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

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

## Idea

An object in a [[model category]] is fibrant if all morphisms into it have extensions along acyclic cofibrations. An **algebraic fibrant object** is a fibrant object equipped with a _choice_ of such extensions.

Under mild conditions, the category $Alg C$ of algebraic fibrant objects in a model category $C$ forms itself naturally a model category which is [[Quillen equivalence|Quillen equivalent]] to $C$, and in which all objects are fibrant.  Notably, $Alg C$ is always a [[category of fibrant objects]].

## Definition

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

Let $C$ be a [[cofibrantly generated model category]] such that

* all acyclic cofibrations are [[monomorphism]]s.

Choose a set $\{A_j \to B_j\}_{j \in J}$ of acyclic cofibrations such that

* the [[domain]]s $A_j$ are [[small object]]s;

* an object $X \in C$ is fibrant precisely if it has the [[right lifting property]] against these morphisms.

An **algebraic fibrant object** in $C$ is a fibrant object of $C$ together with a choice of lifts ("fillers") $\sigma_j : B_j \to X$ in

$$
  \array{
    A_j &\to& X
    \\
    \downarrow & \nearrow_{\mathrlap{\sigma_j}}
    \\
    B_j
  }
$$

for each morphism $A_j \to X$, $j \in J$.

Write $Alg C$ for the category whose objects are algebraic fibrant objects in $C$ and whose morphisms are morphisms in $C$ that respect the chosen lifts.

=--

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

The set $J$ can be taken to be that of all generating acyclic cofibrations, if their domains are small. But often there are smaller subsets that still characterize all fibrant objects.

=--

+-- {: .un_theorem #ModelStructureTheorem}
###### Theorem

The [[forgetful functor]] [[adjunction]]

$$
  (F \dashv U) : Alg C \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C
$$

induces the [[transferred model structure]] on $Alg C$: the fibrations
and weak equivalences in $Alg C$ are those of the underlying morphisms in $C$.

In particular, every object in $Alg C$ is fibrant.

The [[Quillen adjunction]] $(F \dashv U)$ is a [[Quillen equivalence]].

=--

This is ([Nikolaus, theorem 2.20](#Nikolaus))

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

Since fibrations in $Alg C$ are created in $C$, and any algebraically fibrant object is, in particular, fibrant, every object in the model category $Alg C$ is fibrant.  Thus almost any model category is equivalent to one in which all objects are fibrant.  However, in general not all objects in $Alg C$ will be cofibrant, even if this was true in $C$ itself.

=--

## Details

We spell out the constructions and lemmas that yield the [theorem on the model structure](#ModelStructureTheorem). In the course of this we also discuss a few results of interest in their own right, such as the monadicity, local presentability and combinatoriality of $Alg C$.

### The left adjoint

We describe explicitly the [[left adjoint]] $F : C \to Alg C$ to the [[forgetful functor]] $U : Alg C \to C$. This follows [[Richard Garner]]'s improved version of the [[small object argument]] ([Garner](#Garner)).

For $X \in C$ define a new object $X_\infty$ inductively as follows.

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

$X_1$ is the [[pushout]] (in $C$)

$$
  \array{
     \coprod A_j &\to& X
     \\
     \downarrow && \downarrow
     \\
     \coprod B_j &\to& X_1
  }
  \,,
$$

here the disjoint unition is taken over all $j$ and all morphisms $A_j \to X$.

=--

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

This pushout adds all possible "[[horn]]" fillers to $X$, given by the $B_j \to X_1$. However, after adding the new fillers there may also appear new horns. So we continue this procedure iteratively by filling these new horns.

Since $\coprod A_j \to \coprod B_j$ is an acyclic cofibration in $C$, so is its its pushout, hence by assumption the pushout $X \to X_1$ is a monomorphism. This implies that if an $A_j \to X_1$ factors through $X \to X_1$ then it does so uniquely.

=--

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

Let $X_2$ be the [[pushout]]

$$
  \array{
     \coprod A_j &\to& X_1
     \\
     \downarrow && \downarrow
     \\
     \coprod B_j &\to& X_2
  }
  \,,
$$

where now the coproduct is over those morphisms $A_j \to X$ that do _not_ factor throuth $X \to X_1$.

Proceeding this way yields a sequence of acyclic cofibrations

$$
  X \to X_1 \to X_2 \to X_3 \to \cdots
  \,.
$$

We set

$$
  X_\infty :=  \lim_\to (X \to X_1 \to X_2 \to X_3 \to \cdots )
$$

be the [[colimit]] over this sequence.

=--

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

The object $X_\infty$ inherits canonical fillers: since by all $A_j$ are [[small object]]s we have that every morphism $A_j \to X_\infty$ factors through one of the intermediate $X_i$. Then for any of the morphisms $A_j \to B_j$ in $J$ the corresponding morphism $B_j \to X_{i+1} \to X_\infty$ provides a filler.

=--

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

The assignment  $F : X \to X_\infty$ with $X_\infty$ regarded as an algebraic fibrant objects as above constitutes a left [[adjoint functor]] $F : C \to Alg C$ to the [[forgetful functor]] $U : Alg C \to C$.

The unit of this [[adjunction]] is the inclusion $X \to X_\infty$ given by the above construction and is hence a weak equivalence.

=--

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

We establish the hom-isomorphism of the [[adjunction]] and its construction by the unit of the adjunction: For any $Z \in Alg C$ and $\phi : X \to U(Z)$ any morphism in $C$, we need to show that there is a unique [[commuting diagram]]

$$
  \array{
    X &\stackrel{\phi}{\to}& Z
    \\
    \downarrow & \nearrow_{\exists ! \phi_\infty}
    \\
    X_\infty
  }
  \,.
$$

Since by assumption $Z$ has all fillers we have commuting diagrams

$$
  \array{
      A_j &\to& X
     \\
     \downarrow && \downarrow^{\mathrlap{\phi}}
     \\
     B_j &\to& Z
  }
  \,.
$$

By the universality of the pushout these induce unique morphisms

$$
  \array{
    X &\stackrel{\phi}{\to}& Z
    \\
    \downarrow & \nearrow_{\phi_1 }
    \\
    X_1
  }
  \,.
$$

By induction this induces morphisms $\phi_n : X_n \to Z$ that form a [[cocone]] under $(X \to X_1 \to X_2 \to \cdots )$. Therefore there is a unique morphusm $\phi_\infty$ as indicated.

=--

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

The morphisms $F(j) : F A_j \to F B_j$ have [[retract]]s.

=--

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

Since $F$ is [[left adjoint]] to $U$ the condition that $F A_j \to F B_j \stackrel{r}{\to} F A_j$ is the [[identity]] is equivalent to its [[adjunct]], the diagonal in

$$
  \array{
    A_j &\stackrel{j}{\to}& B_j &\stackrel{r}{\to}& A_j
    \\
    \downarrow && \downarrow &\searrow^{\mathrlap{\tilde r}}& \downarrow
    \\
    U F A_j &\stackrel{U F (j)}{\to}& U F B_j &\stackrel{U F r}{\to} & U F A_j
  }
$$

being the unit of the adjunction $A_j \to U F A_j$. Take $\tilde r$ to be the (unique) filler for this morphism.

=--

### Monadicity of the forgetful functor {#Monadicity}

We now give a formal justification for calling the objects of $Alg C$ _algebraic_ objects by showing that $Alg C$ is the category of algebras over an [[algebraic theory]] in $C$, or more precisely that it is the category of [[algebras over a monad|algebra over the monad]] $F \circ U$ on $C$

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

The functor $U : Alg C \to C$ is [[monadic]].

=--

This is ([Nikolaus, prop. 24](#Nikolaus))

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

By one of the equivalent statements of [[Beck's monadicity theorem]] we have to check that

1. $U$ has a [[left adjoint]],
1. $U$ reflects [[isomorphisms]] (i.e. it is [[conservative functor|conservative]]), and
1. $D$ has, and $U$ [[preserved limit|preserves]], coequalizers of $U$-split pairs.

The first item we already checked above.

For the second item it is sufficient to observe that if $f : X \to Y$ is a morphism in $Alg C$ such that $U f$ has an inverse $g : U Y \to U X$, then it follows that $g$ must preserve fillers, hence itself constitutes the underlying morphism of an inverse if $f$ in $Alg C$.

For the third item, let $f,g : X \to Y$ be two morphisms in $Alg C$ such that the [[coequalizer]]

$$
  U X \stackrel{\overset{U f}{\to}}{\underset{U g}{\to}}
  U Y
  \stackrel{\pi}{\to}
  Q
$$

exists in $C$, together with [[section]]s $s$ of $\pi$ and $t$ of $U(f)$ such that $s \circ \pi = U(t) \circ t$. We need to equip $Q$ with fillers such that it becomes the coequalizer of $f,g$ in $Alg C$.

For $h : A_j \to Q$ a horn, declare that its filler is the the chosen filler of $s \circ h$

$$
  \array{
    A_j &\stackrel{h}{\to}& Q
    \\
    \downarrow && \downarrow^{\mathrlap{s}} & \searrow^{\mathrlap{Id}}
    \\
    B_j &\to& Y &\stackrel{\pi}{\to} & Q
  }
  \,.
$$

We check now that this choice indeed makes $Q$ the coequalizer in $Alg C$.

First of all we need to check that $\pi$ preserves the chosen fillers.

So given a filler

$$
  \array{
    A_j &\stackrel{k}{\to}& Y
    \\
    \downarrow & \nearrow_{\mathrlap{\hat k}}
    \\
    B_j
  }
$$

it is sent by $\pi$ to the filler

$$
  \array{
    A_j &\stackrel{k}{\to}& Y &\stackrel{\pi}{\to}& Q
    \\
    \downarrow & \nearrow && \nearrow_{\pi \hat k}
    \\
    B_j
  }
$$

of $\pi k $. This is supposed to be the same as

$$
  \array{
    A_j &\stackrel{k}{\to}& Y &\stackrel{\pi}{\to}& Q &\stackrel{s}{\to}& Y &\stackrel{\pi}{\to}& Q
    \\
    \downarrow && && & \nearrow_{\hat r} && \nearrow_{\pi \hat r}
    \\
    B_j
  }
  \,.
$$

Now use the relation $U(f) t = Id$ to rewrite the first diagram equivalently as

$$
  \array{
    A_j &\stackrel{k}{\to}& Y &\stackrel{t}{\to}& X &\stackrel{U(f)}{\to}&
    Y &\stackrel{\pi}{\to}& Q
    \\
    \downarrow &&&&&  \nearrow_{\mathrlap{\hat g}} &&  \nearrow_{\pi \hat k}
    \\
    B_j
  }
$$

and the relation $U(g) t =  s \pi$ to rewrite the second diagram equivalently as

$$
  \array{
    A_j &\stackrel{k}{\to}& Y &\stackrel{t}{\to}& X &\stackrel{U(g)}{\to}& Y &\stackrel{\pi}{\to}& Q
    \\
    \downarrow && && & \nearrow_{\hat r} && \nearrow_{\pi \hat r}
    \\
    B_j
  }
  \,.
$$

This show that the fillers $\hat k$ and $\hat r$ are images under $U(f)$ and $U(g)$, respectively, of a single filler in $X$. Since $\pi$ coequalizes, it follows that $\pi \hat k = \pi \hat r$.

Finally we need to show that the coequalizing morphism $\pi$ in $Alg C$ thus established is indeed universal with this property.

For $\phi : Y \to Z$ a coequalizing morphism in $Alc C$, the fact that $Q$ is a coequalizer in $C$ already gives a unique morphism $\phi_Q$ in

$$
  \array{
    X & \stackrel{\overset{f}{\to}}{\underset{g}{\to}} & Y
    &\stackrel{\pi}{\to} & Q
    \\
    && {}^{\mathllap{\phi}}\downarrow & \swarrow_{\mathrlap{\exists ! \phi_Q}}
    \\
    && Z
  }
  \,.
$$

So it is sufficient to observe that $\phi_Q$ preserves chosen fillers. But this is true be the above construction, which says that the chosen fillers in $Q$ are the images of the chosen fillers in $Y$.

=--

### Solidity of the forgetful functor {#Solidity}

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

The [[forgetful functor]] $U : Alg C \to C$ is a [[solid functor]].

=--

We discuss the proof for a slightly simpler statement, from which the full statement follows easily.

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

Let $Y$ be an algebraic fibrant object, $X$ an object in $C$ and

$$
  f : Y \to X
$$

a morphism in $C$ (really: $U(Y) \to X$). Then there is an algebraic fibrant object $X^f_\infty$ and a morphism $X \to X^f_\infty$ such that the composite

$$
  Y\to X \to X^f_{\infty}
$$

is a morphism of algebraic fibrant objects and it is [[initial object|initial]] with this property:

for every morphism $\phi : X \to Z$ in $C$ with $Z$ an algebraic fibrant object such that the composite $Y \to X \to Z$ preserves distinguished fillers, there exists a unique morphism $\phi_\infty : X^f_\infty \to Z$ such that we have a [[commutative diagram]]

$$
  \array{
    Y
    \\
    {}^{\mathllap{f}}\downarrow & \searrow^{\mathrlap{alg}}
    \\
    X &\stackrel{\phi}{\to}& Z
    \\
    \downarrow & \nearrow_{\mathrlap{\exists ! \phi_\infty}}
    \\
    X^f_\infty
  }
  \,.
$$

Moreover, if $f$ is a [[monomorphism]] in $C$, then $X \to X^f_\infty$ is an acyclic cofibration.

=--

This is ([Nikolaus, prop. 2.6](#Nikolaus)).

The naive idea would be to equip $X$ with distinguished fillers $f \hat k$

$$
  \array{
    A_j &\stackrel{k}{\to}& Y &\stackrel{f}{\to}& X
    \\
    \downarrow & \nearrow_{\mathrlap{\hat k}} && \nearrow_{f \mathrlap{\hat k}}
    \\
    B_j
  }
$$

for each distinguished filler $\hat k$. But since the composite may factor through $Y$ in many ways, this will not give a unique notion of filler. Therefore we shall iteratively form colimits that equate these potentially different fillers.

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

Let $H$ be the set of morphisms of the form $h : A_j \to X$ that factor through $f : Y \to X$. For each $h \in H$ let $F_h$ be the set of images of distinuished fillers $B_j \to X$.

Set

$$
  X_H := \lim_\to
  \left(
     \array{
       (B_j)_h && (B_{j'})_{h'} && \cdots
       \\
       & {}_{\mathllap{\phi \in F_h}}\searrow & \downarrow^{\mathrlap{\phi' \in F_{h'}}}
       & \cdots
       \\
       && X
     }
  \right)
$$

where on the right we take the [[colimit]] over the diagram with one object $(B_j)_h$ per morphism $h : A_j \to X$ that factors through $Y$,  and one morphism $(B_j)_h \to X$ per induced distinguished filler obtained from a choice of factorization through $Y$.

This comes with a canonica [[cocone]]-morphism $X \to X_H$. Continue this way to build $X_{H'}$ by coequalizing the different ways morphisms into $X_H$ may factor through $X$, etc. to obtain a sequence

$$
  X \to X_H \to X_{H'} \to X_{H''} \to \cdots
$$

and set

$$
  X_0^f := \lim_\to (X \to X_H \to X_{H'} \to X_{H''} \to \cdots)
  \,.
$$

Notice that if $f$ is a [[monomorphism]] then all factorizations are unique and hence $X \to X_0^f$ is an [[isomorphism]].

The object $X_0^f$ can be seen to have the desired universal factorization property. But it may not yet be itself algebraically fibrant.  So we conclude by essentially applying the construction of the left adjoint $F$. But we start with letting $X_1^f$ be the pushout

$$
  \array{
    \coprod_j A_j &\to& X_0^f
    \\
    \downarrow && \downarrow
    \\
    \coprod_j B_j &\to& X_1^f
  }
$$

where now the coproduct is over all morphisms that do not factor through $Y$. After that we proceed exactly as we did for the construction for $F$ and obtain a sequence

$$
  X_0^f \to X_1^f \to X_2^f \to \cdots
  \,.
$$

Finally we set

$$
  X_\infty^f := \lim_\to(X_0^f \to X_1^f \to X_2^f \to \cdots)
  \,.
$$

One checks that this has the claimed properties (...).

As before, the inclusion $X_0^f \to X_\infty^f$ is an acyclic cofibration. Hence by the above remarks in the case that $f$ is a monomorphism also $X \to X_\infty^f$ is an acyclic cofibration.

=--

By replacing in this proof factorization through a single $Y$ by factorization through a family of $Y$s one finds that $U$ is a [[solid functor]].

### Limits and colimits of algebraic objects {#LimitsAndColimits}

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

The category $Alg C$ has all [[limit]]s and [[colimit]]s in $C$.

The limits and the [[filtered colimit]]s (but not the general colimit)s are [[created limit|created]] by $U$.

=--

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

Consider first limits: for $K : J \to Alg C$ a [[diagram]] and $\lim_{\leftarrow} U K$ its limit in $C$, let $A_j \to \lim_{\leftarrow} U K$ be a horn. Since all the composite maps $A_J \to \lim_{\leftarrow} U K \to U K(j)$ have distinguished fillers and since the morphisms between these respect these fillers, we get a cone of fillers with tip $B_j$ over $U K$. The universal cone morphism $B_j \to \lim_{\leftarrow} U K$ provides then a distinguished filler for the limit. By the same argument, any cone $const_T \to \lim_{\leftarrow} U K$ in $Alg_C$ whose underlying cone in $C$ is a limiting cone is also limiting in $Alg C$.

Now consider [[filtered colimit]]s. By assumption the domains $A_j$ are [[small object]]s. Therefore for $K : I \to C$ a [[filtered diagram]] and $L := \lim_{\to} K$ its [[filtered colimit]], any morphism $A_j \to L$ factors through one of the $K_i$.  This provides a filler $B_j \to K_i \to L$. Observe that while neither $i$ nor the filler $B_j \to K_i$ are uniquely determined, its image in $L$ is. This makes $L$ an object in $Alg C$. By the same argument one finds that it is the universal cocone under $K$ in $Alg C$.

Now for $K : J \to Alg C$ a general diagram with colimit $X := \lim_\to U K$ in $C$, we use the above fact that $U$ is a [[solid functor]] to deduce that $X_\infty^f$ is the colimit of $K$ in $Alg C$.

=--

While this gives a general prescription for computing colimits, the following lemma asserts a slightly simpler way for computing certain [[pushout]]s in $Alg C$. This will be needed for establishing the model category structure.

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

Let $i : A \to B$ be a morphism in $C$ and

$$
  \array{
    F A &\to& Y
    \\
    {}^{\mathllap{F i}}\downarrow
    \\
    F B
  }
$$

a diagram in $Alg C$. Then

* its [[pushout]] in $Alg C$ is $(B \coprod_A U Y)_\infty^f$, where $f : U Y \to B \coprod_A U Y$ is the canonical morphism;

* if $i : A \to B$ is an acyclic cofibration then so is $Y\to (B \coprod_A U Y)_\infty^f$.
=--

This is ([Nikolaus, prop 2.14](#Nikolaus)).

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

First check that $(B \coprod_A U Y)_\infty^f$ has the universal property of the pushout: for any $Z \in Alg C$ we have by the [above proposition](#SolidityLemma) that morphisms $(B \coprod_A U Y)_\infty^f \to Z$ are in bijection to morphisms $B \coprod_A U Y \to U Z$ in $C$ such that the composition

$$
  U Y \stackrel{f}{\to} B \coprod_A U Y \to U Z
$$

preserves distinguished fillers.

This in turn is the same as a morphism $g_1 : Y \to Z$ in $Alg C$ and a morphism $g_2 : B \to U Z$ in $C$ that agree on $A$. By [[adjunction]] $(F \dashv U)$ this corresponds to an [[adjunct]] $\tilde g_2: F B \to Z$. This establishes the pushout property of $(B \coprod_A U Y)_\infty^f$.

To show that $Y \to (B \coprod_A U Y)_\infty^f$ is an acyclic cofibration of $i : A \to B$ is, recall that this morphism is the composite

$$
  Y \stackrel{f}{\to} B \coprod_A U Y \to (B \coprod_A U Y)_\infty^f
  \,.
$$

Here the first morphism $f$ is an acyclic cofibration because it is the [[pushout]] of the acyclic cofibration $i$. Therefore by assumption $f$ is a [[monomorphism]] and by the [remarks in the solidity lemma](#SolidityLemma) the morphism $B \coprod_A U Y \to (B \coprod_A U Y)_\infty^f$ is an acyclic cofibration. Hence so is the composite of the two.

=--

### Transferred model structure

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

Under the above conditions, $Alg C$ becomes a [[model category]] with the $U$-[[transferred model structure]]: weak equivalences and fibrations in $Alg C$ are those morphisms that are so in $C$.

=--

This appears as ([Nikolaus, def. 2.15](#Nikolaus)).

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

This follows from the [above result](#Solidity) that $U$ is a _solid functor_ as described in detail at [[solid functor]] by observing that the _acyclicity condition_ required there follows as in the discussion of the solidity of $U$ [above](#SolidityLemma), using the assumption that all acyclic cofibrations in $C$ are [[monomorphism]]s.

We spell out the argument, following ([Nikolaus](#Nikolaus)).

By the conditions listed at [[transferred model structure]] it is sufficient to check that

1. $U$ preserves [[small object]], which is the case in particular if it preserves [[filtered colimit]]s;

1. $U$ sends [[sequential colimit]]s of [[pushout]]s of images under $F$ of generating acyclic cofibrations to weak equivalences.

The first item is true by the [above discussion](#LimitsAndColimits) of filtered colimits in $Alg C$.

That the second item holds follows from the above [pushout lemma](#PushoutLemma) which asserts that $U$-images of pushouts of $F$-images of generating acyclic cofibrations are acylic cofibrations, and again by the fact that $U$ preserves filtered colimits, which implies that it preserves the [[transfinite composition]] of these acyclic cofibrations.

=--

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

The functor $U$ preserves acyclic cofibrations.

=--

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

The [[Quillen adjunction]]

$$
  (F \dashv U) : Alg C \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  C
$$

is a [[Quillen equivalence]].
=--

This appears as ([Nikolaus, theorem 2.18](#Nikolaus))

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

We observe that the unit and counit of $(F \dashv U)$ are both weak equivalences:

the unit of the adjunction is $X \to X_\infty$ is by the [above construction](#DefOfXInfty) a [[transfinite composition]] of acyclic cofibrations (in fact a fibrant replacement) hence in particular a weak equivalence. Moreover, the unit is a [[section]] of the counit in that

$$
  \array{
     X &\to& F X
     \\
     & {}_{\mathllap{Id}}\searrow & \downarrow
     \\
     && X
  }
  \,.
$$

Hence by [[category with weak equivalences|2-out-of-3]] also $X \to F X$ is a weak equivalence.

This implies that $(F \dashv U)$ induces an [[equivalence of categories]] on the [[homotopy categories]]. Since $R$ equals its total right [[derived functor]] (since every object in $Alg C$ is fibrant) this means that $(F \dashv)$ is a Quillen equivalence.

=--

### Local presentability and combinatoriality {#LocalPresentability}

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

If $C$ is a [[locally presentable category]] then so is $Alg C$.

=--

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

By the [above](#Monadicity) we know that $Alg C$ is the category of [[algebras over a monad]] $U\circ F : C \to C$.

We invoke the <a href="http://ncatlab.org/nlab/show/locally%20presentable%20category#AlgebrasOverAnAccessibleMonad">theorem on local presentability of categories of algebras</a> discussed at [[locally presentable category]]. This says that it is sufficient to check that $U \circ F$ is an [[accessible functor]], hence that it preserves [[filtered colimit]]s. Now $F$ is a left adjoint and hence preserves all colimits, while we checked [above](#LimitsAndColimits) that $U$ preserves filtered colimits.

=--

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

If $C$ is a [[combinatorial model category]] then so is $Alg C$.

=--

## Examples

### Algebraic $\infty$-groupoids {#AlgebaicHigherCategories}

The standard [[model structure on simplicial sets]] $sSet_{Quillen}$ models the [[(∞,1)-category]] [[∞Grpd]] of [[∞-groupoid]]s: its fibrant objects are precisely the [[Kan complex]]es. But a Kan complex is a model for an $\infty$-groupoid in which composites and inverses of [[k-morphism]]s are only guaranteed to exist, but are not specifically _chosen_ .

An algebraic fibrant object in $sSet_{Quillen}$ is a [[Kan complex]] with a chosen filler for each [[horn]]: an **[[algebraic Kan complex]]** (see [[simplicial T-complex]] for a related but much stricter notion). This means precisely that all possible composites and all possible inverses are chosen. So the Quillen equivalence

$$
  sSet_{Quillen} \stackrel{\leftarrow}{\to} Alg sSet_{Quillen}
$$

induces an equivalence from an [[algebraic definition of higher category|algebraic definition of ∞-groupoids]] to a [[geometric definition of higher categories|geometric definition]].

### Algebraic $(\infty,1)$-categories

Similarly, the [[model structure for quasi-categories]] $sSet_{Joyal}$ models [[(∞,1)-categories]]: its fibrant objects are precisely the [[quasi-categories]]: an **[[algebraic quasi-category]]**. Again, these form a model for $(\infty,1)$-categories in which composition is only a relation, not an operation.

But equipping a quasi-category with the structure of an algebraic fibrant object precisely means choosing such composites. Accordingly, the [[Quillen equivalence]]

$$
  sSet_{Joyal} \stackrel{\leftarrow}{\to} Alg sSet_{Joyal}
$$

establishes an equivalence of an algebraic with the standard geometric model for $(\infty,1)$-categories.

## Related concepts

* [[algebraic model category]]

## References

The model structure on algebraic fibrant objects was introduced and discussed in

* [[Thomas Nikolaus]], _Algebraic models for higher categories_ ([arXiv](http://arxiv.org/abs/1003.1342))
{#Nikolaus}

A survey is in

* [[Thomas Nikolaus]] Talk on _Algebraic models for higher categories_ ([here](http://www.math.uni-hamburg.de/home/nikolaus/AlgMod.pdf))

The refined [[small object argument]] that is being used in the construction of the left adjoint $F : C \to Alg C$ is along the lines of the discussion in

* [[Richard Garner]], _Understanding the small object argument_ Applied Categorical Structures, 17(3):247–285, 2009 ([pdf](http://www.dpmms.cam.ac.uk/~rhgg2/CT07/CT07.pdf))
{#Garner}

[[!redirects algebraic fibrant object]]
[[!redirects algebraically fibrant object]]
[[!redirects algebraic fibrant objects]]
[[!redirects algebraically fibrant objects]]
[[!redirects model structure on algebraic fibrant objects]]
[[!redirects model structure on algebraically fibrant objects]]
[[!redirects model structures on algebraic fibrant objects]]
[[!redirects model structures on algebraically fibrant objects]]
[[!redirects model structure for algebraic fibrant objects]]
[[!redirects model structure for algebraically fibrant objects]]
[[!redirects model structures for algebraic fibrant objects]]
[[!redirects model structures for algebraically fibrant objects]]


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

#Contents#
* table of contents
{:toc}

## Idea

For $\mathcal{C}$ a [[cofibrantly generated model category]] and $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ a [[monad]] on $\mathcal{C}$, there is under mild conditions a natural [[model category]] structure on the category of [[algebras over a monad]] over $T$.

## Definition

Let $\mathcal{C}$ be a [[cofibrantly generated model category]] and $T \colon \mathcal{C} \longrightarrow \mathcal{C}$ a [[monad]] on $\mathcal{C}$.

Then under mild conditions there exists the [[transferred model structure]] on the category of [[algebras over a monad]], transferred along the [[free functor]]/[[forgetful functor]] [[adjunction]]

$$
  (F \dashv U)
      \;\colon\;
   Alg_T(\mathcal{C})
     \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}}
   \mathcal{C}
  \,.
$$

See ([Schwede-Shipley 00, lemma 2.3](#SchwedeShipley00)).

## Related concepts

* [[model structure on monoids in a monoidal model category]]

* [[model structure on algebras over an operad]]

* **model structure on algebras over a monad**

* [[model structure on coalgebras over a comonad]]

* [[model structure on simplicial T-algebras]]

## References

* {#SchwedeShipley00} [[Stefan Schwede]], [[Brooke Shipley]], _Algebras and modules in monoidal model categories_, Proc. London Math. Soc. (2000) 80(2): 491-511  ([pdf](http://www.math.uic.edu/~bshipley/monoidal.pdf))


+-- {: .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

A [[model category]] structure on a [[category]] of [[algebras over an operad]] enriched in some suitable [[homotopical category]] $\mathcal{E}$ is supposed to be a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] of [[∞-algebras over an (∞,1)-operad]].

## Definition

## Existence

### Cofibrant operads

+-- {: .un_theorem}
###### Theorem
Let $C$ be a [[cofibrantly generated model category|cofibrantly generated]] [[symmetric monoidal model category]].
Let $O$ be a [[cofibrant object|cofibrant]] [[operad]].
If $C$ satisfies the [[monoid axiom in a monoidal model category]], then there is an induced [[model structure]] on the category $Alg_C(O)$ of [[algebras over an operad]].
=--

See ([Spitzweck 01](#Spitzweck01), Theorem 4).

### $G$-objects

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

Let $\mathcal{E}$ be a [[category]] equipped with the structure of

* a [[closed monoidal category|closed]] [[symmetric monoidal category]];

* a [[monoidal model category]];

such that

* the model structure is [[cofibrantly generated model category|cofibrantly generated]];

* the tensor unit $I$ is cofibrant.

=--

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

Under these conditions there is for each [[finite group]] $G$ the structure of a [[monoidal model category]] on the category $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-[[action]], for which the [[forgetful functor]]

$$
  \mathcal{E}^{\mathbf{B}G} \to \mathcal{E}
$$

preserves and reflects fibrations and weak equivalences.

=--

This is discussed in the examples at _[[monoidal model category]]_.

### Coloured operads

For $C \in $ [[Set]] a set of colours and $P$ a $C$-[[coloured operad]] in $\mathcal{E}$ we write $Alg_{\mathcal{E}}(P)$ for the category of $P$-[[algebras over an operad]]. There is a [[forgetful functor]]

$$
  U_P \;\colon\; Alg_{\mathcal{E}}(P) \to \mathcal{E}^C
$$

from the category of [[algebra over an operad|algebras over the operad]] in $\mathcal{E}$ to the underlying $C$-colored objects of $\mathcal{E}$.

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

A $C$-[[coloured operad]] $P$ is called **admissible** if the [[transferred model structure]] on $Alg_{\mathcal{E}}(P)$ along the [[forgetful functor]]

$$
  U_P : Alg_{\mathcal{E}}(P)
  \to \mathcal{E}^{C}
$$

exists.

=--

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

So if $P$ is admissible, then $Alg_{\mathcal{E}}(P)$ carries the model structure where a morphism of $P$ algebras $f : A \to B$ is a fibration or weak equivalence if the underlying morphism in $\mathcal{E}$ is, respectively.

Below we discuss general properties of $P$ under which this model structure indeed exists.

=--

## Properties

### Existence by coalgebra intervals

The above [[transferred model structure]] on [[algebras over an operad]] exists if there is a suitable [[interval object]] in $\mathcal{E}$.

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

A **cocommutative coalgebra [[interval object]]** $H\in \mathcal{E}$ is

* a cocommutative co-unital [[comonoid]] in $\mathcal{E}$

* equipped with a factorization

  $$
    \nabla : I \coprod I \hookrightarrow H \to I
  $$

  of the [[codiagonal]] on $I$ into two [[homomorphism]]s of comonoids with the first a cofibration and the second a weak equivalence in $\mathcal{E}$.

=--

+-- {: .num_example}
###### Examples

Such cocommutative coalgebra intervals exist in

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

* the [[model structure on simplicial sets]];

* the [[model structure on symmetric spectra]].

In

* the [[model structure on chain complexes]]

there is a coalgebra interval.

=--

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

If $\mathcal{E}$ has a [[symmetric monoidal functor|symmetric monoidal]] fibrant replacement functor and a coalgebra [[interval object]] $H$ then every non-symmetric [[coloured operad]] in $\mathcal{E}$ is admissible, def. \ref{admissible}:  the [[transferred model structure]] on algebras exists.

If the interval is moreover cocommutative, then the same is true for every symmetric coloured operad.

=--

This is ([BergerMoerdijk, theorem 2.1](#BergerMoerdiskAlgebras)), following ([BergerMoerdijk-Homotopy, theorem 3.2](#BergerMoerdijkHomotopy)). For more details see at _[[model structure on operads]]_.

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

Since the coalgebra interval in the [[category of chain complexes]] is not cocommutative, this case requires special discussion, as some of the statements below will not apply to it. For more on this case see _[[model structure on dg-algebras over an operad]]_.

=--

### Rectification of algebras
 {#RectificationOfAlgebras}

Recall the notion of resolutions of operads and of the _[[Boardman-Vogt resolution]]_ $W(H,P)$ from [[model structure on operads]].

We now discuss conditions under which model categories of algebras over a resolved operad is Quillen equivalent to that over the original operad. This yields general _[[rectification]]_ results for homotopy-algebras over an operad (see also the _[Examples](#Examples)_ below.)

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

Let $\mathcal{E}$ be in addition a [[left proper model category]].

Then for $\phi : P \to Q$ a weak equivalence between admissible $\Sigma$-cofibrant well-pointed $C$-coloured operads in $\mathcal{E}$, the [[adjunction]]

$$
  (\phi_! \dashv \phi^*) :
  Alg_\mathcal{E}(P)
  \stackrel{\leftarrow}{\to}
  Alg_\mathcal{E}(Q)
$$

is a [[Quillen equivalence]].

=--

This is  ([BergerMoerdijk, theorem 4.1](#BergerMoerdijkAlgebras)).

+-- {: .num_theorem #RectificationTheorem}
###### Theorem
**(rectification of homotopy $T$-algebras)**

Let still $\mathcal{E}$ be left proper.

Let $P$ be an admissible $\Sigma$-cofibrant operad in $\mathcal{E}$ such that also $W(H,P)$ is admissible.

Then

$$
  (\epsilon_! \dashv \epsilon^*) :
  Alg_\mathcal{E}(P)
  \stackrel{\leftarrow}{\to}
  Alg_\mathcal{E}(W(H,P))
$$

is a [[Quillen equivalence]].

=--

## Equivalence of model-categorical algebras and quasicategorical algebras

The results of [Lurie](#Lurie.HA), [Pavlov–Scholbach](#PavlovScholbach18), and [Haugseng](#Haugseng) establish an equivalence of [[quasicategories]] between the underlying quasicategory of the model category of algebras over an operad and the quasicategory of quasicategorical algebras over the underlying quasicategorical operad, provided some mild conditions are met.

\begin{theorem}
(Theorem 7.11 in [Pavlov–Scholbach](#PavlovScholbach18), Theorem 4.10 in [Haugseng](#Haugseng).)
Suppose $V$ is a [[symmetric monoidal model category]] equipped with a subcategory $V^\flat$ of flat objects.
Given a flat admissible Σ-cofibrant $V$-operad $O$,
the canonical comparison functor
$$Alg_O(V)^c[W_O^{-1}]\to Alg_O(V[W^{-1}])$$
is an equivalence of [[quasicategories]].
\end{theorem}

Here a full subcategory $V^\flat\subset V$
is a __subcategory of flat objects__
([Haugseng](#Haugseng), Definition 4.1)
if it contains all cofibrant objects of $V$,
is closed under monoidal products,
and tensoring a weak equivalence with an object
produce a weak equivalence (in $V^\flat$).

Here a $V$-operad is __flat__ if it is enriched in the subcategory $V^\flat\subset V$.

An operad is __admissible__ if the category of algebras admits a [[transferred model structure]].

An operad $O$ is __Σ-cofibrant__ if the unit map $1\to O(1)$ is a cofibration and the object $O(n)$ is cofibrant in the projective model structure on $\Sigma_n$-objects in $V$.

By Remark 4.9 in [Haugseng](#Haugseng),
a Σ-cofibrant operad is flat whenever the objects of unary endomorphisms $O(x,x)$ are flat.

## Examples
 {#Examples}

### Monoids (associative algebras)

For $P = Assoc$ the [[associative operad]] it category of algebras $Alg_{\mathcal{E}} P $ is the [[category of monoids]] in $\mathcal{E}$. The above model structure on $Alg_{\mathcal{E}} P$ is the standard [[model structure on monoids in a monoidal model category]].

### $A_\infty$-Algebras {#AInfAlgebras}

Let $Assoc$ be the [[associative operad]] in [[Set]] regarded as an operad in [[Top]] under the [[discrete space]] embedding $Disc : Set \to Top$.

Let $I_*$ be the operad whose algebras are pointed objects. There is a canonical morphism $i : I_* \to Assoc$.

+-- {: .num_lemma}
###### Claim

The [[relative Boardman-Vogt resolution]]

$$
  I_* \hookrightarrow I_*[i] \hookrightarrow W([0,1], I_* \to Assoc)  \stackrel{\simeq}{\to} Assoc
$$

produces precisely [[Jim Stasheff|Stasheff]]'s [[A-∞ operad]].

=--

This is ([BergerMoerdijk, page 13](#BergerMoerdijkAlgebras))

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

Every [[A-∞ space]] is equivalent as an $A_\infty$-space to a topological [[monoid]].

=--

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

This follows from the [rectification theorem](#RectificationTheorem), using that by the above algebras over $W([0,1], I_* \to Assoc)$ are precisely [[A-∞ space]]s.

=--

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

This is a classical statement. See [[A-∞ algebra]] for background and references.

=--

### $L_\infty$-algebras and simplicial Lie algebras

Let $Lie$ be the [[Lie operad]].

A cofibrant resolution is $L_\infty$, the operad whose algebras in [[chain complex]]es are [[L-infinity algebras]].

Now (...)

### Homotopy coherent diagrams {#HomotopyCoherentDiagrams}

Let $C$ be a small $\mathcal{E}$-[[enriched category]] with set of objects $Obj(C)$. There is an operad $Diag_{C}$

$$
  Diag_C(c_1, \cdots, c_n;c) =
  \left\{
    \array{
      C(c_1, c) & if n = 1
      \\
      \emptyset & otherwise
    }
  \right.
$$

whose algebras are [[enriched functor]]s

$$
  F : C \to \mathcal{E}
  \,,
$$

hence [[diagram]]s in $\mathcal{E}$. Then the [[Boardman-Vogt resolution]]

$$
  HoCoDiag_C := W(H,Diag_C)
$$

is the operad for [[homotopy coherent diagram]]s over $C$ in $\mathcal{E}$.

The [[rectification]] theorem above now says that every homotopy coherent diagram is equivalent to an ordinary $\mathcal{E}$-diagram. For $\mathcal{E} = $ [[Top]] this is known as [[Vogt's theorem]].

### $(\infty,1)$-Categories of algebras and bimodules over an operad {#InfCatOfMods}

The constuction $Alg_{\mathcal{E}}(P)$ of a category of [[algebras over an operad]] is contravariantly functorial in $P$. Therefore if $P^\bullet$ is a [[cosimplicial object]] in the category of operads, we have that $Alg_{\mathcal{E}}(P^\bullet)$ is a (large) [[simplicial category]] of algebras. Moreover, the [[Boardman-Vogt resolution]] $W(P)$ is functorial in $P$.

These two facts together allow us to construct simplicial categories of homotopy algebras.

Specifically, there is a cosimplicial operad $Assoc^\bullet$ which

* in degree 0 is the usual [[associative operad]] $Assoc^0  = Assoc$,

* in degree 1 is the operad whose algebras are triples consisting of two associative monoids and one [[bimodule]] between them;

* in degree 2 it is the operad whose algebras are tuples consisting of three associative algebras $A_0, A_1, A_2$ as well as  one $A_i$-$A_j$-bimodule $N_{ i j}$ for each $0 \leq i \lt j \leq 2$ and a homomorphism of bimodules

  $$
    N_{0 1} \otimes_{A_1} N_{1 2} \to N_{0 2}
  $$

* and so on.

The simplicial category of algebras over $Assoc^\bullet$ is one incarnation of the [[2-category]] of algebras, bimodules and bimodules homomorphisms.

We can pass to the corresponding $\infty$-algebras by applying the [[Boardman-Vogt resolution]] to the entire cosimplicial diagram of operads, to obtain the cosimplicial [[A-∞ operad]]

$$
  A_\infty^\bullet := W(Assoc^\bullet)
  \,.
$$

The [[simplicial category]] of algebras over this has as objects [[A-∞ algebra]]s, as morphism bimodules between these, and so on.

This is discussed in ([BergerMoerdijkAlgebras, section 6](#BergerMoerdijkAlgebras)).

## Related concepts

* [[model structure on monoids in a monoidal model category]]

  * [[model structure on dg-algebras]]

  * [[model structure for ring spectra]]

* [[model structure on modules in a monoidal model category]]

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

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

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* [[(∞,1)-operad]], [[model structure on operads]]

  * [[algebra over an (∞,1)-operad]], **model structure on algebras over an operad**

    * [[module over an algebra over an (∞,1)-operad]], [[model structure on modules over an algebra over an operad]]

## References

A general discussion of the [[model structure on operads]] is in

* [[Clemens Berger]], [[Ieke Moerdijk]], _Axiomatic homotopy theory for operads_ Comment. Math. Helv. 78 (2003), 805–831. ([arXiv:math/0206094](http://arxiv.org/abs/math/0206094))
{#BergerMoerdijkHomotopy}

See also

* {#Spitzweck01} [[Markus Spitzweck]], _Operads, Algebras and Modules in General Model Categories_, [arXiv:math/0101102](http://arxiv.org/abs/math/0101102).

The concrete construction of the specific cofibrant resolutions in these structures going by the name [[Boardman-Vogt resolution]] is in

* {#BergerMoerdijkResolution} [[Clemens Berger]], [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_, Topology 45 (2006), 807–849. ([pdf](http://math.unice.fr/~cberger/BV.pdf))

The discussion of the model structure on algebras over a suitable operad:

* {#BergerMoerdijkAlgebras} [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_, in *Categories in Algebra, Geometry and Mathematical Physics*, Contemp. Math. **431** (2007) 31-58 [[arXiv:math/0512576](http://arxiv.org/abs/math/0512576), [ams:conm/431](www.ams.org/books/conm/431)]

More discussion on the transport of operad algebra structures along [[Quillen adjunctions]]/[[Bousfield localization of model categories|Bousfield localizations]] between the underlying model categories is in

* [[Carles Casacuberta]], [[Javier Gutiérrez]], [[Ieke Moerdijk]], [[Rainer Vogt]], _Localization of algebras over coloured operads_, Proceedings of the London Mathematical Society (3) 101 (2010), no. 1, 105-136 ([arXiv:0806.3983](http://arxiv.org/abs/0806.3983))

* [[Javier Gutiérrez]], _Transfer of algebras over operads along derived Quillen adjunctions_, Journal of the London Mathematical Society 86 (2012), 607-625 ([arXiv:1104.0584](http://arxiv.org/abs/1104.0584))

Discussion with an eye towards [[ring spectra]] realized as [[symmetric spectra]] is in

* {#Schwede12} [[Stefan Schwede]], section III.6 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Discussion with application to [[homotopical algebraic quantum field theory]] is in

* {#BeniniSchenkelWoike18} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Homotopy theory of algebraic quantum field theories_ ([arXiv:1805.08795](https://arxiv.org/abs/1805.08795))

On [[rectification]] of [[symmetric operad|symmetric]] [[colored operads]]:

* {#PavlovScholbach18} [[Dmitri Pavlov]], [[Jakob Scholbach]], *Admissibility and rectification of colored symmetric operads*, Journal of Topology **11** 3 (2018) 559-601 $[$[doi:10.1112/topo.12008](https://doi.org/10.1112/topo.12008), [arXiv:1410.5675](https://arxiv.org/abs/1410.5675)$]$

* {#Haugseng} [[Rune Haugseng]], _Algebras for enriched ∞-operads_, [arXiv](https://arxiv.org/abs/1909.10042).

These are based on an earlier account by Lurie:

* {#Lurie.HA} [[Jacob Lurie]], _[[Higher Algebra]]_.

[[!redirects model structures on algebras over an operad]]

[[!redirects model category of algebras over an operad]]
[[!redirects model categories of algebras over an operad]]


model structure on cellular sets

+-- {: .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

For $\Theta_n$ the $n$-truncation of the [[Theta-category]], an _$n$-[[cellular set]]_, hence a [[presheaf]] on $\Theta_n$, may be viewed as a collection of [[n-morphisms]] of an "$n$-graph" underlying an [[(∞,n)-category]]. The _model structure on cellular sets_ ([Ara](#Ara)) models $(\infty,n)$-categories this way. This model is referred to as _[[n-quasicategories]]_.

## Definition

Let $\Theta_n$ be the [[Theta category]] restricted to $n$-cells. The _model structure on cellular sets_ is the [[Cisinski model structure]] on the [[category of presheaves]] $PSh(\Theta_n)$ defined by the following [localizer](Cisinski+model+structure#Localizer): (...)

In ([Ara](#Ara)) the fibrant objects in this model structure are called **$n$-quasi-categories**, see _[Relation to quasi-categories](#RelationToQuasiCategories) below_.

## Properties

### Relation to quasi-categories
 {#RelationToQuasiCategories}

For $n = 1$ we have $\Theta_1 \simeq \Delta$ is the [[simplex category]]; and the model structure on 1-cellular sets reproduces the [[model structure for quasi-categories]]. ([Ara, theorem 5.26](#Ara))

### Relation to $\Theta_n$-spaces

The model structure on $\Theta_n$-cellular sets is [[Quillen equivalence|Quillen equivalent]] to that [[Theta space|∞-n spaces]].  ([Ara, theorem 7.4](#Ara)).

## Related concepts

* [[test category]]

## References

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