on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
equivalences in/of $(\infty,1)$-categories
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 (otherways 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.
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:
$W$ 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);
$(Cof, Fib \cap W)$ and $(Cof \cap W, Fib)$ are two weak factorization systems on $\mathcal{C}$.
A model category is a complete and cocomplete category $\mathcal{C}$ equipped with a model structure according to def. 1.
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.
(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.
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.
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.
Some authors, notably Mark Hovey, require that the factorizations given by (ii) are actually functorial factorization systems. In practice, Quillen’s small object argument means that many model categories can be made to have functorial factorizations.
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 one that in addition is a locally presentable category.
A left/right proper model category is one where the weak equivalences are stable under pushforward along cofibrations / pullback along fibrations
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.
As a consequence of the definition, the classes $Cof, Fib$, and $W$ are all closed under retracts in the arrow category $Arr C$ and under composition and contain 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:
Given a model category in the sense of def. 2, then its class of weak equivalences is closed under forming retracts (in the arrow category).
(Joyal, prop. E.1.3), highlighted in (Riehl 09)
Let
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
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
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.
It is clear that:
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 $C \cap W$ followed by a morphism in $F \cap W$.
But, in fact, already the cofibrations and the fibrant objects determine the model structure.
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)
Let $\mathcal{E}$ with $C,F,W \subset Mor(\mathcal{E})$ be a model category.
By remark 2 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
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.
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.
See at
Every category with limits and colimits carries the trivial model structure whose weak equivalences are the isomorphisms and all morphisms are cofibrations and fibrations.
The archetypical model structures are the
and the
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
In fact, the original definition of model categories in (Quillen 67) was motivated by the analogy between constructions in homotopy theory and homological algebra.
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
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:
There are related model structures for enriched higher categories:
Other “higher categorical structures” can also be expected to form model categories, such as the
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.
The parameterized version of the model structure on simplicial sets is a
which serves as a model for ∞-stack (∞,1)-toposes (for hypercomplete (∞,1)-toposes, more precisely).
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
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.
The concept originates in
An account is in
and appendix E of
The version of the definition in (Joyal) is also highlighted in
An introductory survey of some key concepts is in the set of slides
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:
More recent review includes
William Dwyer, J. Spalinski, Homotopy theories and model categories (pdf) in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology 1995
Paul Goerss, Rick Jardine, chapter 1 of Simplicial homotopy theory, Birkhäuser, 1999, 2009
Paul Goerss, Kristen Schemmerhorn, Model Categories and Simplicial Methods (arXiv)
Monographs:
Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (AMS, pdf toc, pdf)
Mark Hovey, Model Categories Mathematical Surveys and Monographs, Volume 63, AMS (1999) (pdf, Google books)
See
for errata and more.
For yet another introduction to model categories, with an eye towards their use as presentations of $(\infty,1)$-categories see