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\begin{document}

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\section*{model category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{slight_variations_on_the_axioms}{Slight variations on the axioms}\dotfill \pageref*{slight_variations_on_the_axioms} \linebreak
\noindent\hyperlink{enhancements_of_the_axioms}{Enhancements of the axioms}\dotfill \pageref*{enhancements_of_the_axioms} \linebreak
\noindent\hyperlink{weaker_axiom_systems}{Weaker axiom systems}\dotfill \pageref*{weaker_axiom_systems} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ClosureOfMorphisms}{Closure of morphism classes under retracts}\dotfill \pageref*{ClosureOfMorphisms} \linebreak
\noindent\hyperlink{RedundancyInTheAxioms}{Redundancy in the defining factorization systems}\dotfill \pageref*{RedundancyInTheAxioms} \linebreak
\noindent\hyperlink{opposite_model_structure}{Opposite model structure}\dotfill \pageref*{opposite_model_structure} \linebreak
\noindent\hyperlink{homotopy_and_homotopy_category}{Homotopy and Homotopy category}\dotfill \pageref*{homotopy_and_homotopy_category} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classical_model_structures}{Classical model structures}\dotfill \pageref*{classical_model_structures} \linebreak
\noindent\hyperlink{categorical_model_structures}{Categorical model structures}\dotfill \pageref*{categorical_model_structures} \linebreak
\noindent\hyperlink{parametrized_model_structures}{Parametrized model structures}\dotfill \pageref*{parametrized_model_structures} \linebreak
\noindent\hyperlink{functor_and_localized_model_structures}{Functor and localized model structures}\dotfill \pageref*{functor_and_localized_model_structures} \linebreak
\noindent\hyperlink{limit_and_colimit_model_structures}{Limit and colimit model structures}\dotfill \pageref*{limit_and_colimit_model_structures} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{model category} (sometimes called a \emph{Quillen model category} or a \emph{closed model category}, but \textbf{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 space|topological spaces]], the category of [[simplicial set|simplicial sets]], and the category of [[chain complex|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 \emph{[[weak equivalences]]}, \emph{[[fibrations]]} and \emph{[[cofibrations]]}:

\begin{itemize}%
\item 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'.


\item 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]].


\item 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.



\end{itemize}
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, \emph{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 [[simplicially enriched category|simplicial subcategory]] $M_{cf}\subset M$ of cofibrant-fibrant objects. We say this [[(∞,1)-category]] is \emph{presented} (or modeled) by the model category, and that the objects of the model category are \emph{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 [[category with weak equivalences|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 [[equivalence|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 functor|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 [[presentable (infinity,1)-category|presents]] the [[(infinity,1)-category|category]] $\mathbf{A}^\circ$ in the form of a [[simplicially enriched category]] given by the full [[SSet]]-[[enriched category|enriched subcategory]] on objects that are both fibrant and cofibrant.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The following is a somewhat terse account. For a more detailed exposition see at \emph{[[Introduction to Homotopy Theory]]} the section \emph{\href{Introduction+to+Homotopy+Theory#ModelCategoryTheory}{Abstract homotopy theory}}.



\begin{defn}
\label{ModelStructure}\hypertarget{ModelStructure}{}
A \textbf{model structure} on a [[category]] $\mathcal{C}$ is a choice of three distinguished [[classes]] of [[morphisms]]

\begin{itemize}%
\item \emph{cofibrations} $Cof \subset Mor(\mathcal{C})$,


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


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



\end{itemize}
satisfying the following conditions:

\begin{enumerate}%
\item $W$ makes $\mathcal{C}$ into a [[category with weak equivalences]],

(meaning that it contains all [[isomorphisms]] and is closed under \textbf{[[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);


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



\end{enumerate}
\end{defn}
\begin{defn}
\label{ModelCategory}\hypertarget{ModelCategory}{}
A \textbf{model category} is a [[complete category|complete]] and [[cocomplete category]] $\mathcal{C}$ equipped with a model structure according to def. \ref{ModelStructure}.

\end{defn}
This equivalent version of the definition was observed in (\hyperlink{Joyal}{Joyal, def. E.1.2}), highlighted in (\hyperlink{Riehl09}{Riehl 09}). This definition already implies all the closure conditions on classes of morphisms which other definitions in the literature explicitly ask for, see \hyperlink{ClosureOfMorphisms}{below}.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(terminology)}

\begin{itemize}%
\item The morphisms in $W \cap Fib$ (the fibrations that are also weak equivalences) are called \textbf{trivial fibrations} or \textbf{[[acyclic fibrations]]}


\item The morphisms in $W \cap Cof$ (the cofibrations that are also weak equivalences) are called \textbf{trivial cofibrations} or \textbf{[[acyclic cofibrations]]}.


\item An object is called \textbf{[[cofibrant]]} if the unique morphism $\emptyset \to X$ from the [[initial object]] is a cofibration


\item An object is called \textbf{[[fibrant]]} if the unique morphism $X\to *$ to the [[terminal object]] is a fibration.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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.

\end{remark}
\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

\hypertarget{slight_variations_on_the_axioms}{}\subsubsection*{{Slight variations on the axioms}}\label{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.

Some authors, notably [[Mark Hovey]], require that the factorizations given by (ii) are actually \emph{[[functorial factorization systems]]}. In practice, Quillen's [[small object argument]] means that many model categories can be made to have functorial factorizations.

\hypertarget{enhancements_of_the_axioms}{}\subsubsection*{{Enhancements of the axioms}}\label{enhancements_of_the_axioms}

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

\begin{itemize}%
\item A [[monoidal model category]] is [[monoidal category]] that is also a model category in a compatible way.


\item An [[enriched model category]] is an [[enriched category]] over a monoidal category, that is also a model category in a compatible way.


\item An [[algebraic model category]] is one where the two defining [[weak factorization systems]] are refined to [[algebraic weak factorization systems]].


\item A [[cofibrantly generated model category]] is one with a good compatible notion of [[cell complexes]].


\item A [[combinatorial model category]] is a cofibrantly generated one that in addition is a [[locally presentable category]].


\item An [[accessible model category]] is one on a locally presentable category that admits [[accessible functor|accessible]] factorizations, which can therefore be enhanced to [[algebraic weak factorization systems]].


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



\end{itemize}
\hypertarget{weaker_axiom_systems}{}\subsubsection*{{Weaker axiom systems}}\label{weaker_axiom_systems}

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

\begin{itemize}%
\item 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.


\item A [[Waldhausen category]] dually has a notion of weak equivalences and cofibrations, and all of its objects are cofibrant.



\end{itemize}
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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{ClosureOfMorphisms}{}\subsubsection*{{Closure of morphism classes under retracts}}\label{ClosureOfMorphisms}

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 \emph{[[weak factorization system]]} in the section \emph{\href{https://ncatlab.org/nlab/show/weak+factorization+system#ClosureProperties}{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:

\begin{prop}
\label{WeakEquivalencesAreClosedUnderRetracts}\hypertarget{WeakEquivalencesAreClosedUnderRetracts}{}
Given a model category in the sense of def. \ref{ModelCategory}, then its class of weak equivalences is closed under forming [[retracts]] (in the [[arrow category]]).

\end{prop}
(\hyperlink{Joyal}{Joyal, prop. E.1.3}), highlighted in (\hyperlink{Riehl09}{Riehl 09})

\begin{proof}
Let

\begin{displaymath}
\itexarray{
    id \colon & A &\longrightarrow& X &\longrightarrow& A
    \\
    & 
    {}^{\mathllap{f}}
    \downarrow
      &&
    \downarrow^{\mathrlap{w}}
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    id \colon & B &\longrightarrow& Y &\longrightarrow& B
  }
\end{displaymath}
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

\begin{displaymath}
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    & 
    A 
      &\longrightarrow& 
    X 
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    \\
    &
    {}^{\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
  }
  \,,
\end{displaymath}
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 \href{weak+factorization+system#ClosuredPropertiesOfWeakFactorizationSystem}{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

\begin{displaymath}
\itexarray{
    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
  }
  \,,
\end{displaymath}
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 \href{weak+factorization+system#ClosuredPropertiesOfWeakFactorizationSystem}{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.

\end{proof}
\hypertarget{RedundancyInTheAxioms}{}\subsubsection*{{Redundancy in the defining factorization systems}}\label{RedundancyInTheAxioms}

It is clear that:

\begin{remark}
\label{AnyTwooClassesDetermineTheThird}\hypertarget{AnyTwooClassesDetermineTheThird}{}
Given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:

\begin{itemize}%
\item given $W$ and $C$, we have $F = RLP(W \cap C)$;


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


\item 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 $RLP(C)$.



\end{itemize}
(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 $S$.)

\end{remark}
But, in fact, already the cofibrations and the fibrant objects determine the model structure.

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

\end{prop}
This statement appears for instance as (\hyperlink{Joyal}{Joyal, prop. E.1.10})

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

By remark \ref{AnyTwooClassesDetermineTheThird} 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 factorization|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

\begin{displaymath}
Ho(\hat u, X) : Ho(\hat B, X) \to Ho(\hat A, X)
\end{displaymath}
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.

\end{proof}
\hypertarget{opposite_model_structure}{}\subsubsection*{{Opposite model structure}}\label{opposite_model_structure}

\begin{prop}
\label{}\hypertarget{}{}
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.

\end{prop}
\hypertarget{homotopy_and_homotopy_category}{}\subsubsection*{{Homotopy and Homotopy category}}\label{homotopy_and_homotopy_category}

See at

\begin{itemize}%
\item [[homotopy in a model category]]


\item [[homotopy category of a model category]]



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Every category with limits and colimits carries the [[trivial model structure]] whose weak equivalences are the [[isomorphism]]s and all morphisms are cofibrations and fibrations.

\hypertarget{classical_model_structures}{}\subsubsection*{{Classical model structures}}\label{classical_model_structures}

The archetypical model structures are the

\begin{itemize}%
\item [[classical model structure on topological spaces]]

\end{itemize}
and the

\begin{itemize}%
\item [[classical model structure on simplicial sets]].

\end{itemize}
These model categories are [[Quillen equivalence|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 \emph{[[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

\begin{itemize}%
\item [[model structures on chain complexes]]


\item [[model structures on dg-algebras]]



\end{itemize}
In fact, the original definition of model categories in (\hyperlink{Quillen67}{Quillen 67}) was motivated by the [[analogy]] between constructions in [[homotopy theory]] and [[homological algebra]].

\hypertarget{categorical_model_structures}{}\subsubsection*{{Categorical model structures}}\label{categorical_model_structures}

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

\begin{itemize}%
\item [[canonical model structure]]s, including ``categorical'' model structures for\begin{itemize}%
\item categories
\item (strict or weak) [[2-categories]]
\item [[strict ∞-categories]], and
\item [[strict ∞-groupoids]].

\end{itemize}


\end{itemize}
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:

\begin{itemize}%
\item [[simplicially enriched categories]]
\item [[quasi-categories]] (via the Joyal [[model structure on simplicial sets]])
\item [[Segal categories]]
\item [[complete Segal spaces]]

\end{itemize}
There are related model structures for enriched higher categories:

\begin{itemize}%
\item [[model structure on enriched categories]]
\item [[model structure on dg-categories]]

\end{itemize}
Other ``higher categorical structures'' can also be expected to form model categories, such as the

\begin{itemize}%
\item [[model structure on dendroidal sets]]

\end{itemize}
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 structure]]s (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 equivalence|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]].

\hypertarget{parametrized_model_structures}{}\subsubsection*{{Parametrized model structures}}\label{parametrized_model_structures}

The \emph{parameterized} version of the model structure on simplicial sets is a

\begin{itemize}%
\item [[model structure on simplicial presheaves]] or
\item [[model structure on simplicial sheaves]] or

\end{itemize}
which serves as a [[models for infinity-stack (infinity,1)-toposes|model for ∞-stack (∞,1)-toposes]] (for [[hypercomplete (∞,1)-topos]]es, more precisely).

\hypertarget{functor_and_localized_model_structures}{}\subsubsection*{{Functor and localized model structures}}\label{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

\begin{itemize}%
\item [[projective model structure]],
\item [[injective model structure]], or
\item [[Reedy model structure]],

\end{itemize}
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.

\hypertarget{limit_and_colimit_model_structures}{}\subsubsection*{{Limit and colimit model structures}}\label{limit_and_colimit_model_structures}

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

\begin{itemize}%
\item [[Grothendieck construction for model categories]] (lax colimits)
\item [[model structure on sections]] (lax limit)

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[synthetic homotopy theory]]


\item [[homotopy category of a model category]]


\item [[localization of model categories]]


\item [[double category of model categories]]


\item [[Ho(CombModCat)]]



\end{itemize}
[[!include algebraic model structures - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept originates in

\begin{itemize}%
\item [[Daniel Quillen]], chapter I \emph{Axiomatic homotopy theory} in \emph{Homotopical algebra}, Lecture Notes in Mathematics, No. 43 43, Berlin (1967)

\end{itemize}
An account is in

\begin{itemize}%
\item [[joyalscatlab:HomePage|Joyal's CatLab]], \emph{[[joyalscatlab:Model categories]]}

\end{itemize}
and appendix E of

\begin{itemize}%
\item [[André Joyal]], \emph{The theory of quasi-categories and its applications} (\href{http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf}{pdf})

\end{itemize}
The version of the definition in (\hyperlink{Joyal}{Joyal}) is also highlighted in

\begin{itemize}%
\item [[Emily Riehl]], \emph{A concise definition of model category}, 2009 (\href{http://www.math.jhu.edu/~eriehl/modelcat.pdf}{pdf})

\end{itemize}
An introductory survey of some key concepts is in the set of slides

\begin{itemize}%
\item [[Peter May]], \emph{[[ModelCatPrimer.pdf:file]]}

\end{itemize}
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\xspace , and the author has given \href{http://www-math.mit.edu/~psh/#Other%20mathematics}{permission} for it to be posted on the net:

\begin{itemize}%
\item [[Chris Reedy]], \emph{Homotopy theory of model categories} (\href{http://www-math.mit.edu/~psh/reedy.pdf}{pdf})

\end{itemize}
More recent review includes

\begin{itemize}%
\item [[William Dwyer]], J. Spalinski, \emph{[[Homotopy theories and model categories]]} (\href{http://folk.uio.no/paularne/SUPh05/DS.pdf}{pdf}) in [[Ioan Mackenzie James]] (ed.), \emph{[[Handbook of Algebraic Topology]]} 1995


\item [[Paul Goerss]], [[Rick Jardine]], chapter 1 of \emph{[[Simplicial homotopy theory]]}, Birkh\"a{}user, 1999, 2009


\item [[Paul Goerss]], [[Kristen Schemmerhorn]], \emph{Model Categories and Simplicial Methods} (\href{http://arxiv.org/abs/math.AT/0609537}{arXiv})



\end{itemize}
Monographs:

\begin{itemize}%
\item [[Philip Hirschhorn]], \emph{Model Categories and Their Localizations}, AMS Math. Survey and Monographs Vol 99 (2002) (\href{http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99}{AMS}, \href{http://www.gbv.de/dms/goettingen/360115845.pdf}{pdf toc}, \href{http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf}{pdf})


\item [[Mark Hovey]], \emph{Model Categories} Mathematical Surveys and Monographs, Volume 63, AMS (1999) (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf}{pdf}, \href{http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover}{Google books})



\end{itemize}
See

\begin{itemize}%
\item [[Philip Hirschhorn]], personal website: \emph{\href{http://www-math.mit.edu/~psh/#Mathematics}{Mathematics}}

\end{itemize}
for errata and more.

\begin{itemize}%
\item [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], \emph{[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]} , volume 113 of Mathematical Surveys and Monographs

\end{itemize}
For yet another introduction to model categories, with an eye towards their use as [[presentable (infinity,1)-category|presentations]] of $(\infty,1)$-[[(infinity,1)-category|categories]] see

\begin{itemize}%
\item [[Jacob Lurie]], appendix A.2 of \emph{[[Higher Topos Theory]]}

\end{itemize}
[[!redirects model categories]] [[!redirects model structure]] [[!redirects model structures]] [[!redirects Quillen model structure]] [[!redirects Quillen model structures]] [[!redirects Quillen model category]] [[!redirects closed model category]] [[!redirects closed Quillen model category]] [[!redirects Quillen model categories]] [[!redirects closed model categories]] [[!redirects closed Quillen model categories]]



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