric realization adjunction) to the and Quillen model structure on simplicial sets.

Parametrized model structures

The parameterized version of the model structure on simplicial sets is a

• model structure on simplicial presheaves or
• model structure on simplicial sheaves or

which serves as a model for ∞-stack (∞,1)-toposes (for hypercomplete (∞,1)-toposes, more precisely).

Functor and localized model structures

Many model structures, including those for complete Segal spaces, simplicial presheaves, and diagram spectra, are constructed by starting with a model structure on a functor category, such as a

• projective model structure,
• injective model structure, or
• Reedy model structure,

and applying a general technique called Bousfield localization which forces a certain class of morphisms to become weak equivalences. It can also be thought of as forcing a certain class of objects to become fibrant.

Limit and colimit model structures

Model structures can be induced on certain (usually lax) limits and colimits of diagrams of model categories.

• Grothendieck construction for model categories (lax colimits)
• model structure on sections (lax limit)

Related concepts

• 2-category of model categories

• simplicial homotopy theory

• synthetic homotopy theory

• homotopy category of a model category

• localization of model categories

• double category of model categories

• Ho(CombModCat)

• premodel category

• model (∞,1)-category

Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.

structure	small-set-generated	small-category-generated	algebraicized
weak factorization system	combinatorial wfs	accessible wfs	algebraic wfs
model category	combinatorial model category	accessible model category	algebraic model category
construction method	small object argument	same as $\to$	algebraic small object argument

References

The concept of a model category originates with

• Daniel Quillen, Chapter I, Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43 Springer (1967) [doi:10.1007/BFb0097438]

and the modern form of the axioms (replacing requirement of finite (co-)limits by small (co-)limits ) is due to:

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Model Categories and More General Abstract Homotopy Theory (1997) [pdf, pdf]

Monographs and textbooks

• Mark Hovey, Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]

• Mark Hovey, Errata to Model Categories (1999) [pdf, pdf]

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf, pdf]

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113, AMS 2004 (ISBN: 978-1-4704-1340-8, pdf)

• J. Peter May, Kate Ponto, More Concise Algebraic Topology, The University of Chicago Press, 2012. Part 4.

• Emily Riehl, Categorical Homotopy Theory, Cambridge University Press, 2014.

• Scott Balchin, A Handbook of Model Categories, Algebra and Applications, 27 Springer (2021) $[$doi:10.1007/978-3-030-75035-0$]$

Coverage table for major sources

Topic	Quillen	Hovey	Hirschhorn	DHKS	May-Ponto	Riehl	Lurie	Balchin
combinatorial model categories	no	no	no	no	no*	yes	yes	yes
monoidal model categories	no	yes	no	no	yes	yes	yes	yes
enriched model categories	no	no	no	no	yes	yes	yes	yes
homotopy colimits	no	no	yes	yes	no*	yes	yes	yes
Bousfield localizations	no	no	yes	no	yes	no	yes	yes
transferred model structures	yes	no	yes	no	no	yes	yes	yes
Reedy model structures	no	yes	yes	yes	no	yes	yes	yes

Book chapters

• Paul Goerss, Rick Jardine, Chapters 1 and 2 of Simplicial homotopy theory, Birkhäuser, 1999, 2009.

For yet another introduction to model categories, with an eye towards their use as presentations of $(\infty,1)$-categories see

• Jacob Lurie, Appendix A.2 and A.3 of Higher Topos Theory

Survey articles

• William Dwyer, Jan Spalinski, Homotopy theories and model categories (pdf)

  in: I. M. James, Handbook of Algebraic Topology, North Holland 1995 (ISBN:9780080532981, doi:10.1016/B978-0-444-81779-2.X5000-7)

• Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007(arXiv:math.AT/0609537, doi:10.1090/conm/436)

Other sources

An account is in

• Joyal's CatLab, Model categories

and appendix E of

• André Joyal, The theory of quasi-categories and its applications (pdf, pdf)

The version of the definition in (Joyal) is also highlighted in

• Emily Riehl, A concise definition of model category, 2009 (pdf, pdf)

An introductory survey of some key concepts is in the set of slides

• Peter May, Model Category Primer

There is an unpublished manuscript of Chris Reedy from around 1974 that’s been circulating as an increasingly faded photocopy. It’s been typed into LaTeX, and the author has given permission for it to be posted on the net:

• Chris Reedy, Homotopy theory of model categories (pdf)

See

• Philip Hirschhorn, personal website: Mathematics

for errata and more.