model category

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
* table of contents
{:toc}

## 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 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 _[[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 [[simplicially enriched category|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 [[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.

## Definition

The following is a somewhat terse account. For a more detailed exposition see  at _[[Introduction to Homotopy Theory]]_ the section _[Abstract homotopy theory](Introduction+to+Homotopy+Theory#ModelCategoryTheory)_.

<br>

+-- {: .num_defn #ModelStructure}
###### 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);

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

=--

+-- {: .num_defn #ModelCategory}
###### Definition

A **model category** is a [[complete category|complete]] and [[cocomplete category]] $\mathcal{C}$ equipped with a model structure according to def. \ref{ModelStructure}.

=--

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

+-- {: .num_defn }
###### 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.

=--

+-- {: .num_remark }
###### 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](#DwyerHirschhornKan97), published as [Dwyer, Hirschhorn, Kan & Smith 2004](#DwyerHirschhornKanSmith2004).

[[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](https://arxiv.org/abs/math/0106152).)

### 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 functor|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 category|Semimodel categories]] relax some of the conditions on lifting properties.

* [[weak model category|Weak model categories]] relax these conditions even further.

* [[premodel category|Premodel categories]] are an even weaker notion that allows for a nice 2-categorical treatment.

## Properties

### Closure of morphism classes under retracts
 {#ClosureOfMorphisms}

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](weak+factorization+system#ClosureProperties)_.

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:

+-- {: .num_prop #WeakEquivalencesAreClosedUnderRetracts}
###### Proposition

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]]).

=--

([Joyal, prop. E.1.3](#Joyal)), highlighted in ([Riehl 09](#Riehl09))

+-- {: .proof}
###### 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.](weak+factorization+system#ClosuredPropertiesOfWeakFactorizationSystem).

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.](weak+factorization+system#ClosuredPropertiesOfWeakFactorizationSystem), 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
 {#RedundancyInTheAxioms}

It is clear that:

+-- {: .num_remark #AnyTwooClassesDetermineTheThird}
###### 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.

+-- {: .num_prop}
###### 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](#Joyal))

+-- {: .proof}
###### 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

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

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
 {#BasicClosurePropertiesOfClassOfModelCategories}

\begin{proposition}\label{ProductModelStructure}
**(product model structure)**
\linebreak
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 coproduct|dependent]] $S$-[[tuples]] of objects and morphisms in the $\mathcal{C}_i$.
\end{proposition}
(e.g. [Hovey (1999), Exp. 1.1.6](#Hovey99))

\begin{proposition}\label{OppositeModelStructure}
**([[opposite model structure]])**
\linebreak
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{proposition}
(e.g. [Hovey (1999), Rem. 1.1.7](#Hovey99))

\begin{proposition}
**([[transferred model structure]] along [[adjoint equivalence]])**
\linebreak
  Given an [[right adjoint|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]].
\end{proposition}
(cf. [this Example](transferred+model+structure#RightTransferAlongAdjointEquivalence) 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

\begin{example}\label{TrivialModelStructure}
**(trivial model structure)**
\linebreak
Every [[bicomplete category]] carries its *[[trivial model structure]]* whose weak equivalences are the [[isomorphisms]] and all morphisms are both cofibrations and fibrations.
\end{example}

\begin{remark}
Trivial as this example \ref{TrivialModelStructure} may be, to the extent that model categories are presentations of certain [[(infinity,1)-categories|$\infty$-categories]] it shows that/how ordinary ([[bicomplete category|bicomplete]]) [[1-categories]] are subsumed as a special case.
\end{remark}

### 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 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 _[[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](#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 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

* [[canonical model structure]]s, 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 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]].

### 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 [[models for infinity-stack (infinity,1)-toposes|model for ∞-stack (∞,1)-toposes]] (for [[hypercomplete (∞,1)-topos]]es, 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 limit|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]]

[[!include algebraic model structures - table]]

## References
 {#References}

The concept of a model category originates with

* {#Quillen67} [[Daniel Quillen]], Chapter I, _Axiomatic homotopy theory_ in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics **43** Springer (1967) [[doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)]

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

* {#DwyerHirschhornKan97} [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], *[[Model Categories and More General Abstract Homotopy Theory]]* (1997) [[pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dhk.pdf), [[DwyerHirschhornKan-ModelCategories.pdf:file]]]

#### Monographs and textbooks

* {#Hovey99} [[Mark Hovey]], *[[Model Categories]]*, Mathematical Surveys and Monographs, **63** AMS (1999) [[ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover)]

* {#Hirschhorn02} [[Philip Hirschhorn]], _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

* {#DwyerHirschhornKanSmith04} [[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](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.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](https://doi.org/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

* {#GoerssJardine99} [[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 [[presentable (infinity,1)-category|presentations]] of $(\infty,1)$-[[(infinity,1)-category|categories]] see

* [[Jacob Lurie]], Appendix A.2 and A.3 of _[[Higher Topos Theory]]_

#### Survey articles

* {#DwyerSpalinski95} [[William Dwyer]], [[Jan Spalinski]], *[[Homotopy theories and model categories]]* ([[DwyerSpalinski_HomotopyTheories.pdf:file]])

  in: [[Ioan Mackenzie James|I. M. James]], *[[Handbook of Algebraic Topology]]*, North Holland 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/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](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

#### Other sources

An account is in

* [[joyalscatlab:HomePage|Joyal's CatLab]], _[[joyalscatlab:Model categories]]_

and appendix E of

* {#Joyal} [[André Joyal]], _The theory of quasi-categories and its applications_ ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasicategories.pdf:file]])

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

* {#Riehl09} [[Emily Riehl]], _A concise definition of model category_, 2009 ([pdf](http://www.math.jhu.edu/~eriehl/modelcat.pdf), [[RiehlConciseDefinitionOfModelCategory.pdf:file]])

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

* [[Peter May]], _[[ModelCatPrimer.pdf:file]]_

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](http://www-math.mit.edu/~psh/#Other%20mathematics) for it to be posted on the net:

* [[Chris Reedy]], _Homotopy theory of model categories_ ([pdf](http://www-math.mit.edu/~psh/reedy.pdf))

See

* [[Philip Hirschhorn]], personal website: _[Mathematics](http://www-math.mit.edu/~psh/#Mathematics)_

for errata and more.

[[!redirects model categories]]

[[!redirects model structure]]
[[!redirects model structures]]

[[!redirects model category theory]]

[[!redirects model category structure]]
[[!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]]


model category theory - contents

**[[model category]]**, [[model (infinity,1)-category|model $\infty$-category]]

**Definitions**

* [[category with weak equivalences]]

  ([[relative category]], [[homotopical category]])

* [[fibration]], [[cofibration]]

* [[weak factorization system]]

* [[resolution]]

* [[cell complex]]

* [[small object argument]]

* [[homotopy (as an operation)|homotopy]]

* [[homotopy category]]$\;$[[homotopy category of a model category|of a model category]]

**Morphisms**

* [[Quillen adjunction]]

  * [[Quillen equivalence]]

  * [[Quillen bifunctor]]

  * [[derived functor]]

**Universal constructions**

* [[homotopy Kan extension]]

* [[homotopy limit]]/[[homotopy colimit]]

  [[homotopy weighted colimit|homotopy weighted (co)limit]]

  [[homotopy coend|homotopy (co)end]]

* [[Bousfield-Kan map]]

**Refinements**

* [[monoidal model category]]

  * [[monoidal Quillen adjunction]]

* [[enriched model category]]

  * [[enriched Quillen adjunction]]

* [[monoidal enriched model category]]

* [[simplicial model category]]

  * [[simplicial Quillen adjunction]]

  * [[simplicial monoidal model category]]

* [[cofibrantly generated model category]]

  * [[combinatorial model category]]

  * [[cellular model category]]

* [[algebraic model category]]

* [[compactly generated model category]]

* [[proper model category]]

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* [[stable model category]]

**Producing new model structures**

* [[global model structures on functor categories|on functor categories (global)]]

  * [[Reedy model structure]]

* [[model structure on an overcategory|on slice categories]]

* [[Bousfield localization of model categories|Bousfield localization]]

* [[transferred model structure]]

  * [[model structure on algebraic fibrant objects|on algebraic fibrant objects]]

* [[Grothendieck construction for model categories]]

**Presentation of $(\infty,1)$-categories**

* [[(∞,1)-category]]

* [[simplicial localization]]

* [[(∞,1)-categorical hom-space]]

* [[presentable (∞,1)-category]]

**Model structures**

* [[Cisinski model structure]]

*for $\infty$-groupoids*

[[model structure for ∞-groupoids|for ∞-groupoids]]

* [[model structure on topological spaces|on topological spaces]]

  * [[classical model structure on topological spaces|classical model structure]]

  * [[model structure on compactly generated topological spaces|on compactly generated spaces]]

  * [[model structure on Delta-generated topological spaces|on Delta-generated spaces]]

  * [[model structure on diffeological spaces|on diffeological spaces]]

  * [[Strøm model structure]]

* [[Thomason model structure]]

* [[model structure on presheaves over a test category]]

* [[model structure on simplicial sets|on simplicial sets]], [[model structure on semi-simplicial sets|on semi-simplicial sets]]

  * [[classical model structure on simplicial sets|classical model structure]]

  * [[constructive model structure on simplicial sets|constructive model structure]]

  * [[model structure for right fibrations|for right/left fibrations]]

* [[model structure on simplicial groupoids]]

* [[model structure on cubical sets|on cubical sets]]

* [[model structure on strict ∞-groupoids|on strict ∞-groupoids]], [[natural model structure on groupoids|on groupoids]]

* [[model structure on chain complexes|on chain complexes]]/[[model structure on cosimplicial abelian groups]]

   related by the [[Dold-Kan correspondence]]

* [[model structure on cosimplicial simplicial sets]]

*for equivariant $\infty$-groupoids*

* [[fine model structure on topological G-spaces]]

* [[coarse model structure on topological G-spaces]]

  ([[Borel model structure]])

*for rational $\infty$-groupoids*

* [[model structure on dgc-algebras]]

*for rational equivariant $\infty$-groupoids*

* [[model structure on equivariant chain complexes]]

* [[model structure on equivariant dgc-algebras]]

*for $n$-groupoids*

* [[model structure for n-groupoids|for n-groupoids]]/[[model structure for n-groupoids|for n-types]]

* [[canonical model structure on groupoids|for 1-groupoids]]

*for $\infty$-groups*

* [[model structure on simplicial groups]]

* [[model structure on reduced simplicial sets]]

*for $\infty$-algebras*

*general $\infty$-algebras*

* [[model structure on monoids|on monoids]]

* [[model structure on simplicial T-algebras|on simplicial T-algebras]], on [[homotopy T-algebra]]s

* [[model structure on algebras over a monad|on algebas over a monad]]

* [[model structure on algebras over an operad|on algebras over an operad]],
  [[model structure on modules over an algebra over an operad|on modules over an algebra over an operad]]

*specific $\infty$-algebras*

* [[model structure on dg-algebras|model structure on differential-graded commutative algebras]]

* [[model structure on differential graded-commutative superalgebras]]

* [[model structure on dg-algebras over an operad|on dg-algebras over an operad]]

  * [[model structure on dg-algebras|on dg-algebras]] and on [[simplicial ring|on simplicial rings]]/[[model structure on cosimplicial rings|on cosimplicial rings]]

    related by the [[monoidal Dold-Kan correspondence]]

  * [[model structure for L-∞ algebras|for L-∞ algebras]]: [[model structure on dg-Lie algebras|on dg-Lie algebras]],  [[model structure on dg-coalgebras|on dg-coalgebras]], [[model structure on simplicial Lie algebras|on simplicial Lie algebras]]

* [[model structure on dg-modules]]

*for stable/spectrum objects*

* [[model structure on spectra]]

* [[model structure on ring spectra]]

* [[model structure on parameterized spectra]]

* [[model structure on presheaves of spectra]]

*for $(\infty,1)$-categories*

* [[model structure on categories with weak equivalences|on categories with weak equivalences]]

* [[model structure on quasi-categories|Joyal model for quasi-categories]] (and its [[model structure for cubical quasicategories|cubical version]])

* [[model structure on sSet-categories|on sSet-categories]]

* [[model structure for complete Segal spaces|for complete Segal spaces]]

* [[model structure for Cartesian fibrations|for Cartesian fibrations]]

*for stable $(\infty,1)$-categories*

* [[model structure on dg-categories|on dg-categories]]

*for $(\infty,1)$-operads*

* [[model structure on operads|on operads]], [[model structure for Segal operads|for Segal operads]]

  [[model structure on algebras over an operad|on algebras over an operad]],
  [[model structure on modules over an algebra over an operad|on modules over an algebra over an operad]]

* [[model structure on dendroidal sets|on dendroidal sets]], [[model structure for dendroidal complete Segal spaces|for dendroidal complete Segal spaces]], [[model structure for dendroidal Cartesian fibrations|for dendroidal Cartesian fibrations]]

*for $(n,r)$-categories*

* [[Theta space|for (n,r)-categories as ∞-spaces]]

* [[model structure for weak complicial sets|for weak ∞-categories as weak complicial sets]]

* [[model structure on cellular sets|on cellular sets]]

* [[canonical model structure|on higher categories in general]]

* [[model structure on strict ∞-categories|on strict ∞-categories]]

*for $(\infty,1)$-sheaves / $\infty$-stacks*

* [[model structure on homotopical presheaves|on homotopical presheaves]]

  * [[model structure on simplicial presheaves|on simplicial presheaves]]

    [[global model structure on simplicial presheaves|global model structure]]/[[Cech model structure on simplicial presheaves|Cech model structure]]/[[local model structure on simplicial presheaves|local model structure]]

    [[model structure on simplicial sheaves|on simplicial sheaves]]

    [[model structure on presheaves of simplicial groupoids|on presheaves of simplicial groupoids]]

    [[model structure on sSet-enriched presheaves|on sSet-enriched presheaves]]

* [[model structure for (2,1)-sheaves]]/for stacks


model complete theory

[[!redirects model complete theory ]]
[[!redirects model complete theory ]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
Model completeness is a [[property]] of [[first-order theories]] (in the same way that [[existential closedness]] is a [[property]] of [[first-order structures]]) meant to generalize the properties of the theory [[ACF]] of [[algebraically closed fields]].

As discussed in Hodges' [Model theory](#Hodges1993):

> In the early 1950s [[Abraham Robinson]] noticed that certain maps studied by algebraists are in fact [[elementary embeddings]]. If you choose a map at random, the chances of it being an elementary embedding are negligible. So Robinson reckoned that there must be a systematic reason for the appearance of these elementary embeddings, and he set out to find what the reason was.

> In the course of this quest he introduced the notions of model complete theories, companionable theories and [[model companion|model companions]]. These notions have become essential tools for the model theory of algebra.

## Definition

We say a [[first-order theory]] $T$ is *model complete* if every embedding (not necessarily elementary) between models of $T$ _is_ elementary.

The following characterization appears as Theorem 8.3.1 in ([Hodges93](#Hodges1993)):

+-- {: .num_theorem}
###### Theorem

Let $T$ be as above. The following are equivalent:

1. $T$ is model complete.

2. Every model of $T$ is [[existentially closed model|existentially closed]].

3. For every embedding $A \hookrightarrow B$ where $A$ and $B$ model $T$, there exists an [[elementary extension]] $D$ of $A$ and an embedding $g : B \to D$ such that the triangle commutes.

4. Every [[existentially closed model|existential formula]] is equivalent (mod $T$) to a [[existentially closed model|universal formula]].

5. Every formula is equivalent (mod $T$) to a universal formula.

=--

## Examples

* Since [[quantifier elimination]] is equivalent to [[substructure completeness]], which is stronger than model completeness, every theory which eliminates quantifiers is model complete. So, for example, [[ACF]], [[DLO]], [[RCF]], and the theory of the [[countable random graph]] are all model complete (and so their models are [[existentially closed]].)

## Remarks

* A theory $T$ is model complete if and only if for every _model_ $A$ of $T$, the [[quantifier-free diagram]] $T_{\mathsf{Diag}(A)}$ (whose models are precisely the models of $T$ containing $A$ as an embedded substructure) is _complete_.

* Analogously, a theory $T$ is [[substructure complete]] if and only if for every model $A$ of $T$ and every _substructure_ $B \subseteq A$, the [[quantifier-free diagram]] $T_{\mathsf{Diag}(B)}$ is _complete_.

* A model complete theory is an "existentially closed theory" in the sense that all of its models are [[existentially closed model|existentially closed models]].

* As mentioned above, [[substructure completeness]] implies model completeness because every elementary submodel is also a substructure. The proposition below shows that the converse holds with some additional assumptions.

## Model completeness + amalgamation property implies substructure completeness

+-- {: .num_prop #substructureconverse}
###### Proposition

Suppose the [[first-order theory]] $T$ in the language $\mathcal{L}$ is model complete and has the property that for any two models $X,Y \models T$ and a mutual $\mathcal{L}$-substructure $A \hookrightarrow X,Y$, the latter [[span]] in the category of $\mathcal{L}$-structures and embeddings has a [[cocone]] $Z$ which is also a model of $T$.

Then $T$ is [[substructure complete]].

=--

This property of being able to "amalgamate" the models $X$ and $Y$ over the common substructure $A$ is sometimes called the **amalgamation property**, though this terminology is rather overloaded (c.f. [[Fraisse limit]]).

+-- {: .proof}
###### Proof

Let $A$ be a substructure of some model $M \models T$. Append the [[quantifier-free diagram]] of $A$ to $T$ to form $T_{\mathsf{Diag}(A)}$. Suppose towards a contradiction that $\varphi$ is a sentence which is undecided by $T_{\mathsf{Diag}(A)}$; let $X$ and $Y$ be models of $T_{\mathsf{Diag}(A)}$ which witness this, so that $X \models \varphi$ while $Y \models \neg \varphi$.

Let $Z$ amalgamate $X$ and $Y$ over $A$.

By [[model completeness]], the theory $T_{\mathsf{Diag}(X)}$ of $T$-models (not necessarily [[elementary embedding|elementarily]]) embedding $X$ is complete. Abuse notation and replace the instances of constants from $A$ in $\varphi$ with their images in $X$. Either $T_{\mathsf{Diag}(X)} \models \varphi$ or $T_{\mathsf{Diag}(X)} \models \neg \varphi$. Since $X$ certainly embeds itself and $X \models \varphi$, $T_{\mathsf{Diag}(X)} \models \varphi$.

Similarly, $T_{\mathsf{Diag}(Y)} \models \neg \varphi.$

Now, on one hand, the theory $T_{\mathsf{Diag}(X)} \cup T_{\mathsf{Diag}(Y)}/A$ (where we glue along the copies of $A$ in $\mathsf{Diag}(X)$ and $\mathsf{Diag}(Y)$) entails both $\varphi$ and $\neg \varphi$, which makes it inconsistent. This theory therefore has no models.

On the other hand, this theory axiomatizes the class of amalgams of $X$ and $Y$ over $A$ in the category of $\mathcal{L}$-structures and embeddings, so in particular is modelled by $Z$, a contradiction.
=--

## Related concepts

* [[quantifier elimination]]

* [[model companion]]

* [[substructure completeness]]

* [[existentially closed model]]

## References

* {#Hodges1993}Wilfrid Hodges, _Model theory_, Cambridge Univ. Press 1993

[[!redirects model completeness]]
[[!redirects model-complete theory]]
[[!redirects model-completeness]]
[[!redirects model complete]]


model invariance problem

Show that the interpretation of type theory is independent of the [[model category]] chosen to present an [[(infinity,1)-category]].  Of course, the details depend on the chosen [[type theories|type theory]].


model of type theory in an (infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Overview

It is expected that every [[elementary (infinity,1)-topos]] may serve as a [[categorical semantics|model]] of [[homotopy type theory]] with the [[univalence axiom]] and [[higher inductive types]].  However, so far there is not a standard definition of elementary $(\infty,1)$-topos.  We do *almost* know that every [[Grothendieck (infinity,1)-topos]] admits such a model; the caveat is that we only get univalent [[universes]] that are *weakly a la Tarski* (see [[universe]]).

## Right proper Cisinski model categories

A [[Cisinski model category]] is a [[model category]] on a Grothendieck 1-topos whose cofibrations are the monomorphisms.  It is [[proper model category|right proper]] if weak equivalences are preserved by pullback along fibrations.  This ensures that the adjunctions $f^* \dashv f_*$ for a fibration $f$ are [[Quillen adjunctions]], so that the $(\infty,1)$-category presented by the model category is locally cartesian closed.

The fibrant objects in a right proper Cisinski model category can be shown to form a comprehension category, one of the structures used for [[semantics]] of type theory, which admits all the structure necessary to model the type-forming operations of the [[dependent type|dependent sum type]], [[dependent type|dependent product type]], and [[identity type|identity type]] (the latter by using path objects).  By applying the [[local universe model]] or [[natural model]] coherence theorem, we see that any right proper Cisinski model category models type theory.

Cisinski and Gepner-Kock have also shown that any locally presentable, locally cartesian closed $(\infty,1)$-category can be presented by a right proper Cisinski model category.  Therefore, any such $(\infty,1)$-category admits a model of type theory.

## Higher inductive types

Models of higher inductive types can be constructed in any simplicial combinatorial model category; for now, see the note at [[higher inductive types]].

## Strong univalent universes

In some cases we can construct a univalent universe which satisfies the rules to be a "strong" universe a la Tarski.  This means we have an object $U$ together with a fibration $\tilde{U}\to U$, which is "closed under the type-forming operations" up to isomorphism.  For instance, if $B\to A$ is a pullback of $\tilde{U}$ along some map $A\to U$, and likewise $C\to B$ is a pullback of $\tilde{U}$ along some map $B\to U$, then the composite $C\to A$ is a pullback of $\tilde{U}$ along some map $A\to U$.  In particular, we can apply this to the "generic" composable pair of fibrations over $U^{(1)}$, the "type of composable fibrations", to get a classifying map $U^{(1)}\to U$ of the composite that interprets the $\Sigma$-type former on the universe.  The requirements for $\Pi$-types and identity types are similar.

In practice, the way we ensure these requirements is to show that a fibration is a pullback of $\tilde{U}\to U$ if and only if its "fibers" are bounded by some cardinal number $\kappa$ (used in defining $U$).  Then as long as $\kappa$ is inaccessible, such fibrations will be closed under the type-forming operations.

Strong univalent universes are known to exist in the Reedy model category $sSet^{R^op}$ of simplicial presheaves on an [[elegant Reedy category]] $R$; see [this paper](http://arxiv.org/abs/1203.3253) and [this paper](http://arxiv.org/abs/1307.6248).

## Weak univalent universes

Suppose we have any right proper Cisinski model category that presents a Grothendieck $(\infty,1)$-topos; we will show that it models a [[univalent]] [[universe]] which is "[[weakly Tarski universe|weakly a la Tarski]]".  (Since any Grothendieck $(\infty,1)$-topos is locally cartesian closed, such a model category always exists as remarked above.)

It is known that for any regular $\kappa$, an $(\infty,1)$-topos has an [[object classifier]] for $\kappa$-small morphisms, i.e. a $\kappa$-small morphism $\tilde{V}\to V$ such that for any object $A$, the space of maps $A\to V$ is naturally equivalent to the core of the category of $\kappa$-small morphisms into $A$.

Let $\kappa$ be inaccessible, and let $\tilde{U} \to U$ be a fibration between fibrant objects of the model category that represents $\tilde{V}\to V$.  Then since $\kappa$-small morphisms in the $(\infty,1)$-topos are closed under composition, diagonals, and dependent products, the fibrations in the model category that are homotopy pullbacks of $\tilde{U}\to U$ are also so closed.  In particular, we can again build the universal composable pair over $U^{(1)}$ and obtain a map $U^{(1)}\to U$ which classifies its composite *up to homotopy* (i.e. so that the composite is a homotopy pullback of $\tilde{U}\to U$ along it).  Dependent products and identity types are similar.

Thus, if we use the local universe coherence theorem, $U$ represents a "universe" which comes with type-forming "operations", but these operations do not literally respect the actual type-formers on actual types.  Instead we can say only that $El(\Sigma A B)$ is *equivalent* to $\Sigma (El A) (El B)$, and so on.  However, we can still define a map $Id_U(A,B) \to Equiv(El(A),El(B))$, and the full universal property of the object classifier implies that it is an equivalence (see for instance [Gepner-Kock](http://arxiv.org/abs/1208.1749)).  Thus, we have a univalent universe which is "weakly a la Tarski".

## Related entries

* [[internal logic of an (infinity,1)-topos|internal logic of an $(\infty,1)$-topos]]

* [[Awodey's proposal]]

* [[relation between type theory and category theory]]

* [[semantics of Set in homotopy type theory]]

## References

* [[Denis-Charles Cisinski]], _Univalent universes for elegant models of homotopy types_, 2014 [PDF](http://arxiv.org/pdf/1406.0058)

Full proof of [[Awodey's conjecture]] that $\infty$-stack $\infty$-toposes model univalent homotopy type theory:

* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004))

Review:

* {#Riehl22} [[Emily Riehl]], *On the $\infty$-topos semantics of homotopy type theory*, lecture at *[Logic and higher structures](https://conferences.cirm-math.fr/2689.html)* CIRM (Feb. 2022) $[$[pdf](https://emilyriehl.github.io/files/semantics.pdf), [[Riehl-InfinityToposSemantics.pdf:file]]$]$


model site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **model site** is a [[model category]] model for the notion of [[(∞,1)-site]].

An [[(∞,1)-category]] equipped with the structure of a [[site]] on its [[homotopy category of an (∞,1)-category|homotopy category]] serves as a domain for [[∞-stack]]s. It may be modeled by a [[model category]].

A [[model category]] with a [[site]] structure on its [[homotopy category]] is called a pseudo **model site** ([def 4.1.1](http://arxiv.org/PS_cache/math/pdf/0207/0207028v4.pdf#page=43) in [ToVeI](http://arxiv.org/abs/math/0404373)).

## Examples

* The [[model structure on simplicial algebras]] equipped with the covering of the [[etale (infinity,1)-site]].

## Related concepts

* [[site]]

* [[2-site]]

* [[(∞,1)-site]]

  * **model site**, [[simplicial site]]

  * [[model topos]]

## References

* [[Bertrand Toën]], [[Gabriele Vezzosi]],  _Homotopical Algebraic Geometry I: Topos theory_ ([arXiv](http://arxiv.org/abs/math/0207028))
{#ToenVezzosi}

[[!redirects model topology]]
[[!redirects model pretopology]]

[[!redirects model sites]]


model stack

The notion of model stack is an adaptation of [[model categories]] to [[stacks]]. [[Nikolai Durov]] ([arXiv](http://arxiv.org/abs/0704.2030)) has studied them in passing and noted that a more flexible notion for that context is the weaker notion of [[pseudomodel stack]] which he introduced there.

A [[stack]] $C$ over a [[site]] $S$ is a __model stack__ if in every fiber three local classes of morphisms are distinguished: fibrations, cofibrations and weak equivalences (with the terminology that an acyclic (co)fibration is a (co)fibration which is also a weak equivalence) and

(MS1) all fiber categories are [[finitely complete category|finitely complete]] and [[finitely cocomplete category|finitely cocomplete]] and all [[cartesian morphism]]s in $C$ admit [[left adjoint|left]] and [[right adjoint|right]] adjoints,

(MS2) each distinguished class is local and stable under global [[retracts]],

(MS3) (2-out-of-3) If $f$, $g$, $g\circ f$ are defined morphisms in a fixed fiber, then if any two of the three are weak equivalences then so is the third,

(MS4) any cofibration has the [[local left lifting property]] with respect to any acyclic fibration, and any acyclic fibration has the local left lifting property with respect to any fibration,

(MS5) (factorization) Any morphism can be globally factored as a cofibration followed by a fibration, where one can choose either one of the two acyclic.

[[!redirects model stacks]]


model structure for (2,1)-sheaves

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _model structure of $(2,1)$-sheaves_ is a [[model category]] [[presentable (infinity,1)-category|presentation]] of the [[(2,1)-category of (2,1)-sheaves]] over some [[site]] or [[(2,1)-site]].

## Definition

There are several equivalent ways to set up a [[model category]] structure for $(2,1)$-sheaves.

Suppose first that the [[(2,1)-site]] $C$ is just a 1-[[category]], hence just a [[site]].

The following definition first defines a model presentation for [[(2,1)-presheaves]] ([[1-truncated]] [[(∞,1)-presheaves]]) and then localizes at the [[covering]] morphisms in order to obtain the $(2,1)$-sheaves.

+-- {: .un_defn}
###### Definition

Write [[Grpd]] for the [[category]] of [[small category]] [[groupoid]]s and [[functor]]s between them. Write $Grpd_{nat}$ for the [[natural model structure on groupoids]].

Write $[C^{op}, Grpd_{nat}]_{proj}$ for the projective [[model structure on functors]] on the [[functor category]] $[C^{op}, Grpd]$.

Let $W = \{ C(\{U_i\})\to j(U) \}$ be the set of [[Cech nerve]] projections in $[C^{op}, Grpd]$ for each [[covering]] family $\{U_i \to U\}$ in the [[site]] $C$.

Then let finally

$$
  [C^{op}, Grpd_{nat}]_{proj,loc}
$$

be the [[Bousfield localization of model categories|left Bousfield localization]] at the set of morphisms $W$.

=--

The following definition first gives the presentations for [[(∞,1)-sheaves]] and then further restricts the [[1-truncated]] objects in there, preseting the [[(n,1)-topos]] inside the full [[(∞,1)-topos]] over $C$, for $n = 2$.

+-- {: .un_defn}
###### Definition

Write $[C^{op}, sSet_{Quillen}]_{loc}$ for a local [[model structure on simplicial presheaves]] on $C$, the one which presents the [[(∞,1)-category of (∞,1)-sheaves]] on $C$.

Let $W = \{\partial \Delta[n] \cdot U \to \Delta[n] \cdot U| n \geq 2 \in \mathbb{N}, U \in C\}$ be the set of generating morphisms of weak equivalences on [[homotopy n-type|homotopy 1-type]]s.

Write

$$
  [C^{op}, sSet_{Quillen}]_{loc,W}
$$

for the [[Bousfield localization of model categories|left Bousfield localization]] of the model structure for [[(∞,1)-sheaves]] at the morphisms $W$. Then this is a model structure for $(2,1)$-sheaves on $C$.

=--

These two model structures are equivalent:

+-- {: .un_prop}
###### Proposition

Let

$$
  (\tau \dashv N) : Grpd \stackrel{\overset{\tau}{\leftarrow}}{\underset{N}{\to}}
  sSet
$$

be the [[nerve]] functor and its [[left adjoint]] $\tau$. Postcomposition with this induces a [[Quillen adjunction]]

$$
  (\tau_* \dashv N_*)
  :
  [C^{op}, Grpd_{nat}]_{loc}
  \underoverset{\underset{N_*}{\to}}{\overset{\tau_*}{\leftarrow}}{\simeq}
  [C^{op}, sSet_{Quillen}]_{loc, W}
$$

that is a [[Quillen equivalence]].

=--

This appears as ([Hollander, theorem 5.4](#Hollander)).

## Related concepts

* **model structure for $(2,1)$-sheaves**

* model structures for $(\infty,1)$-sheaves

  * [[model structure on simplicial presheaves]]

  * [[model structure on simplicial sheaves]]

## References

A model structure on presheaves of groupoids Quillen equivalent to the [[left Bousfield localization]] of the local [[model structure on simplicial presheaves|model structure for (∞,1)-sheaves]] at morphisms that are weak equivalences of [[homotopy n-type|homtopy 1-type]]s is in.

* [[Sharon Hollander]], _A homotopy theory for stacks_ ([arXiv:math.AT/0110247](http://arxiv.org/abs/math.AT/0110247))
{#Hollander}

A discussion of $(2,1)$-sheaves/stacks as [[1-truncated]] objects in the full [[model structure on simplicial presheaves|model structure for (∞,1)-sheaves]] is in

* J. F. Jardine, _Stacks and the homotopy theory of simplicial sheaves_ ,  Homology Homotopy Appl. Volume 3, Number 2 (2001), 361-384. ([project euclid](http://projecteuclid.org/euclid.hha/1139840259))
{#Jardine}

[[!redirects (2,1)-sheaves]]
[[!redirects (2,1)-category of (2,1)-sheaves]]


model structure for Cartesian fibrations

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The [[model category]] structure on the [[category]] $SSet^+/S$ of [[marked simplicial set]]s over a given [[simplicial set]] $S$ is a [[presentable (infinity,1)-category|presentation]] for the [[(∞,1)-category]] of [[Cartesian fibration]]s over $S$.
Every object is cofibrant and the fibrant objects of $SSet^+/S$ are precisely the [[Cartesian fibration]]s over $S$.

Notably for $S = {*}$ this is a presentation of the [[(∞,1)-category of (∞,1)-categories]]: as a plain [[model category]] this is [[Quillen equivalence|Quillen equivalent]] to the [[model structure for quasi-categories]], but it is indeed an $sSet_{Quillen}$-[[enriched model category]] (i.e. enriched over the ordinary [[model structure on simplicial sets]] that models [[∞-groupoid]]s).

The $(\infty,1)$-categorical [[Grothendieck construction]] that exhibits the correspondence between [[Cartesian fibration]]s and [[(∞,1)-presheaf|(∞,1)-presheaves]] is in turn modeled by a [[Quillen equivalence]] between the model structure on marked simplicial over-sets and the projective [[global model structure on simplicial presheaves]].

## Marked simplicial sets

Marked simplicial sets are [[simplicial set]]s with a little bit of extra [[stuff, structure, property|structure]]: a marking that remembers which edges are supposed to be [[Cartesian morphism]]s.

### Definition

#### In components

+-- {: .un_def }
###### Definition

A **marked simplicial set** is

* a pair $(S,E)$ consisting of

  * a [[simplicial set]] $S$

  * and a subset $E \subset S_1$ of edges of $S$, called the _marked edges_,

* such that

  * all degenerate edges are marked edges.

A morphism $(S,E) \to (S',E')$ of marked simplicial sets is a morphism $f : S \to S'$ of [[simplicial set]]s that carries marked edges to marked edges in that $f(E) \subset E'$.

=--

+-- {: .un_def }
###### Notation

* The category of marked simplicial sets is denoted $sSet^+$.

* for $S$ a [[simplicial set]] let

  * $S^\flat$ or $S^{min}$ be the minimally marked simplicial set: only the degenerate edges are marked;

  * $S^\sharp$ or $S^{max}$ be the maximally marked simplicial set: every edge is marked.

* for $p : X \to S$ a [[Cartesian fibration]] of [[simplicial set]]s let

  * $X^\natural$ or $X^{cart}$ be the cartesian marked simplicial set: precisely the $p$-[[cartesian morphism]]s are marked

=--

#### As a quasi-topos

Simple as the above definition is, for seeing some of its properties it is useful to think of $sSet^+$ in a more abstract way.

+-- {: .un_def }
###### Definition

Let $\Delta^+$ be the [[category]] defined as the [[simplex category]] $\Delta$, but with one more object $[1^+]$ that factors the unique morphism $[1] \to [0]$ in $\Delta$

$$
  \array{
    [0] &\stackrel{\to}{\to}& [1]
    \\
    {}^{\mathllap{=}}\downarrow &\swarrow& \downarrow^{\mathrlap{p}}
    \\
    [0] &\leftarrow& [1^+]
}
  \,.
$$

Equip this category with a [[coverage]] whose only non-trivial covering family is  $\{p : [1] \to [1]^+\}$.

=--

+-- {: .un_lemma #Quasitopos}
###### Observation

The category $sSet^+$ is the [[quasi-topos]] of [[separated presheaves]] on $\Delta^+$:

$$
  sSet^+ \simeq SepPSh(\Delta^+)
  \,.
$$

=--

+-- {: .proof}
###### Proof

A [[presheaf]] $X : (\Delta^+)^{op} \to Set$ is separated precisely if the morphism

$$
  X(p) : X_{1^+} \to X_1
$$

is a [[monomorphism]], hence if $X_{1^+}$ is a subset of $X_1$. By functoriality this subset contains all the degenerate 1-cells

$$
  \array{
      X_0
      \\
      \downarrow & \searrow^{\mathrlap{\sigma}}
      \\
      X_{1^+} &\hookrightarrow& X_1
  }
  \,.
$$

Therefore we may naturally identify $X$ as a simplicial set equipped with a subset of $X_1$ that contains all degenerate 1-cells.

Moreover, a morphism of separated preseheaves on $\Delta^+$ is by definition just a [[natural transformation]] between them, which means it is under this interpretation precisely a morphism of simplicial sets that respects the marked 1-simplices.

=--

Notice that $sSet^+$ is a genuine quasi-topos:

+-- {: .un_lemma}
###### Observation

$sSet^+$ is not a [[topos]].

=--

+-- {: .proof}
###### Proof

The canonical morphisms $X^\flat \to X^\sharp$ are [[monomorphism]]s and [[epimorphism]]s, but not [[isomorphism]]s. Therefore $sSet^+$ is not a [[balanced category]], hence cannot be a topos.

=--

### Cartesian closure {#CartesianClosure}

+-- {: .un_lemma }
###### Lemma

The category $sSet^+$ is a [[cartesian closed category]].

=--

+-- {: .proof}
###### Proof

This is an immediate consequence of the above [observation](#Quasitopos) that $sSet^+$ is a [[quasitopos]]. But it is useful to spell out the Cartesian closure in detail.

By the general logic of the [[closed monoidal structure on presheaves]] we have that $PSh(\Delta^+)$ is cartesian closed. It remains to check that if $X,Y \in PSh(\Delta^+)$ are marked simplicial sets in that $X_{1^+} \to X_1$ is a monomorphism and similarly for $Y$, that then also $Y^X$ has this property.

We find that the marked edges of $Y^X$ are

$$
  (Y^X)_{1^+} \simeq
  Hom_{PSh(\Delta^+)}([1^+], Y^X) \simeq Hom_{PSh(\Delta^+)}([1^+] \times X, Y)
$$

and the morphism $(Y^X)_{1^+} \to (Y^X)_1$ sends $X \times [1^+] \stackrel{\eta}{\to} Y$ to

$$
  X \times [1] \stackrel{(Id,p)}{\to} X \times [1^+]
  \stackrel{\eta}{\to} Y
  \,.
$$

Now, by construction, every non-identity morphism $U \to [1^+]$ in $\Delta^+$ factors through $U \to [1]$, which implies that if the components of $p^* \eta_1$ and $p^* \eta_2$ coincide on $U \neq [1^+]$, then already the components of $\eta_1$ and $\eta_2$ on $U$ coincided. By assumption on $X$ the values of $\eta_1$ and $\eta_2$ on $U = [1^+]$ are already fixed, due to the inclusion $X_{1^+} \times [1^+]_{1^+} \hookrightarrow X_{1} \times [1^+]_{1}$. Hence $p^*$ is injective, and so $Y^X$ formed in $PSh(\Delta^+)$ is itself a marked simplicial set.

=--

+-- {: .un_def }
###### Definition

* For $X$ and $Y$ marked simplicial sets let

  * $Map^\flat(X,Y)$ be the [[simplicial set]] underlying the [[cartesian closed category|cartesian]] [[internal hom]] $Y^X \in sSet^+$

  * $Map^\sharp(X,Y)$ the simplicial set consisting of all simplices $ \sigma \in Map^\flat(X,Y)$ such that every edge of $\sigma$ is a marked edge of $Y^X$.

=--

+-- {: .un_corolary }
###### Corollary

These mapping complexes are characterized by the
fact that we have natural bijections

$$
  Hom_{sSet}(K, Map^\flat(X,Y))
  \simeq
  Hom_{sSet^+}(K^\flat, Y^X)
  \simeq
  Hom_{sSet^+}(K^\flat \times X, Y)
$$

and

$$
  Hom_{sSet}(K, Map^\sharp(X,Y))
   \simeq
  Hom_{sSet^+}(K^\sharp, Y^X)
   \simeq
  Hom_{sSet^+}(K^\sharp \times X, Y)
$$

for $K \in sSet$ and $X,Y \in sSet^+$. In particular

$$
  Map^\flat(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\flat, Y)
$$

and

$$
  Map^\sharp(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\sharp, Y)
  \,.
$$

=--

In words we have

* The $n$-simplices of the internal hom $Y^X$ are simplicial maps $X \times \Delta[n] \rightarrow Y$ such that when you restrict $X_1 \times \Delta[n]_1 \rightarrow Y_1$ to $E \times \Delta[n]_0$ (where $E$ is the set of marked edges of $X$), this morphism factors through the marked edges of $Y$.

* The marked edges of $Y^X$ are those simplicial maps $X \times \Delta[1] \rightarrow Y$ such that the restriction of $X_1 \times \Delta[1]_1 \rightarrow Y_1$ to $E \times \Delta[1]_1$ factors though the marked edges of $Y$. In the presence of the previous condition, this says that when you apply the homotopy $X \times \Delta[1] \rightarrow Y$ to a marked edge of $X$ paired with the identity at $[1]$, the result should be marked.

+-- {: .un_corolary }
###### Corollary

$Map^\flat(X,Y)$ is full simplicial subset of the internal hom of the underlying simplicial sets spanned by the vertices giving mark preserving maps, in the sense that $\Map^\flat(X,Y)$ contains precisely the simplices whose vertices are mark-preserving.

=--

+-- {: .proof}
###### Proof

The $n$-simplices of this type are the simplicial maps
$X \times \Delta[n] \to Y$ such that, for each point $\Delta[0] \to \Delta[n]$,
the composite $X \times \Delta[0] \to X \times \Delta[n] \to Y$ is mark-preserving.

=--

+-- {: .un_def }
###### Definition

We generalize all this notation from $sSet^+$ to the [[overcategory]] $sSet^+/S := sSet^+/(S^\sharp)$ for any given (plain) simplicial set $S$, by declaring

$$
  Map_S^\flat(X,Y) \subset Map^\flat(X,Y)
$$

and

$$
  Map_S^\sharp(X,Y) \subset Map^\sharp(X,Y)
$$

to be the subcomplexes spanned by the cells that respect that map to the base $S$.

=--

+-- {: .un_lemma }
###### Observation

Let $Y \to S$ be a [[Cartesian fibration]] of simplicial sets, and $X^\natural$ as above the marked simplicial set with precisely the [[Cartesian morphism]]s marked.

Then

* $Map_S^\flat(X,Y^\natural)$ is an [[quasi-category]];

* $Map_S^\sharp(X, Y^\natural)$ is its [[core]], the maximal [[Kan complex]] inside it.

=--

This is [[Higher Topos Theory|HTT, remark 3.1.3.1]].

+-- {: .proof}
###### Proof

The $n$-cells of $Map_S^\flat(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\flat \to Y^\natural$ over $S$.  This means that for fixed $x \in X_0$, $\Delta[n]$ maps into a [[fiber]] of $Y\to S$. But fibers of Cartesian fibrations are fibers of [[inner fibration]]s, hence are quasi-categories.

Similarly, the $n$-cells of $Map_S^\sharp(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\sharp \to Y^\natural$ over $S$.  Again for fixed $x \in X_0$, $\Delta[n]$ maps into a [[fiber]] of $Y\to S$, but now only hitting Cartesian edges there. But (as discussed at [[Cartesian morphism]]), an edge over a point is Cartesian precisely if it is an equivalence.

=--

+-- {: .un_prop }
###### Proposition

We have a sequence of [[adjoint functor]]s

$$
  (-)^{\flat} \dashv (-)_{\flat} \dashv (-)^{\sharp} \dashv (-)_{\sharp}
  :
  \array{
    & \stackrel{(-)^{\flat}}{\to} &
    \\
    & \stackrel{(-)_{\flat}}{\leftarrow} &
    \\
    sSet & \stackrel{(-)^{\sharp}}{\to} & sSet^+
    \\
    & \stackrel{(-)_{\sharp}}{\leftarrow} &
  }
$$

=--

## Model structure on marked simplicial sets

### Cartesian weak equivalences

Observe that weak equivalences in the [[model structure for quasi-categories]] may be characterized as follows.

+-- {: .un_lemma }
###### Lemma

A morphism $f : C \to D$ between simplicial sets that are [[quasi-categories]] is a weak equivalence in the [[model structure for quasi-categories]] precisely if the following equivalent coditions hold:

* For every simplicial set $K$, the morphism $sSet(K,f) : sSet(K,C) \to sSet(K,D)$ is a weak equivalence in the model structure for quasi-categories.

*  For every simplicial set $K$, the morphism $Core(sSet(K,f)) : Core(sSet(K,C)) \to Core(sSet(K,D))$ on the [[core]]s, the maximal [[Kan complex]]es inside, is a weak equivalence in the standard [[model structure on simplicial sets]], hence a [[homotopy equivalence]].

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, lemma 3.1.3.2]].

=--

This may be taken as motivation for the following definition.

+-- {: .un_def }
###### Definition/Proposition

For every [[Cartesian fibration]] $Z \to S$, we have that

$$
  Map_S^\flat(X, Z^{\natural})
$$

is a [[quasi-category]] and

$$
  Map_S^\sharp(X,Z^\natural) = Core(Map_S^\flat(X,Z^\natural))
$$

is the maximal Kan complex inside it.

A morphism $p : X \to Y$ in $sSet^+/S$ is a **Cartesian equivalence** if for every Cartesian fibration $Z$ we have

* The induced morphism $Map_S^\flat(Y,Z^{\natural}) \to Map_S^\flat(X,Z^{\natural})$ is  an [[equivalence of quasi-categories]];

Or equivalently:

* The induced morphism $Map_S^\sharp(Y,Z^{\natural}) \to Map_S^\sharp(X,Z^{\natural})$ is a [[model structure on simplicial sets|weak equivalence of Kan complexes]].

=--

This is [[Higher Topos Theory|HTT, prop. 3.1.3.3]] with [[Higher Topos Theory|HTT, remark 3.1.3.1]].

+-- {: .un_prop }
###### Proposition

Let

$$
  \array{
    X &&\stackrel{p}{\to}&& Y
    \\
    & \searrow && \swarrow
    \\
    && S
  }
$$

be a morphism in $sSet/S$ such that both vertical maps to $S$ are
Cartesian fibrations. Then the following are equivalent:

* $p$ is a [[homotopy equivalence]].

* The induced morphism $X^\natural \to Y^\natural$ in $sSet^+/S$ is a Cartesian equivalence.

* The induced morphism on each fiber $X_s \to Y_{p(s)}$ is a
  weak equivalence in the [[model structure for quasi-categories]].

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, lemma 3.1.3.5]].

=--

### The model structure

The **model structure on marked simplicial over-sets** $Set^+/S$ over $S \in SSet$ -- also called the **Cartesian model structure** since it models [[Cartesian fibration]]s -- is defined as follows.

+-- {: .un_def }
###### Definition/Proposition
**(Cartesian model structure on $sSet^+/S$)**

The category $SSet^+/S$ of [[marked simplicial set]]s over a marked simplicial set $S$ carries a structure of a [[proper model category|left proper]] [[combinatorial simplicial model category]] defined as follows.

The [[SSet]]-[[enriched category|enrichment]] is given by

$$
  sSet^+/S(X,Y) := Map_S^\sharp(X,Y)
  \,.
$$

A morphism $f : X \to X'$ in $SSet^+/S$ of [[marked simplicial set]]s is

* a _cofibration_ precisely if underlying morphism of [[simplicial set]]s is a cofibration in the standard [[model structure on simplicial sets]] (i.e. a [[monomorphism]]).

  (def. 3.1.2.2 of [[Higher Topos Theory|HTT]])

* a weak equivalences precisely if it is a Cartesian equivalence, as defined above.
=--
+-- {: .proof}
###### Proof
The model structure is proposition 3.1.3.7 in [[Higher Topos Theory|HTT]]. The simplicial enrichment is corollary 3.1.4.4.
=--

+-- {: .un_remark}
###### Remark
Using $Map_S^\flat(X,Y)$ for the mapping objects makes $sSet^+/S$ a
$sSet_{Joyal}$-[[enriched model category]] (i.e. enriched in the [[model structure for quasi-categories]]).  This is [[Higher Topos Theory|HTT, remark 3.1.4.5]].
=--

Notice that trivially every object in this model structure is cofibrant.
The following proposition shows that the above model structure indeed
presents the $(\infty,1)$-category $CartFib(S)$ of [[Cartesian fibration]]s.

+-- {: .num_prop #FibrantObjects}
###### Proposition
An object $p : X \to S$ in $sSet^+/S$ is _fibrant_ with respect to the above model structure precisely if it is isomorphic to an object of the
form $Y^\natural$, for $Y \to S$ a [[Cartesian fibration]] in [[sSet]].
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, prop. 3.1.4.1]].
=--

In particular, the fibrant objects of $sSet^+ \cong sSet^+/*$ are precisely the quasicategories in which the marked edges are precisely the [[equivalence in a quasi-category|equivalences]].  Note that the Cartesian model structure on $sSet^+/S$ is *not* the [[model structure on an over category]] induced on $sSet^+/S$ from the Cartesian model structure on $sSet^+$!

+-- {: .un_def }
###### Definition/Proposition
**(coCartesian model structure on $sSet^+/S$)**

There is another such model structure, with [[Cartesian fibration]]s replaced everywhere by **coCartesian fibrations.

=--

### Marked anodyne morphisms {#MarkedAnodyne}

A class of morphisms with [[left lifting property]] again some class of fibrations is usually called _anodyne_ . For instance a left/right/inner anodyne morphism of simplicial sets is one that has the left lifting property against all [[left fibration|left/right fibration]]s or [[inner fibration]]s, respectively.

The class of _marked anodyne_ morphisms in $sSet^+$ as defined in the following is something that comes close to having the left lifting property against all Cartesian fibrations. It does not quite, but is still useful for various purposes.

+-- {: .un_defn}
###### Definition ([[Higher Topos Theory|HTT, Def 3.1.1.1]])

The collection of **marked anodyne morphisms** in $SSet^+/S$ is the class of morphisms $An^+ = LLP(RLP(An^+_0))$ where the generating set $An^+_0$ consists of

* for $0 \lt i \lt n $ the minimally marked [[horn]] inclusions

  $$
    (\Lambda[n]_i)^\flat \to \Delta[n]^\flat
  $$

* for $i = n$ the horn inclusion with the last edge marked:

  $$
    (\Lambda[n]_n, \mathcal{E} \cap (\Lambda[n]_n)_1)
    \to
    (\Delta[n], \mathcal{E} )
    \,,
  $$

  where $\mathcal{E}$ is the union of all degenerate edges in $\Delta[n]$ together with the edge $\Delta^{\{n-1,n\}} \to \Delta[n]$.

* the inclusion

  $$
    (\Lambda[2]_1)^\sharp \coprod_{(\Lambda[2]_1)^\flat}
    (\Delta[2])^\flat
    \to
    (\Delta[2])^\sharp
    \,.
  $$

* for every [[Kan complex]] $K$ the morphism $K^\flat \to K^\sharp$.

=--

The crucial property of marked anodyne morphisms is the following characterization of morphisms that have the right lifting property with respect to them.

+-- {: .un_prop}
###### Proposition

A morphism $p : X \to S$ in $SSet^+$ has the [[right lifting property]] with respect to the class $An^+$ of marked anodyne maps precisely if

1. $p$ is an [[inner fibration]]

1. an edge $e$ of $X$ is marked precisely if it is a $p$-[[Cartesian morphism]] and $p(e)$ is marked in $S$

1. for every object $y$ of $X$ and every marked edge $\bar e : \bar x \to p(y)$ in $S$ there exists a marked edge $e : x \to y$ of $X$ with $p(e) = \bar e$.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 3.1.1.6]]

=--

+-- {: .un_remark}
###### Remark

Thus, if $(X, E_X) \to (S,E_S)$ is a morphism in $sSet^+$ with RLP against marked anodyne morphisms, then its underlying morphism $X\to S$ in $sSet$ is almost a [[Cartesian fibration]]: it may fail to be such only due to missing markings in $E_S$.

However, if _all_ morphisms in $S$ are marked, then $(X,E_X) \to S^\sharp$ has the RLP against marked anodyne morphisms precisely when the underlying morphism $X\to S$ is a [[Cartesian fibration]] and exactly the [[Cartesian morphism]]s are marked in $X$, $(X,E_X) = X^\natural$ --- in other words, precisely if it is a fibrant object in the model structure on $sSet^+/S$.

=--

See also [[Higher Topos Theory|HTT, remark. 3.1.1.11]].

The following stability property of marked anodyne morphisms is important in applications. Recall that a cofibration in $sSet^+$ is a morphism whose underlying morphism in [[sSet]] is a [[monomorphism]].

+-- {: .un_prop}
###### Proposition
**(stability under smash product with cofibrations)**

Marked anodyne morphisms are stable under "[[smash product]]" with cofibrations:

for $f : X \to X'$ marked anodyne, and $g : Y \to Y'$ a cofibration, the induced morphism

$$
  (X \times Y') \coprod_{X \times Y} (X' \times Y)
  \to
  X' \times Y'
$$

out of the [[pushout]] in $sSet^+$ is marked anodyne.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 3.1.2.3]].

=--

### As a model for the $(\infty,1)$-category of $(\infty,1)$-categories

The Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$ is an [[enriched category]] enriched over itself. So it is _not_ a [[simplicial model category]] in the standard sense, which means $sSet_{Quillen}$-enriched.

Indeed, the full sSet-enriched subcategory $(sSet_{Joyal})^\circ$ on fibrant-cofibrant objects is a model for the [[(∞,2)-category]] [[(∞,1)Cat]] of [[(∞,1)-categories]]. For many applications it is more convenient to work just with the [[(∞,1)-category of (∞,1)-categories]] inside that, obtained by taking in each [[hom-object]] [[quasi-category]] the maximal [[Kan complex]].

The resulting [[(∞,1)-category]] should have a presentation by a [[simplicial model category]]. And the model structure on marked simplicial sets does accomplish this.

## Related concepts

* [[model structure for dendroidal Cartesian fibrations]]

## References

Marked simplicial sets are introduced in section 3.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

The model structure on marked simplicial oversets is described in section 3.1.3

[[!redirects marked simplicial set]]
[[!redirects marked simplicial sets]]
[[!redirects model structure for coCartesian fibrations]]
[[!redirects model structure on marked simplicial over-sets]]
[[!redirects model structure on marked simplicial oversets]]

[[!redirects model structure on marked simplicial sets]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _[[complete Segal space]]_ is a model for the notion of [[(∞,1)-category]] regarded as an [[internal category in an (∞,1)-category]] in [[∞Grpd]].

The _[[model category]] structure_ on the collection of all [[bisimplicial sets]] is a [[presentable (∞,1)-category|presentation]] the [[(∞,1)-category]] of all complete Segal spaces, hence for the [[(∞,1)-category of (∞,1)-categories]].

## Definition

We first discuss the model structure

* [for complete Segal spaces](#ForCompleteSegalSpaces)

as such, and then more generally model structures

* [for complete Segal space objects](#ForCompleteSegalSpaceObjects)

in a suitable ambient model category $\mathcal{C}$, which reproduce
the previous case when $\mathcal{C} = sSet_{Quillen}$.

### For complete Segal spaces
 {#ForCompleteSegalSpaces}

Write [[sSet]] for the [[category]] of [[simplicial sets]], which here we always think of as equipped with the standard [[model structure on simplicial sets]]  that is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] [[∞Grpd]].

Write $[\Delta^{op}, sSet]$ for the category of [[simplicial objects]] in $sSet$, hence for the category of [[bisimplicial sets]]. This inherits from $sSet$ in particular the

* [[Reedy model structure]] $[\Delta^{op}, sSet]_{Reedy}$,

which is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] of [[simplicial objects]] in [[∞Grpd]]. In all of the following in its place one can also use the

* projective or injective [[model structure on functors]] $[\Delta^{op}, sSet]_{proj}$, $[\Delta^{op}, sSet]_{inj}$.

Write $c : Set \to sSet$ for the inclusion of sets as [[discrete objects]] into [[simplicial sets]]. Write

$$
   [\Delta^{op}, c] : sSet \to [\Delta^{op}, sSet]
$$

for the corresponding inclusion of simplicial sets into bisimplicial sets.

When we think in the following of a simplicial set in the context of $[\Delta^{op}, sSet]$, we always do so via this inclusion (and not via the _other_ natural such inclusion!).

+-- {: .num_defn #RezkModelStructure}
###### Definition

The _model structure for complete Segal spaces_ is the [[Bousfield localization of model categories|left Bousfield localization]] of $[\Delta^{op}, sSet]_{Reedy}$ at the set of morphisms

$$
  W
  :=
  \{
     Sp[n] \hookrightarrow \Delta[n]
  \}_{n \in \mathbb{N}}
  \cup
  \{
    N(\{a \stackrel{\simeq}{\to} b\}) \to *
  \}
  \,,
$$

where the first summand is the set of [[spine]] inclusions, while the second summand is the singleton containing the morphism from the [[nerve]] of the [[groupoid]] with two objects and precisely one non-identity morphism (and its inverse) from one to the other.

=--

This appears originally in section 12 of ([Rezk](#Rezk)).

+-- {: .num_remark}
###### Remark

Equivalently one can replace here the [[spine]] inclusions with the
[[inner Kan fibration|inner]] [[horn]] inclusions.

=--

+-- {: .num_remark}
###### Remark

An object $X \in [\Delta^{op}, sSet]$ being a [[local object]] with respect to the $n$th [[spine]] inclusion says that the morphism

$$
  X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1
$$

(with $n$ factors on the right) is a [[weak homotopy equivalence]] of simplicial sets. Therefore the objects which are local with respect to all spine inclusions are precisely the [[Segal spaces]].

Accordingly, an object is furthermore local also with respect to $J \to *$ if it is a [[complete Segal space]].

=--
The model category structure thus obtained is characterized as follows.

+-- {: .num_prop}
###### Proposition

The category $[\Delta^{op}, sSet] = [\Delta^{op}\times \Delta^{op}, Set]$ of [[bisimplicial sets]] carries the structure of an [[sSet]]-[[enriched category]] with [[hom-object]] for two bisimplicial sets $X$ and $Y$ given by

$$
  hom(X,Y) := i_2^*(Y^X)
  \,,
$$

where

* $Y^X$ is the [[internal hom]] in the standard [[closed monoidal structure on presheaves]];

* $i_2 : \Delta \to \Delta \times \Delta$ is the [[right adjoint]] to the projection $\Delta \times \Delta \to \Delta$ on the second factor, given by $i_2 : [n] \mapsto ([0],[n])$.

This refines to the structure of a

* [[left proper model category|left proper]]

* [[cartesian monoidal category|cartesian]] [[closed monoidal category|closed]] [[monoidal model category|monoidal]]

* [[simplicial model category]]

by setting

* cofibrations are the [[monomorphisms]]

* weak equivalences are the **Rezk weak equivalences**:. those morphisms $u : A \to B$ such that for all [[complete Segal space]]s $X$ the morphism $hom(u,X) : hom(B,X) \to hom(A,X)$ is a weak equivalence in the standard [[model structure on simplicial sets]].

The fibrant objects in the structure are precisely the [[complete Segal spaces]].

=--

This is essentially ([Rezk, theorem 7.2](#Rezk)).
See also ([Joyal-Tierney, theorem 4.1](#JoyalTierney)).

### For complete Segal space objects
 {#ForCompleteSegalSpaceObjects}

We may generalize from complete Segal spaces to complete Segal space objects in an ambient context other tham [[∞Grpd]] $\simeq (sSet_{Quillen})^\circ$:

Let $C$ be a [[simplicial combinatorial model category]]. Write $C^\circ$ for the [[(∞,1)-category]] [[presentable (∞,1)-category|presented]] by it. Write $[\Delta^{op}, C]$ for the [[functor category]]/category of [[simplicial objects]] in $C$.

+-- {: .num_prop}
###### Proposition

There is the structure of a  [[simplicial model category]] on $[\Delta^{op}, C]$ such that

1. is is a [[Bousfield localization of model categories|left Bousfield localization]] of the injective [[model structure on functors]] $[\Delta^{op}, C]_{inj}$;

1. such that the [[fibrant objects]] $X \colon \Delta^{op} \to C$ are precisely the injectively fibrant objects such that furthermore there image under $[\Delta^{op}, C] \to ([\Delta^{op},C])^\circ$ is an [[internal (∞,1)-category]] in $C^\circ$.

=--

([Lurie, prop. 1.5.4, remark 1.5.6](#Lurie))

## Properties

### Cartesian monoidal model structure
 {#CartesianMonoidalModelStructure}

+-- {: .num_prop}
###### Proposition

The category $[\Delta^{op}, sSet]$ is a [[cartesian closed category]]. This [[closed monoidal category]]-structure is compatible with the model category structure in that it makes $[\Delta^{op}, sSet]_{cSegal}$ into a [[monoidal model category]].

=--

This is the last clause of ([Rezk, theorem 7.2](#Rezk)). The key lemma for establishing this clause is ([Rezk, prop. 9.2](#Rezk)).

### Relation to other model structures

#### Model structure for quasi-categories
  {#RelQCat}

We discuss the relation to the [[model structure for quasi-categories]].

(See also at _[[model structure for dendroidal complete Segal spaces]]_ the section _[Relation to quasi-operads](/model+structure+for+dendroidal+complete+Segal+spaces#RelationToDendroidalSets)_ .)

A quick way to say the following turns out to be to say that the model structure for complete Segal spaces is the [simplicial completion of Cisinski model structures](Cisinski+model+structure#SimplicialCompletion) of the [[model structure for quasi-categories]] (see ([Ara](#Ara))).

+-- {: .num_defn #GroupoidalSimplices}
###### Definition

For $n \in \mathbb{N}$, write

$$
  \Delta_J[n] := N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n)
  \in sSet
$$

for the [[nerve]] of the [[free construction|free]] [[groupoid]] on $\Delta[n]$
(the [[codiscrete groupoid]] in $(n+1)$ objects.) This extends to a functor

$$
  \Delta_J : \Delta \to sSet
  \,.
$$

=--

+-- {: .num_prop #ComparisonAdjunctions}
###### Proposition

We have the following pair of [[adjoint functors]] between [[simplicial sets]] and [[bisimplicial sets]].

The first is

$$
  (p_1^* \dashv i_1^*)
  :
  [\Delta^{op}, Set]
  \stackrel{\overset{i_1^*}{\leftarrow}}{\underset{p_1^*}{\to}}
  [\Delta^{op} \times \Delta^{op}, Set]
$$

where $i_1^*$ sends a bisimplicial set to the simplicial set of its first row

$$
  i_1^* : X_{\bullet,\bullet} \mapsto X_{\bullet, 0}
  \,.
$$

The other is

$$
  (t_! \dashv t^!)
  :
  [\Delta^{op} \times \Delta^{op}, Set]
  \stackrel{\overset{t^!}{\leftarrow}}{\underset{t_!}{\to}}
  [\Delta^{op}, Set]
$$

where

* $t_!$ assigns the [[total simplicial set]] to a [[bisimplicial set]]: it is the left [[Kan extension]] of the functor $t : \Delta \times \Delta \to ssSet$ given by $([k],[l]) \mapsto \Delta[k]\times \Delta_J[l]$ (with $J$ from def. \ref{GroupoidalSimplices}) along the [[Yoneda embedding]]

  $$
    t_!(X_{\bullet,\bullet})
    =
    \int^{[k],[l]}
      X_{k,l} \cdot \Delta[k] \times \Delta_J[l]
      \,;
  $$

* and where $t^!$ forms in each degree the mapping space

  $$
    t^! : X_\bullet \mapsto
    \left(
     [m,n] \mapsto
      Hom_{sSet}( \Delta[m] \times \Delta_J[n], X)
    \right)
    \,.
  $$

=--

+-- {: .num_prop}
###### Proposition

The composite of the two adjunctions from prop. \ref{ComparisonAdjunctions}

$$
  sSet
    \stackrel{\overset{i_1^* t^!}{\leftarrow}}{\underset{t_! p_1^*}{\to}}
  sSet
$$

is the identity adjunction: both functors are [[isomorphic]] to the identity functor.

=--

This appears at the end of the proof of ([Joyal-Tierney, theorem 4.12](#JoyalTierney)).

+-- {: .num_prop}
###### Proposition

Both adjunctions of prop. \ref{ComparisonAdjunctions} are [[Quillen equivalences]] between the [[model structure for quasi-categories]] on simplicial sets and the Rezk model structure for complete Segal spaces on bisimplicial sets, def. \ref{RezkModelStructure}.

=--

This appears ([Joyal-Tierney, theorem 4.11, 4.12](#JoyalTierney)).

#### Model structure for dendroidal complete Segal spaces

The [[operad|operadic]] analog of [[simplicial sets]] / [[bisimplicial sets]] are [[dendroidal sets]] / dendroidal spaces, parameterized over the [[tree category]] $\Omega$

$$
  dsSet := [\Omega^{op}, sSet]
  \,.
$$

Write $\eta \in \Omega \hookrightarrow dSet \hookrightarrow dsSet$ for the [[tree]] with a single edge and no non-trivial vertex.

Then [[slice category]] of $dsSet$ over $\eta$ is evidently equivalent to that of [[bisimplicial sets]]

$$
  ssSet \simeq dsSet_{/\eta}
  \hookrightarrow
  dsSet
  \,.
$$

By restriction along this inclusion, the
[[model structure for dendroidal complete Segal spaces]] reproduces the above model structure. See there for more details.

## Related concepts

* [[model structure for dendroidal complete Segal spaces]]

* [[table - models for (infinity,1)-operads]]

## References

Complete Segal spaces were originally defined in

* {#Rezk} [[Charles Rezk]], _A model for the homotopy theory of homotopy theory_ , Trans. Amer. Math. Soc., 353(3), 973-1007 ([pdf](http://www.math.uiuc.edu/~rezk/rezk-ho-models-final-changes.pdf))

The [[Quillen equivalence]] with the [[model structure for quasi-categories]] is discussed in

* {#JoyalTierney} [[André Joyal]], [[Myles Tierney]], _Quasi-categories vs Segal spaces_ ([arXiv:0607820](http://arxiv.org/abs/math/0607820))

A survey of the model structures and their relations is in

* {#Bergner} [[Julia Bergner]], _A survey of $(\infty, 1)$-categories_ ([arXiv](http://arxiv.org/abs/math.AT/0610239)).

The generalization to _complete Segal objects_ in model categories other than $sSet$ was considered in

* {#Lurie} [[Jacob Lurie]], _$(\infty,2)$-Categories and the Goodwillie calculus_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))

Discussion in terms of [[Cisinski model structures]] is in

* {#Ara}[[Dimitri Ara]], _Higher quasi-categories vs higher Rezk spaces_ ([arXiv:1206.4354](http://arxiv.org/abs/1206.4354))

A model structure for [[(infinity,2)-sheaves]] of complete Segal spaces is discussed in

* [[Nicholas Meadows]], _Local Complete Segal Spaces_ ([arXiv:1607.05794](http://arxiv.org/abs/1607.05794))


model structure for connective spectra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] [[presentable (infinity,1)-category|presenting]] the [[homotopy theory]] of [[connective spectra]].

## Related entries

* [[model structure for spectra]]

## References

A model structure on [[Gamma spaces]] was given in.

* {#BousfieldFriedlander77} [[A. K. Bousfield]], [[E. M. Friedlander]], _Homotopy Theory of Γ-spaces, Spectra, and Bisimplicial Sets_, Geometric Applications of Homotopy Theory (1977) ([pdf](http://dodo.pdmi.ras.ru/~topology/books/bousfield-friedlander.pdf))

[[!redirects model structure on Gamma-spaces]]


model structure for cubical quasicategories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analogue of the [[Joyal model structure]] for [[cubical sets]], with or without [[cubical set with connections|connections]].

## Related concepts

* [[Joyal model structure]], [[quasicategory]]

* [[cubical set]], [[cubical set with connections]]

## References

With connections (positive, negative, or both):

* [[Brandon Doherty]], [[Chris Kapulkin]], [[Zachery Lindsey]], [[Christian Sattler]], _Cubical models of (∞,1)-categories_, Memoirs of the AMS ([accepted 2022](https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo)) [[arXiv:2005.04853](https://arxiv.org/abs/2005.04853)]

Without connections:

* [[Brandon Doherty]], _Cubical models of higher categories without connections_ [[arXiv:2207.03636](https://arxiv.org/abs/2207.03636)]

[[!redirects model structures for cubical quasicategories]]

[[!redirects model structure for cubical quasi-categories]]
[[!redirects model structures for cubical quasi-categories]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _model structure for dendroidal (co)Cartesian fibrations_ is an [[operad|operadic]] analog of the _[[model structure for Cartesian fibrations]]_. Its fibrant objects are (co)[[Cartesian fibrations of dendroidal sets]]. These in turn model (Grothendieck-)fibrations of [[(∞,1)-operads]].

In particular, over the [[terminal object]], the [[E-∞ operad]], this is a model for the collection [[symmetric monoidal (∞,1)-categories]]. Over an arbitrary [[(∞,1)-operad]], this is a model for the [[(∞,1)-category]] [[OMon(∞,1)Cat]] of [[O-monoidal (∞,1)-categories]].

## Related concepts

* [[model structure for Cartesian fibrations]]

* [[model structure for dendroidal left fibrations]]

For an overview of models for [[(∞,1)-operads]] see
_[[table - models for (infinity,1)-operads]]_.

## References

The model structure for dendroidal Cartesian fibrations is due to

* [[Gijs Heuts]], _Algebras over infinity-Operads_ ([arXiv:1110.1776](http://arxiv.org/abs/1110.1776))

Its further localization to the [[model structure for dendroidal left fibrations]] is discussed in

* [[Gijs Heuts]], _An infinite loop space machine for infinity-operads_ ([arXiv:1112.0625](http://arxiv.org/abs/1112.0625))

[[!redirects model structure for dendroidal coCartesian fibrations]]