model structure for L-infinity algebras

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

There exist various [[model category]] structures which [[presentable (infinity,1)-category|present]] the [[homotopy theory]] of [[L-∞ algebras]].

By definition [[L-∞ algebras]] are the [[∞-algebras]] in the [[category of chain complexes]] over the [[Lie operad]]. As such they carry a [[model structure on algebras over an operad]]. There is a strictification which leads equivalently to a [[model structure on dg-Lie algebras]].

A more geometric way is to think of [[L-∞ algebras]] as being the [[tangent spaces]] to connected [[smooth ∞-groupoids]], hence to the [[delooping]]/[[moduli ∞-stacks]] $\mathbf{B}G$ of [[smooth ∞-groups]], hence as the first order infinitesimal neighbourhood

$$
  \mathbf{B}\mathfrak{g} \hookrightarrow \mathbf{B}G
$$

of the essentially unique point $* \to \mathbf{B}G$ (see at _[[Lie differentiation]]_). Since this is equivalently the first order neighbourhood of the _[[formal geometry|formal]]_ neighbourhood (the [[jets]]) and the formal neighbourhood can be described purely in terms of [[associative algebra]]/[[coalgebra]] (see at [[smooth algebra]]) many models for $L_\infty$-algebras are formulated in terms of such data.

In particular, one succinct way to present [[L-∞ algebras]] (as discussed there) is as [[dg-coalgebras]]:

+-- {: .num_prop  #LInfinityAlgebraIsQuasiFreeDgCoalgebra}
###### Proposition/Definition

An [[L-∞ algebra]] $(\mathfrak{g}, [-], [-,-], [-,-,-], \cdots)$ structure on a [[graded vector space]] $\mathfrak{g}$ is equivalently a [[dg-coalgebra]] structure on the graded-commutative [[cofree coalgebra]] over $\mathfrak{g}$.

Conversely, the [[category]] of [[L-∞ algebras]] (and general "weak" morphisms between them) is the [[full subcategory]] of that of counital cocommutative [[dg-coalgebras]] on those whose underlying bare graded-commutative [[coalgebra]] (forgetting the codifferential) is free

$$
  L_\infty Alg \hookrightarrow dgCoCAlg
  \,.
$$

=--

Accordingly it is of interest to have also [[model structure on dg-coalgebras]] or dually (on [[pro-objects]] of) [[dg-algebras]] which presents $L_\infty$-algebras. Prop. \ref{LInfinityAlgebraIsFibrantObjectIndgFormalSpace} below identifies the category $L_\infty Alg$ as the ([[full subcategory|full sub-]])[[category of fibrant objects]] inside such a model structure of "differential graded formal spaces", which in turn is related by a [[zig-zag]] of [[Quillen equivalence]] to various other models

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

So we write "$L_\infty Alg$" here for the _[[1-category]]_ of $L_\infty$-algebras and general morphisms between them, since this is an entry on model category presentations. If we want to refer to the [[(∞,1)-category]] of $L_\infty$-algebras we here write explicitly "$L_W( L_\infty Alg)$", referring to the [[simplicial localization]] of this 1-category.

=--

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

All gradings in the following are $\mathbb{Z}$-gradings, unless explicitly stated otherwise. In terms of the underlying [[geometry]] this means that we are dealing with [[derived geometry]] (see below the section _[Simplicial sheaves over comsimplicial formal spaces](##SheavesOverCosimplicialInfinitesimallyThickenedPoints)_ for details): the algebra elements in positive degree correspond to [[category theory|categorical]]/[[simplicial object|simplicial]]/[[∞-groupoid]]/[[∞-stack]]-degree, and those in negative degree to the [[cosimplicial object|cosimplicial]] degree of the derived site of cosimplicial formal spaces.

Technically this affects for instance the nature of [[fibrations]]: for instance the [[model structure on dg-Lie algebras]] [below](#OndgLieAlgebras) is [[transferred model structure|transferred]] from a [[model structure on chain complexes]]. For unbounded chain complexes this is the "[Categorical projective class structure](model+structure+on+chain+complexes#CategoricalProjectiveClass)" whose fibrations are the [[chain maps]] that are surjective in every degree. This appears for instance in prop. \ref{ModelStructureOnDgLieAlgebras} and prop. \ref{LInfinityAlgebrasAsACategoryOfFibrantObjects} below.

On the other hand, if one considered chain complexes in non-negative degree (for tangent complexes in "higher but non-derived geometry"), then one would use the [Projective structure on chain complexes in non-negative degree](model+structure+on+chain+complexes#ProjectiveStructureOnChainComplexes). This has as fibrations precisely the chain maps that are surjective in every _positive_ degree. This case is (currently) *not* discussed in the following.

=--

+-- {: .num_remark #NonWeakMaps}
###### Remark/Warning

Some of the model structures below are on the category of $L_\infty$-algebras with "strict" morphisms between them, namely for those morphisms which are morphisms of [[algebras over an operad]] for an $L_\infty$-algebra regarded as an algebra over a [[cofibrant resolution]] of the [[Lie operad]]. We write

$$
  L_\infty Alg_{str} \to L_\infty Alg
$$

for the [[wide subcategory]] on the strict $L_\infty$-morphisms.

=--

## Definition as algebras over an operad

As their name indicates, [[L-∞ algebras]] are the [[homotopy algebras]] [[algebra over an operad|over]] the [[Lie operad]] (in a [[category of chain complexes]]). As such, the general theory of [[model structures on algebras over an operad]] provides a [[model category]] structure on the category of $L_\infty$-algebras.

This we discuss here. But there is also a natural identification of $L_\infty$-algebras with [[infinitesimal object|infinitesimal]] [[derived stack|derived]] [[∞-stacks]]. For expressing this a host of other, Quillen equivalent model structures are available. These we discuss below in _[Definitions as formal/infinitesimal ∞-stacks](#DefinitionsAndQuillenEquivalences)_.

By the general discussion at _[[model structure on dg-algebras over an operad]]_, if $k$ is a [[field]] which contains the field of [[rational numbers]], then for every [[symmetric operad]] (uncolored) $\mathcal{O}$ in the [[category of chain complexes]] (unbounded) $Ch_\bullet(k)$, the [[free-forgetful adjunction]]

$$
  Alg(\mathcal{O})
  \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  Ch_\bullet(k)
$$

between the [[algebras over an operad]] and the underlying chain complexes induces a [[transferred model structure]] from the projective unbounded [[model structure on chain complexes]] where hence on both sides

* the weak equivalences are the morphisms that are [[quasi-isomorphisms]] on the (underlying) chain complexes;

* the fibrations are the morphisms that are degreewise [[surjections]] on the (underlying) chain complexes.

Hence in particular $(F \vdash U)$ is [[Quillen adjunction]] between these model structures.

([Hinich97, theorem 4.1.1](#Hinich97))

So this is in particular true for $\mathcal{O} = \widehat Lie$ the standard [[cofibrant resolution]] of the [[Lie operad]]. In this case $Alg(\widehat Lie) \simeq L_\infty Alg_{str}$ is the category of (unbounded) $L_\infty$-algebras (with strict $L_\infty$-maps between them as in remark \ref{NonWeakMaps} above) and hence is equipped with a [[transferred model structure]] this way

$$
  L_\infty Alg_{str}(k)
  \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  Ch_\bullet(k)
  \,.
$$

Moreover, by the rectification result discussed at _[[model structure on dg-algebras over an operad]]_, the resolution map $\widehat Lie \stackrel{\simeq}{\to} Lie$ induces a [[Quillen equivalence]]

$$
  L_\infty Alg_{str}(k) \stackrel{\simeq}{\to} dgLieAlg(k)
$$

with the [[model structure on dg-Lie algebras]], similarly [[transferred model structure|transferred]] from the [[model structure on chain complexes]].

## Definitions as formal/infinitesimal $\infty$-stacks
 {#DefinitionsAndQuillenEquivalences}

We list here definitions of various further [[model category]] structures that all [[presentable (infinity,1)-category|present]] the [[(∞,1)-category]] of [[L-∞ algebras]] and describe a web of [[zig-zags]] of [[Quillen equivalences]] between them. These Quillen equivalences may be thought of as presenting an equivalence between the [[(∞,1)-category]] of $L_\infty$-algebras and that of [[infinitesimal object|infinitesimal]] [[derived stack|derived]] [[∞-stacks]] ("[[formal moduli problems]]").

* [Summary](#DefinitionsAndQuillenEquivalencesSummary)

* [On dg-Lie algebras](#OndgLieAlgebras)

* [On dg-Coalgebras](#OndgCoalgebras)

* [On simplicial sheaves over cosimplicial formal spaces](#SheavesOverCosimplicialInfinitesimallyThickenedPoints)

* [On dg-formal spaces](#OnDgInfinitesimallyThickenedPoints)

### Summary
 {#DefinitionsAndQuillenEquivalencesSummary}

The following tabulates the main categories considered below, the functors relating them and their homotopy theoretic nature. The last row points to the relevant definitions and propositions of the following text.

|  [[L-∞ algebras]] | form [[Chevalley-Eilenberg algebra]] | [[pro-objects]] in commutative [[Artin algebra|Artin]] [[dg-algebras]] | dualize | commutative [[dg-coalgebra]] | form [[tangent space|tangents]] | [[dg-Lie algebras]] |
|--|--|--|--|--|--|--|
| $L_\infty Alg$ | $\stackrel{CE}{\hookrightarrow}$ | $Pro(dgArtinCAlg)^{op} $ | $\underoverset{\;\simeq\;}{(-)^*}{\longrightarrow}$ | $dgCoCAlg$ | $\stackrel{\mathcal{L}}{\to}$ | $dgLieAlg$ |
|    |     | $=: dgFormalSpace$ |   |  | | |
| [[category of fibrant objects]] | [[equivalence of (∞,1)-categories]] under [[simplicial localization]] |  [[opposite model structure]] of [[cofibrantly generated model category]] |  [[left adjoint|left]] [[Quillen equivalence]] | [[model category]] |  [[left adjoint|left]] [[Quillen equivalence]] | [[cofibrantly generated model category|cofibrantly generated]] [[model category]] |
| prop. \ref{LInfinityAlgebraIsQuasiFreeDgCoalgebra} | def. \ref{ChevalleyEilenbergAlgebraConstruction} | def. \ref{dgFormalSpace} | prop. \ref{DualizingInclusionOfDGFormalSpaceIntoDgCoalgebras} | def. \ref{DGCoalgebrasCategory} |  prop. \ref{LeftAdjointFromDgCoAlgToDgAlg} | def. \ref{dgLieAlgebraCategory} |

Here we are trying to use suggestive names of the categories involved. The notation used here corresponds to that in ([Pridham](#Pridham)) by the following dictionary

> (handle with care, may still need attention)

| notation  used here | notation in [Pridham](#Pridham) |
|--|--|
| $DerivedFormalSpace$, def. \ref{DerivedFormalSpace} | $scSp$, def. 1.32 |
| $dgFormalSpace$, def. \ref{dgFormalSpace} | $DG_\mathbb{Z}Sp$. def. 3.1 |
| $FormalSpace^{\Delta^{op}}$ | $sDGSp$, def. 4.6 |

### On dg-Lie algebras
 {#OndgLieAlgebras}

Let $k$ be a [[field]] of [[characteristic]] 0.

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

Write $dgLieAlg_k \in Cat$ for the [[category]] of [[dg-Lie algebras]]
over $k$.

=--

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

The category $dgLieAlg_k$ carries a
[[cofibrantly generated model category|cofibrantly generated]]
[[model category]] structure in which

* [[fibrations]] are the degreewise [[surjective maps]];

* [[weak equivalences]] are the [[quasi-isomorphisms]]

on the underlying [[chain complexes]]. This is the
[[transferred model structure]] of the corresponding
[[model structure on chain complexes]] along the [[forgetful functor]]
to the [[category of chain complexes]].

=--

([Pridham, lemma 3.24](#Pridham))

We call this the **[[model structure on dg-Lie algebras]]**.

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

Write

$$
  \mathcal{C} \;\colon\;  dgLieAlg_k \to dgCoCAlg_k
$$

for the [[functor]] which sends a [[dg-Lie algebra]] $(\mathfrak{g},d,[-,-])$ to the [[dg-coalgebra]] whose underlying [[coalgebra]] is free on the underlying [[graded vector space]] $\mathfrak{g}$ and whose [[coderivation]] is given by

$$
  \delta \colon v_1 \mapsto d v_1
$$

$$
  \delta \colon (v_1, v_2) \mapsto [v_1, v_2]
$$

and then extended as a coderivation.

(The _Chevalley-Eilenberg_ dg-coalgebra.)

=--

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

The functor from def. \ref{SendingDGLieAlgebraToDgCoalgebra} has
a [[left adjoint]]

$$
  \mathcal{L} \;\colon\;   dgCoCAlg_k \to dgLieAlg_k
  \,.
$$

=--

([Quillen 69, App. B6](#Quillen69), [Hinich98, 1.2.1, 2.2.5](#Hinich98), see also [Pridham, def. 3.23](#Pridham)).

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

If one thinks of a [[dg-coalgebra]] as presenting a
a derived formal space, as discuss [below](#SheavesOverCosimplicialInfinitesimallyThickenedPoints)
then its image under $\mathcal{L}$, prop. \ref{LeftAdjointFromDgCoAlgToDgAlg},
may be thought of as its _tangent dg-Lie algebra_. Therefore $\mathcal{L}$
is also called the **tangent Lie algebra functor**.

=--

([Hinich98, 1.2.1](#Hinich98))

### On dg-coalgebras
 {#OndgCoalgebras}

Let $k$ be a [[field]] of [[characteristic]] 0.

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

Write $dgCoCAlg_k \in Cat$ for the [[category]] of
co-commutative counital $\mathbb{Z}$-[[graded object|graded]]
[[dg-coalgebras]] over $k$.

=--

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

There exists a [[model category]] structure on $dgCoCAlg_k$
for which

* the [[cofibrations]] are the morphisms that are degreewise [[injections]];

* the [[weak equivalences]] are the morphisms $f$ whose corresponding morphisms of [[dg-Lie algebras]] $\mathcal{L}(f)$, prop. \ref{LeftAdjointFromDgCoAlgToDgAlg}, is a [[quasi-isomorphism]] on the underlying [[chain complexes]].

=--

([Hinich98, theorem 3.1](#Hinich98)) See also ([Pridham, lemma 3.25](#Pridham)).

We call this the **[[model structure on dg-coalgebras]]**.

+-- {: .num_remark #OndgCoAlgWEs}
###### Remark/Warning

Beware that the class of weak equivalences in prop. \ref{ModelStructureOnCocommutativeDGCoalgebras} is _not_ that of quasi-isomorphisms on the chain complexes underlying the dg-coalgebras.

=--

But they form a sub-class:

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

The Chevalley-Eilenberg functor

$$
  \mathcal{C} \;\colon\;  dgLieAlg_k \to dgCoCAlg_k
$$

from def. \ref{SendingDGLieAlgebraToDgCoalgebra} sends [[quasi-isomorphisms]] on the [[chain complexes]] underlying [[dg-Lie algebras]] to [[quasi-isomorphisms]] on the chain complexes underlying their Chevalley-Eilenberg [[dg-coalgebras]].

=--

([Lurie, prop. 2.2.6](#Lurie))

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

The pair of [[adjoint functors]]

$$
  (\mathcal{L} \dashv \mathcal{C})
  \;\colon\;
  dgLie_k
   \stackrel{\overset{\mathcal{L}}{\leftarrow}}{\underset{\mathcal{C}}{\to}}
  dgCoCAlg_k
$$

from prop. \ref{LeftAdjointFromDgCoAlgToDgAlg} constitutes a [[Quillen equivalence]] between the [[model structure on dg-Lie algebras]], prop. \ref{ModelStructureOnDgLieAlgebras}, and the [[model structure on dg-coalgebras]], prop. \ref{ModelStructureOnCocommutativeDGCoalgebras}.

=--

([Hinich 1998, theorem 3.2](#Hinich98))

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

Since every object in $dgCoCAlg_k$ is a [[cofibrant object]] and every object in $dgLie_k$ is a [[fibrant object]], the composite

$$
  \mathcal{C}\mathcal{L}
  \colon
  dgCoCAlg_k
  \to
  dgCocAlg_k
$$

is already its [[derived functor]] and the [[unit of an adjunction|unit]]

$$
  \mathfrak{g} \to \mathcal{C}\mathcal{L}\mathfrak{g}
$$

is a [[weak equivalence]] that exhibits $\mathcal{C}\mathcal{L}\mathfrak{g}$ as a [[fibrant resolution]] and moreover, if $\mathfrak{g}$ was already fibrant, hence by prop. \ref{LInfinityAlgebraIsFibrantObjectIndgFormalSpace} below an [[L-∞ algebra]], as a **strictification** of $\mathfrak{g}$: because a [[dg-Lie algebra]] is an $L_\infty$-algebra in which the Lie bracket satisfies its [[Jacobi identity]] strictly (not just up to a [[homotopy]] measured by the trinary bracket) and in which the "Jacobiator identity" holds strictly, etc.

=--

### On simplicial presheaves over cosimplicial formal spaces
 {#SheavesOverCosimplicialInfinitesimallyThickenedPoints}

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

Write

$$
  InfThPoint \hookrightarrow Alg^{op}
$$

for the category of [[infinitesimally thickened points]], the [[full subcategory]] of the [[opposite category]] of
[[Artin algebras]] ("Weil alebras" in the language of [[synthetic differential geometry]]). The category of _[[infinitesimally thickened points]]_.

Write

$$
  cInfThPoint \hookrightarrow sAlg^{op} \simeq (Alg^{op})^{\Delta}
$$

for the [[full subcategory]] of the [[opposite category]] on [[simplicial algebras]] on those which are [[Artin algebra|Artinian]] (or "Weil" ): the category of [[cosimplicial object|cosimplicial]] [[infinitesimally thickened points]].

=--

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

Write

$$
  FormalSpace \coloneqq Fun^{lex}(InfThPoints^{op}, Set)
$$

for the [[full subcategory]] of the [[category of presheaves]] over [[infinitesimally thickened points]] on those given by [[left exact functors]].

=--

In ([Pridham](#Pridham)) this is def. 1.18.

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

Write

$$
  DerivedFormalSpace \coloneqq Fun^{lex}(cInfThPoints^{op}, sSet)
$$

for the [[full subcategory]] of the category of [[simplicial presheaves]] over [[cosimplicial object|cosimplicial]] infinitesimally thickened points on those given by [[left exact functors]].

=--

In ([Pridham](#Pridham)) this is def. 1.32.

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

There exists a [[cofibrantly generated model category|cofibrantly generated]] [[model category]] structure on $DerivedFormalSpace$ whose

* [[weak equivalences]] are those morphisms $f \colon X \to Y$ which are [[local morphisms]] with respect to quasi-smooth maps $p \colon E \to Y$ in the [[homotopy category]] of the [[slice category]] over $Y$.

(...)

=--

This is ([Pridham, def. 2.7, theorem 2.14](#Pridham))

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

Between quasi-smooth objects in $DerivedFormalSpace$ the weak equivalences are precisely the morphisms which are [[weak homotopy equivalences]] of simplicial sets over each object in $cInfThPoint$.

=--

This is ([Pridham, cor. 2.16](#Pridham)).

### On dg formal spaces
 {#OnDgInfinitesimallyThickenedPoints}

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

Let $\Lambda$ be a [[commutative ring]] which is

1. [[noetherian ring|noetherian]]

1. [[local ring|local]] with

   1. maximal ideal $\mu$

   1. [[residue field]] $k$.

Write

$$
  ArtCAlg_{\Lambda}
$$

for the [[category]] of commutative [[local Artin algebras]] over $\Lambda$ whose residue field is also $k$.

=--

[Pridham, 1.2](#Pridham)

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

If $\mu = 0$ then $\Lambda = k$ and we just write

$$
  ArtCAlg_k
  \,.
$$

=--

In all of the following this is assumed to be the case, with $k$ of [[characteristic zero]].

([Pridham, below def. 3.3](#Pridham))

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

Write

$$
  dgArtCAlg_k \in Cat
$$

for the [[category]] of $\mathbb{Z}$-[[supercommutative superalgebra|graded-commutative]] [[Artin algebra|Artin]] [[dg-algebras]] over $k$ with [[residue field]] $k$.

Write

$$
  Pro(dgArtCAlg_k) \in Cat
$$

for its category of [[pro-objects]] and write

$$
  dgFormalSpace \coloneqq Pro(dgArtCAlg_k)^{op}
$$

for the [[opposite category]] of that.

=--

This is ([Pridham, def. 3.1](#Pridham)) following ([Manetti 02](#Manetti02)).

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

While it so happens that every [[nLab:coalgebra]] and [[dg-coalgebra]] is the [[filtered colimit]] of its [[finite number|finite]]-[[dimension|dimensional]] subalgebras (see at _[coalgebra -- As filtered colimits](coalgebra#AsFilteredColimits)_), this is not in general the case for [[algebras]]. But it follows that the [[linear dual]] of a general coalgebra is a [[filtered limit]] of finite-dimensional algebras, hence a [[pro-object]] in finite dimensional algebras. This is the reason for the appearance of [[pro-objects]] in def. \ref{dgFormalSpace}.

=--

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

There is a [[cofibrantly generated model category|cofibrantly generated]]
[[model category]] structure on $Pro(dgArtinCAlg_k)$, def. \ref{dgFormalSpace} --
hence an [[opposite model category|opposite model structure]] on $dgFormalSpaces$ -- whose [[weak equivalences]] are those morphisms that are [[local morphisms]] relative to quasi-smooth maps in the [[homotopy category]] of the [[slice category]] over their codomain.

=--

This is ([Pridham, prop. 4.36](#Pridham)).

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

Write

$$
  CE
  \;\colon\;
  L_\infty Alg \stackrel{}{\to} Pro(dgArtinCAlg_k)^{op}
$$

for the [[functor]]
which regards an [[L-infinity algebra]] $\mathfrak{g}$ as a [[dg-coalgebra]] by prop. \ref{LInfinityAlgebraIsQuasiFreeDgCoalgebra} and then forms the [[dual vector space|linear dual]] [[dg-algebra]], the **[[Chevalley-Eilenberg algebra]]** $CE(\mathfrak{g})$ of $\mathfrak{g}$ (a pro-dg-algebra according to def. \ref{dgFormalSpace}).

=--

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

$L_\infty$-algebras are precisely the [[fibrant objects]] in $dgFormalSpace$ (hence in the [[model structure on dg cocommutative coalgebras]], by [Pridham Cor. 4.56](#Pridham)): the [[Chevalley-Eilenberg algebra]] [[functor]] of def. \ref{ChevalleyEilenbergAlgebraConstruction},

$$
  L_\infty Alg
    \stackrel{CE}{\to}
  Pro(dgArtinCAlg_k)^{op}
    \simeq
  dgFormalSpace
$$

is an [[equivalence of categories]] onto its [[essential image]], which are the [[fibrant objects]] of $dgFormalSpace$:

$$
  CE
   \;\colon\;
  L_\infty Alg
  \stackrel{\simeq}{\to}
  dgFormalSpace_{fib}
  \hookrightarrow
  dgFormalSpace
  \,.
$$

=--

This is proven inside the proof of ([Pridham, prop. 4.42](#Pridham)).

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

Prop. \ref{LInfinityAlgebraIsFibrantObjectIndgFormalSpace}
shows in particular that  the category $L_\infty Alg$
of prop/def. \ref{LInfinityAlgebraIsQuasiFreeDgCoalgebra} carries the structure of a _[[category of fibrant objects]]_ that presents the [[homotopy theory]] of $L_\infty$-algebras. Notice that, of course, passing to the full subcategory of fibrant objects does not change the [[homotopy theory]] presented by the underlying [[category with weak equivalences]] in that we have an [[equivalence of (∞,1)-categories]] between the [[simplicial localizations]]

$$
  L_W(dgFormalSpace)
  \simeq
  L_W(L_\infty Alg)
  \,.
$$

=--

The following proposition characterizes the structure of this [[category of fibrant objects]].

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

The induced structure of a [[category of fibrant objects]] on $L_\infty Alg$ under the inclusion of prop. \ref{LInfinityAlgebraIsFibrantObjectIndgFormalSpace} has

1. [[weak equivalences]] are precisely the maps that are [[quasi-isomorphisms]] on the underlying chain complexes;

1. [[fibrations]] include in particular the maps that are surjections on the underlying chain complexes.

=--

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

The first statement is proven in the proof of ([Pridham, prop. 4.42](#Pridham)).

The second statement follows by ([Pridham, def. 4.34](#Pridham)) with the existence of the model structure on $dgFormalSpaces$.

> Should spell out how this follows, using lifting.

=--

+-- {: .num_prop #OnWEsOnLInfinity}
###### Remark/Warning

Beware, as in remark \ref{OndgCoAlgWEs}, that the class of weak equivalences in prop. \ref{LInfinityAlgebrasAsACategoryOfFibrantObjects} differs from that of those maps on associated [[Chevalley-Eilenberg algebras]] which are quasi-isos on the underlying chain complexes of the [[dg-algebra]] (which instead are the weak equivalences in the standard [[model structure on dg-algebras]], hence in particular those used in Sullivan [[rational homotopy theory]]). Instead the weak equivalences correspond to the maps of CE-algebra that are quasi-isomorphisms only on the chain complexes given by the co-unary component of the differential of the CE-algebra.

=--

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

There is an [[equivalence of categories]] between the [[homotopy category]] of $dgFormalSpace$ hence of $L_\infty Alg$ according to remark \ref{CategoryOfFibrantObjectsLInfinity}, and the homotopy category of $L_\infty$-algebras according to ([Kontsevich 94](#Kontsevich94)).

=--

([Pridham, prop. 4.42, see above def. 4.29](#Pridham))

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

If $k$ is of [[characteristic]] 0 then there is a [[zig-zag]] of [[Quillen equivalences]] between
$DerivedFormalSpace$, def. \ref{DerivedFormalSpace} and $dgFormalSpace$, def. \ref{dgFormalSpace}, hence an [[equivalence of (∞,1)-categories]] between their [[simplicial localizations]]

$$
  L_W DerivedFormalSpace \simeq L_W dgFormalSpace
  \,.
$$

=--

This is ([Pridham, cor. 4.49](#Pridham)).

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

For arbitrary $k$, there is a [[Quillen equivalence]]

$$
  dgLie_k \stackrel{\simeq}{\to} dgFormalSpace
  \,.
$$

=--

([Pridham, theorem 4.55](#Pridham))

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

The inclusion

$$
  dgFormalSpace \to dgCoCAlg_k
$$

given by sending an object in $Pro(dgArticCAlg_k)^{op} \coloneqq dgFormalSpace$, hence an [[dg-algebra]] $A$, to its dual dg-coalgebra $A^*$,
is the [[left adjoint]] part of a [[Quillen equivalence]] between the model structure on $dgFormalSpace$, prop. \ref{ModelStructureOnDgFormalSpace}, and the [[model structure on dg-coalgebras]], prop. \ref{ModelStructureOnCocommutativeDGCoalgebras}.

=--

([Pridham, cor. 4.56](#Pridham))

### On cosimplicial algebras (and dual Dold-Kan correspondence)
 {#OnCosimplicialAlgebras}

Also a version of the "[dual monoidal Dold-Kan correspondence](monoidal+Dold-Kan+correspondence#DualDoldKanQuillenEquivalences)" gives a Quillen equivalence between two model structures for $L_\infty$-algebras. This is ([Pridham, section 4.4](#Pridham)). This we discuss now

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

This equivalence has the nice property that starting with the [[Chevalley-Eilenberg algebra]] and then "denormalizing" it under dual monoidal Dold-Kan to a cosimplicial nilpotent algebra yields manifestly an incarnation of the $L_\infty$-algebra in terms of simplicial complexes of infinitesimal simplices as is implicit in the work of [[Anders Kock]] in [[synthetic differential geometry]]. This is spelled out further in [[schreiber:differential cohomology in a cohesive topos|dcct, section 4.5.1]].

=--

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

Write

$$
  (dg\hat {\mathcal{C}})^{\Delta}
$$

for [[cosimplicial object|cosimplicial]] [[pro-objects]] of [[dg-algebra|dg]]-[[Artin algebras]] ($\mathbb{N}$-graded).

=--

([Pridham, def. 4.6](#Pridham))

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

The category $(dg\hat{\mathcal{C}})^{\Delta}$ of def. \ref{CosimplicialDgAlgebras} carries a [[model category]] structure where

(...)

=--

This is ([Pridham, def. 4.11, prop. 4.12](#Pridham)).

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

Write

$$
  DGdg\hat \mathcal{C}
$$

for [[pro-objects]] in [[dg-algebras]] ($\mathbb{N}$-graded) in [[dg-algebras|dg]]-[[Artin algebras]] ($\mathbb{N}$-graded).

=--

([Pridham, def. 4.19](#Pridham))

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

The dual [[monoidal Dold-Kan correspondence]] functor
from [[dg-algebras]] to [[cosimplicial algebras]] (the inverse equivalence to the [[normalized cochain complex]] functor)

$$
  D
  \;\colon\;
  DGdg\hat \mathcal{C}
  \to
  (dg\hat \mathcal{C})^{\Delta}
$$

induces on $DGdg\hat \mathcal{C}$ the [[transferred model structure]] from that of prop. \ref{ModelStructureOnCosimplicialDgAlgebras} and is the [[right adjoint]] of a [[Quillen equivalence]] with respect to these model structures

=--

This is ([Pridham, theorem 4.26](#Pridham)).

## Properties

### General

We discuss some further properties of the [above](DefinitionsAndQuillenEquivalences) model category structures.

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

The model category $dgFormalSpace$, def. \ref{dgFormalSpace}, is a [[right proper model category]].

=--

This observation has been communicated privately by [[Jonathan Pridham]]

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

We need to show that the [[pullback]] of a weak equivalence $w$ along a fibration $f$ is again a weak equivalence. If $w$ is a fibration, this is automatic, so by factorisation we reduce to the case where $w$ is a cofibration. Now, every trivial cofibration is $Spf$ of a composition of acyclic small extensions, so we may take $w$ to be $Spf$ of an acyclic small extension $A \to B$ with [[kernel]] $I$. Then $f$ is $Spf$ of a quasi-free map $A \to R$, so the pullback is $Spf$ of $R \to R/IR$, and $IR =I\hat{\otimes}_A R$, so $R \to R/IR$ is also an acyclic small extension.

=--

### Homotopies and derived hom spaces
 {#HomotopiesAndDerivedHomSpaces}

In any [[model category]] we have a notion of [[homotopy]] between [[1-morphisms]]. In any [[category of fibrant objects]] we still have a notion of [[right homotopy]], given by maps into a [[path space object]]. So all of the above model category/fibrant object category structures yield models for [[homotopies]] between morphisms of $L_\infty$-algebras.

A discussion of [[path space objects]] of and hence of [[right homotopies]] between $L_\infty$-algebras (in the category of def. \ref{LInfinityAlgebraIsQuasiFreeDgCoalgebra}) is for instance in ([Dolgushev 07, section 5](#Dolgushev07)).

More generally, a description of the full [[derived hom space]] between two $L_\infty$-algebras is obtained via remark \ref{CategoryOfFibrantObjectsLInfinity} from the description of [derived hom-spaces in categories of fibrant objects](category+of+fibrant+objects#DerivedHomSpaces).

### Homotopy fiber products
 {#HomotopyFiberProducts}

Recognizing [[homotopy fiber products]] in any of the model structure above can be a bit subtle. A recognition principle of [[homotopy fibers]] over abelian $L_\infty$-algebras , hence useful for discussion of [∞-Lie algebra extensions](infinity-Lie+algebra+cohomology#Extensions)), is described in ([Fiorenza-Rogers-Schreiber 13, theorem 3.1.13](#FiorenzaRogersSchreiber13)).

## Related concepts

* [[relation between L-∞ algebras and dg-Lie algebras]]

* [[model structure on dg-coalgebras]]

* [[model structure on dg-Lie algebras]]

* [[model structure on simplicial Lie algebras]]

* [[L-∞ operad]], [[model structure on algebras over an operad]]

* [[deformation theory]]

  * [[Lie differentiation]]

  * [[deformation context]]

## References

Precursors for 2-reduced dg-algebras are dicussed in

* {#Quillen69} [[Dan Quillen]], _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([jstor:1970725](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

The homotopy-theoretic nature of $L_\infty$-algebras and their relation to deformation problems was then notably amplified in

* {#Kontsevich94} [[Maxim Kontsevich]], _Topics in algebra — deformation theory_ Lecture Notes (1994) ([ps](http://www.math.brown.edu/∼abrmovic/kontsdef.ps))

On [[model category]] [[structures]] on [[algebra over an operad|algebras over operads]] in [[chain complexes]]:

* {#Hinich97} [[Vladimir Hinich]],  *Homological algebra of homotopy algebras*, Communications in Algebra **25** 10   (1997) 3291-3323 [[arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), [doi:10.1080/00927879708826055](https://doi.org/10.1080/00927879708826055), Erratum: ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453))]

The full model structure on [[dg-coalgebras]] (in characteristic 0) as a [[model structure for L-infinity algebras|model structure for $L_\infty$-algebras]] and the [[Quillen equivalence]] between [[dg-Lie algebras]] as well as the interpretation in terms of formal $\infty$-stacks ([[L-infinity algebras|$L_\infty$-algebras]]):

* {#Hinich98} [[Vladimir Hinich]], _DG coalgebras as formal stacks_, Journal of Pure and Applied Algebra **162** 2 (2001) 209-250 [[arXiv:9812034](http://arxiv.org/abs/math/9812034), <a href="https://doi.org/10.1016/S0022-4049(00)00121-3">doi:10.1016/S0022-4049(00)00121-3</a>]

Also:

* {#Valette14} Theorem 2.1 in: [[Bruno Vallette]], _Homotopy theory of homotopy algebras_, Annales de l'Institut Fourier __70__:2 (2020)  683--738 [doi](https://10.5802/aif.3322) [Zbl:1335.18001](https://zbmath.org/?q=an:1335.18001) [arXiv:1411.5533](https://arxiv.org/abs/1411.5533)

The structure of a [[category of fibrant objects]] on connective [[finite type]] $L_\infty$-algebras:

* {#Rogers18} [[Christopher L. Rogers]], _An explicit model for the homotopy theory of finite type Lie $n$-algebras_, Algebr. Geom. Topol. 20 (2020) 1371-1429 ([arXiv:1809.05999](https://arxiv.org/abs/1809.05999), [doi:10.2140/agt.2020.20.1371](https://doi.org/10.2140/agt.2020.20.1371))

(For relation to [Valette 14](#Valette14) see [Rogers 18, below Theorem 5.9](#Rogers18))

In

* {#Lurie} [[Jacob Lurie]], _[[Formal Moduli Problems]]_

the relation to $\infty$-stacks is discussed more in detail.

More model category theoretic developments relating various of the previous approaches and generalizing to arbitrary characteristic are in

* {#Pridham} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

in parts based on

* {#Manetti02} [[Marco Manetti]], _Extended deformation functors_, Int. Math. Res. Not., (14):719–756, 2002 ([arXiv:math/9910071](https://arxiv.org/abs/math/9910071))

A useful summary of that paper is given in the [notes](http://poisson.phc.unipi.it/~maggiolo/wp-content/uploads/2008/12/WDTII_Pridham.pdf), by Stefano Maggiolo.

A discussion of [[path space objects]] for $L_\infty$-algebras is in section 5 of

* {#Dolgushev07} Vasiliy A. Dolgushev, _Erratum to: "A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold"_ ([arXiv:math/0703113](http://arxiv.org/abs/math/0703113))

A discussion of [[homotopy fibers]] of morphisms to abelian $L_\infty$-algebras (and hence [∞-Lie algebra extensions](infinity-Lie+algebra+cohomology#Extensions)) is in section 3.1 of

* {#FiorenzaRogersSchreiber13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_ ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

See also

* [[Alexander Berglund]], _Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras_, Homology, Homotopy and Applications Volume 17 (2015) Number 2 ([arXiv:1110.6145](https://arxiv.org/abs/1110.6145), [doi:10.4310/HHA.2015.v17.n2.a16]( https://dx.doi.org/10.4310/HHA.2015.v17.n2.a16))

[[!redirects model structure for L-∞ algebras]]

[[!redirects model structures for L-∞ algebras]]
[[!redirects model structures for L-infinity algebras]]

[[!redirects homotopy theory of L-∞ algebras]]
[[!redirects homotopy theory of L-infinity algebras]]


model structure for left fibrations

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

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

## Idea

The **model structure for right fibrations** $(SSet/S)_{rfib}$ is a [[model category]] structure on the [[overcategory]] $SSet/S$ of [[simplicial set]]s over a given [[quasi-category]] $S$, that [[presentable (∞,1)-category|presents]] the [[(∞,1)-category]] of [[right Kan fibration]]s [[fibrations of quasi-categories|of quasi-categories]] over $S$

$$
  ((SSet/S)_{rfib})^\circ \simeq RFib(S)
  \,.
$$

This is the $(\infty,1)$-analog of the $(2,1)$-category $Fib_{grpd}(S)$ of [[category fibered in groupoids|categories fibered in groupoids]] over a category $S$.

Similarly there is an analogous **model structure for left fibrations** that models [[left Kan fibration]]s, i.e.  op-fibrations in [[∞-groupoid]]s

$$
  ((SSet/S)_{lfib})^\circ \simeq LFib(S)
  \,.
$$

The extension of this from right fibrations to [[Cartesian fibration]]s and from left fibrations to [[coCartesian fibration]]s is the [[model structure for coCartesian fibrations]].

The [[(∞,1)-Grothendieck construction]] relates this to the [[global model structure on functors]] that presents the [[(∞,1)-category of (∞,1)-functors]] $Fun(S,\infty Grpd)$ (for left fibrations) and $Fun(S^{op},\infty Grpd)$ (for right fibrations).

## Motivation {#Motivation}

The following [[model category]] structure is best understood with the [[(∞,1)-Grothendieck construction]] in mind, which it serves to model.

Recall from the discussion there that given a morphism $p : X \to S$ of [[quasi-categories]], the [[(∞,1)-functor]] $S^{op} \to \infty Grpd$ that the left adjoint to the Grothendieck construction extracts from it is all encoded in the [[pushout]] $X^{\triangleleft} \coprod_X S$ in

$$
  \array{
    X &\stackrel{p}{\to}& S
    \\
    \downarrow && \downarrow
    \\
    X^{\triangleleft} &\to& X^{\triangleleft} \coprod_X S
  }
  \,,
$$

where $X^\triangleleft = (*) \star X $ is the [[join of quasi-categories|join]] of $X$ with the point, i.e. $X$ with an [[initial object]] freely adjoined to it.

More discussion of why this is the case is at [Adjoints to the Grothendieck construction](http://ncatlab.org/nlab/show/Grothendieck+construction#adjunction).

The model category structure described below declares that a morphism in the [[overcategory]] $sSet/S$ is a weak equivalence if it induces a weak equivalence of the quasi-categories given by these pushouts. So this is effectively saying that we regard a morphism of right fibration of quasi-categories as a weak equivalences, if under the left adjoint to the $(\infty,1)$-Grothendieck construction it induces a weak equivalences of the $(\infty,1)$-functors that classify these fibrations.

## Definition

For $f : X \to S$ a morphism of [[simplicial set]]s, write $C^{\triangleleft}(f)$ for the [[pushout]]

$$
  \array{
    X &\hookrightarrow& X^{\triangleleft}
    \\
    \downarrow && \downarrow
    \\
    S &\to& S \coprod_{X} X^{\triangleleft} & =: C^{\triangleleft}(f)
  }
$$

in the category [[sSet]] of simplicial sets. Call this the  **left cone** over $f$.

The [[colimit]]s in [[sSet]] are computed componentwise, so that the set of vertices $C^{\triangleleft}(f)_0$ is the disjoint union of the vertices of $S$ and one extra vertex $v$, the **cone point**.

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

([[Higher Topos Theory|HTT, def. 2.1.4.5]])

The **model structure for left fibrations** or **covariant model structure** $(sSet/S)_{lfib}$ on $SSet/S$ is given by

A morphism $f : X \to Y$ is

* a cofibration if the underlying morphism of simplicial sets is a cofibration in the standard [[model structure on simplicial sets]], i.e. a [[monomorphism]];

* a weak equivalence if the induced morphism of cones

  $$
    X^{\triangleleft} \coprod_X S
    \to
    Y^{\triangleleft} \coprod_Y S
  $$

  is a weak equivalence in the Joyal [[model structure for quasi-categories]], where $X^{\triangleleft}$ is the [[join of simplicial sets|join]] $X^{\triangleleft} := {*} \star X$.

=--

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

This is a

* [[proper model category|left proper]]

* [[simplicial model category|simplicial]]

* [[combinatorial model category]].

([[Higher Topos Theory|HTT, prop 2.1.4.7, 2.1.4.8]])

We have

* The fibrant objects are precisely the [[left fibration]]s
  ([[Higher Topos Theory|HTT, corollary 2.2.3.12]])

* Every left anodyne morphism is an acyclic cofibration.
  ([[Higher Topos Theory|HTT, prop 2.1.4.9]])

=--

An alternative characterization of this model structure is:

+-- {: .num_theorem #AsLocalization}
###### Theorem
The model structure for left fibrations is the left [[Bousfield localization]] of the [[model structure on an overcategory|overcategory]] model structure on $SSet/X$ induced by the [[model structure for quasicategories]] on $SSet$ at the set of maps $\{ \Lambda^n_0 \hookrightarrow \Delta^n | n\ge 0, \Delta^n \to X \}$ indexed by all the simplices of $X$.
=--

This is mentioned in [Heuts-Moerdijk](#HeutsMoerdijk13), p.5; see also [this discussion](http://nforum.ncatlab.org/discussion/915/model-structure-for-left-fibrations/?Focus=53814#Comment_53814).

## Properties

### Weak equivalences {#WeakEquivalences}

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

Let $S$ be any [[simplicial set]]. Every morphism

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

in $sSet_S$ for which $X\to Y$ is [[left fibration|left anodyne]] is a weak equivalence in the model structure for left fibrations.

=--

This is [[Higher Topos Theory|HTT, prop. 2.1.4.6]].

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

Recall from here that left anodyne morphisms are the weakly saturated class generated by the [[horn]] inclusions (i.e. under [[transfinite composition]] of [[retract]]s of [[pushout]]s). Therefore it is sufficient to check the statement for these generating morphisms.

By the definition of weak equivalences above, this means that we need to check that

$$
  (\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda[n]_i} S
  \to
  (\Delta[n])^{\triangleleft} \coprod_{\Delta[n]_i} S
$$

is a weak equivalence in $sSet_{Quillen}$.

Observe that this is a [[pushout]]

$$
   \array{
       \Lambda[n+1]_{i+1} &\to& (\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda^n_i} S
      \\
       \downarrow && \downarrow
       \\
       \Delta[n+1] &\to& (\Delta[n])^{\triangleleft} \coprod_{\Delta^n} S
   }
$$

of the [[inner fibration|inner anodyne morphism]] $\Lambda[n+1]_{i+1} \to \Delta[n+1]$ and therefore a weak equivalence.

To illustrate the above pushout property set $n = 2$ for example. Start with a 2-simplex $\sigma$ in $S$. Then $(\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S$ is the original simplicial set $S$ together with a tetrahedron $\Delta^3$ built over $\sigma$. One face of the tetrahedron is the original 2-simplex $\sigma$ in $S$, the three others "stick out" of $S$:

The simplicial set $(\Lambda^2_1)^{\triangleleft} \coprod_{\Lambda^2_1} S$ is accordingly the simplicial set $S$ with only two of the three faces of this tetrahedron over $\sigma$ erected.

The map $(\Lambda^3_2) \to (\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S $ identifies the horn of this tetrahedron given by these two new faces and the original face $\sigma$.

The pushout therefore glues in the remaining face of the tetrahedron and fills it with a 3-cell.

=--

### Change of base
 {#ChangeOfBase}

For every morphism $j : S \to S'$ we have the corresponding [[adjunction]] on [[overcategories]]

$$
  (j_! \dashv j^*)
  :
  sSet/S \stackrel{\overset{f_!}{\to}}{\underset{j^*}{\leftarrow}}
  sSet/{S'}
  \,,
$$

where

* $j_!$ is given by postcomposition of $j$;

* $j^*$ is given by [[pullback]] along $j$.

+-- {: .num_prop}
###### Proposition
**(change of base)**

This is a [[Quillen adjunction]] with respect to the model structures
for left fibrations over $S$ and $S'$, respectively. ([[Higher Topos Theory|HTT, prop. 2.1.4.10]])

If $j$ is a [[model structure for quasi-categories|weak equivalence in]] $sSet_{Joyal}$, then this is a [[Quillen equivalence]]. ([[Higher Topos Theory|HTT, remark. 2.1.4.11]])

=--

### Grothendieck construction
 {#GrothendieckConstruction}

+-- {: .num_prop}
###### Proposition
**($(\infty,1)$-Grothendieck construction)**

For $C$ a [[simplicially enriched category]] and $S = N(C)$ its [[homotopy coherent nerve]], there is a [[Quillen equivalence]]

$$
  (sSet/S)_{lfib} \stackrel{\leftarrow}{\to}
  [C, sSet_{Quillen}]
$$

between the model structure for left fibrations over $S$ and the [[global model structure on functors|global model structure on sSet-functors]] on $C$ with values in [[sSet]] equipped with the standard [[model structure on simplicial sets]].

=--

For more on this see [[(∞,1)-Grothendieck construction]].

## Related concepts

The [[operad|operadic]] generalization is the

* [[model structure for dendroidal left fibrations]].

## References

This is the content of section 2.1.4 of

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

There the model structure $(sSet/S)_{lfib}$ is called the **covariant model structure** and the model structure $(sSet/S)_{rfib}$  the **contravariant model structure**.

The alternative construction as a localization is mentioned in

* {#HeutsMoerdijk13} [[Gijs Heuts]], [[Ieke Moerdijk]], _Left fibrations and homotopy colimits_, Mathematische Zeitschrift **279** (2015) 723–744 [[doi:10.1007/s00209-014-1390-7](http://dx.doi.org/10.1007/s00209-014-1390-7), [arXiv:1308.0704](http://arxiv.org/abs/1308.0704)]

* [[Gijs Heuts]], [[Ieke Moerdijk]], *Left fibrations and homotopy colimits II* [[arXiv:1602.01274](https://arxiv.org/abs/1602.01274)]

Further discussion:

* {#Stevenson15} [[Danny Stevenson]], *Covariant Model Structures and Simplicial Localization*, North-West. Eur. J. Math. **3** (2017) 141-202 [[arXiv:1512.04815](http://arxiv.org/abs/1512.04815)]

[[!redirects model structure for right fibrations]]
[[!redirects covariant model structure]]
[[!redirects contravariant model structure]]


model structure for n-excisive functors

+-- {: .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|presentation]] for an [[(∞,1)-category]] of [[n-excisive functors]].

For $n = 1$ this is a [[model structure on excisive functors]].

## Related concepts

* [[model structure on functors]]

* [[model structure for spectra]]

* [[excisive (∞,1)-functor]]

## References

* {#BiedermannChornyRondings06} [[Georg Biedermann]], [[Boris Chorny]], Oliver Röndigs, _Calculus of functors and model categories_, Advances in Mathematics 214 (2007) 92-115 ([arXiv:math/0601221](http://arxiv.org/abs/math/0601221))

* [[Georg Biedermann]], Oliver Röndigs, _Calculus of functors and model categories II_ ([arXiv:1305.2834v2](http://arxiv.org/abs/1305.2834v2))

[[!redirects model structures for n-excisive functors]]

[[!redirects model structure on n-excisive functors]]
[[!redirects model structures on n-excisive functors]]


model structure for parameterized spectra

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

#Contents#
* table of contents
{:toc}

## Idea

On [[model category]] [[structures]] [[presentable (infinity,1)-category|presenting]] [[(infinity,1)-category|$\infty$-categories]] of [[parameterized spectra]] (in [[parameterized stable homotopy theory]]), either over a fixed base or over varying bases and then presenting [[tangent (infinity,1)-category|tangent $\infty$-categories]].

(...)

## Properties
 {#Properties}

We consider some properties of model category

\[
  \label{ModelCategoryOfSpectraInISpaces}
  \mathrm{Sp}^\Sigma_{\mathcal{R}}
  \;\;
  \in
  \;\;
  ModCat
\]

of parameterized [[symmetric spectra]] in [[I-space|$\mathcal{I}$-spaces]] based on [[SimplicialSets]] and equipped with the absolute/positive local model structures.

$$
  \mathcal{S} \coloneqq sSet
$$

as given in [Hebestreit, Sagave & Schlichtkrull (2020)](#HebestreitSagaveSchlichtkrull20):

\begin{proposition}
With the [[external tensor product|external]] [[smash product of spectra]] the model category (eq:ModelCategoryOfSpectraInISpaces)

1. is a [[symmetric monoidal category|symmetric]] [[closed monoidal category]]

1. with [[tensor unit]] the [[sphere spectrum]] $\mathcal{S}$,

1. which satisfies the [[pushout-product axiom]]

1. and the [[unit axiom]],

hence is a [[monoidal model category]].
\end{proposition}
\begin{proof}
The first statements are made explicit in [HSS, Prop. 4.5 & Prop. 5.11](#HebestreitSagaveSchlichtkrull20).
To see that also the unit axiom is satisfied it is sufficient to see that $\mathbb{S}$ is a [[cofibrant object]]. For that use the definition of $\mathbb{S}$ as $\mathbb{S}_t^{\mathcal{I}}[\ast]$ (hidden on [p. 38](https://arxiv.org/pdf/1904.01824.pdf#page=38)) and the fact that $\mathbb{S}_t^{\mathcal{I}}$ is a [[left Quillen functor]], by [HSS, Lem. 5.12](#HebestreitSagaveSchlichtkrull20).
\end{proof}

\begin{proposition}
The inclusion of [[I-spaces|$\mathcal{I}$-spaces]] via their [[zero-spectrum bundles]] is a [[bireflective subcategory]] (according to [this Def.](bireflective+subcategory#BireflectiveSubcategory)) which is also a [[Quillen adjoint triple]]:

\[
  \label{BireflectionOfZeroSpectrumBundles}
\]

\begin{tikzcd}
  \phantom{-}
  \\[-70pt]
  \mathcal{S}^{\mathcal{I}}
  \ar[
    from=rr,
    shift left=22pt,
    "{ \beta }"{description}
  ]
  \ar[
    rr,
    hook,
    "{ 0 }"{description}
  ]
  \ar[
    from=rr,
    shift right=22pt,
    "{ \beta }"{description}
  ]
  \ar[
    rr,
    phantom,
    shift left=11pt,
    "{ \scalebox{.6}{$\bot_{{}_{\mathrlap{\mathrm{Qu}}}}$} }"
  ]
  \ar[
    rr,
    phantom,
    shift right=11pt,
    "{ \scalebox{.6}{$\bot_{{}_{\mathrlap{\mathrm{Qu}}}}$} }"
  ]
  &&
  \mathrm{Sp}^\Sigma_{\mathcal{R}}
\end{tikzcd}

\end{proposition}
\begin{proof}
This may be recognized from [HSS, Lem. 3.19 & Cor. 5.13](#HebestreitSagaveSchlichtkrull20).
\end{proof}

\begin{definition}
In the following we will leave the inclusions
$$
  \mathcal{S}
    \overset{const}{\hookrightarrow}
  \mathcal{S}^{\mathcal{I}}
    \overset{0}{\hookrightarrow}
  \mathrm{Sp}^\Sigma_\mathcal{R}
$$
notationally implicit. For example, for $B$ a [[simplicial set]] we write $\big(\mathrm{Sp}^\Sigma_\mathcal{R}\big)_{/B}$ for the [[slice model structure]] of (eq:ModelCategoryOfSpectraInISpaces) over the [[zero-spectrum bundle]] over $B$.
\end{definition}

The model category (eq:ModelCategoryOfSpectraInISpaces) is not quite [[right proper]] (cf. [pp. 40](https://arxiv.org/pdf/1904.01824.pdf#page=40)) but, in its version based on [[simplicial sets]], we have:

\begin{proposition}\label{QuillenRightBaseChange}
Left [[base change]] $f^\ast$ along [[Kan fibrations]] $f \,\colon\, B_1 \to B_2$ of ([[zero-spectrum bundles]] (eq:BireflectionOfZeroSpectrumBundles) over) [[Kan complexes]] *is* a [[left Quillen functor]] between the [[slice model structures]] of the *absolute* local model structure. Since, generally, $f^\ast$ is also a [[right Quillen functor]], we have [[base change]] [[Quillen adjoint triples]] of this form:

\begin{tikzcd}
  \left.
  \begin{array}{r}
    \mbox{$B_2$ Kan complex}
    \\
    \mbox{$f : B_1 \to B_2$ Kan fibration}
  \end{array}
  \right\}
  \;\;\;\;
    \vdash
  \;\;\;\;
  \big(\mathrm{Sp}^\Sigma_{\mathcal{R}}\big)_{/B_1}
  \ar[
    rr,
    shift left=22pt,
    "{ f_! }"{description}
  ]
  \ar[
    from=rr,
    shift left=00pt,
    "{ f^\ast }"{description}
  ]
  \ar[
    rr,
    shift right=22pt,
    "{ f_\ast }"{description}
  ]
  \ar[
    rr,
    phantom,
    shift left=11pt,
    "{ \scalebox{.6}{$\bot_{{}_{\mathrlap{\mathrm{Qu}}}}$} }"
  ]
  \ar[
    rr,
    phantom,
    shift right=11pt,
    "{ \scalebox{.6}{$\bot_{{}_{\mathrlap{\mathrm{Qu}}}}$} }"
  ]
  &&
  \big(\mathrm{Sp}^\Sigma_{\mathcal{R}}\big)_{/B_2}
\end{tikzcd}

\end{proposition}
\begin{proof}
  The left Quillen property of $f^\ast$ is HSS20, Lem. 7.22](#HebestreitSagaveSchlichtkrull20).
The right Quillen property is mentioned inside the proof of Lem. 8.10 there.
\end{proof}

Similarly, but now also for the positive local model structure, we have:

\begin{proposition}
With respect absolute *or* positive local model structure:

\begin{tikzcd}
  \left.
  \begin{array}{r}
    \mbox{$B_2$ Kan complex}
    \\
    \mbox{$f : B_1 \to B_2$ Kan fibration}
  \end{array}
  \right\}
  \;\;\;\;
    \vdash
  \;\;\;\;
  \mathrm{Sp}^\Sigma_{B_1}
  \ar[
    rr,
    shift left=22pt,
    "{ f_! }"{description}
  ]
  \ar[
    from=rr,
    shift left=00pt,
    "{ f^\ast }"{description}
  ]
  \ar[
    rr,
    shift right=22pt,
    "{ f_\ast }"{description}
  ]
  \ar[
    rr,
    phantom,
    shift left=11pt,
    "{ \scalebox{.6}{$\bot_{{}_{\mathrlap{\mathrm{Qu}}}}$} }"
  ]
  \ar[
    rr,
    phantom,
    shift right=11pt,
    "{ \scalebox{.6}{$\bot_{{}_{\mathrlap{\mathrm{Qu}}}}$} }"
  ]
  &&
  \mathrm{Sp}^\Sigma_{/B_2}
\end{tikzcd}

\end{proposition}
\begin{proof}
[HSS, Lem. 5.17 & Lem. 7.25](#HebestreitSagaveSchlichtkrull20)
\end{proof}

(...)

## Related concepts

* [[model structure for spectra]]

* [[model structure on presheaves of spectra]]

* [[parameterized stable homotopy theory]]

* [[tangent (infinity,1)-category]]

## References

### Tools

Tools for creating [[model structures on spectra]] relative to suitable ambient model categories:

* {#Schwede97} [[Stefan Schwede]], Section 3 of: _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra **120** (1997) 104 [[pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf)]

and also for [[model structures on symmetric spectra]]:

* {#Hovey00} [[Mark Hovey]], *Spectra and symmetric spectra in general model categories*, Journal of Pure and Applied Algebra **165** 1 (2001) 63-127 [[arXiv:0004051](http://arxiv.org/abs/math/0004051), doi:10.1016/S0022-4049(00)00172-9]

### Over a fixed base

Model structure for [[genuine equivariant spectra]] parameterized over a fixed base:

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], Chapter 12 of: _[[Parametrized Homotopy Theory]]_, Mathematical Surveys and Monographs **132**, AMS (2006) [[arXiv:math/0411656](https://arxiv.org/abs/math/0411656)]

### Over varying bases
 {#ReferencesOverVaryingBases}

Model structures of parameterized spectra over varying bases:

* {#BraunackMayer19} [[Vincent Braunack-Mayer]], *Combinatorial parametrised spectra*, Algebr. Geom. Topol. **21** (2021) 801-891 [[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), [doi:10.2140/agt.2021.21.801](https://doi.org/10.2140/agt.2021.21.801)]

  > (based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]], 2018)

* {#HebestreitSagaveSchlichtkrull20} [[Fabian Hebestreit]], [[Steffen Sagave]], [[Christian Schlichtkrull]], *Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces*, Forum of Mathematics, Sigma **8** (2020) e16 [[arXiv:1904.01824](https://arxiv.org/abs/1904.01824), [doi:10.1017/fms.2020.11](https://doi.org/10.1017/fms.2020.11)]

[[!redirects model structures for parameterized spectra]]

[[!redirects model structure on parameterized spectra]]
[[!redirects model structures on parameterized spectra]]


model structure for quasi-categories

+-- {: .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#
* table of contents
{:toc}

## Idea

A [[quasi-category]] is a [[simplicial set]] satisfying [[weak Kan complex|weak Kan filler conditions]] that make it behave like the [[nerve]] of an [[(∞,1)-category]].

There is a [[model category]] structure on the category [[SSet]] --  the **[[Andre Joyal|Joyal]] model structure** or **model structure for quasi-categories** -- such that the fibrant objects are precisely the quasi-categories and the weak equivalences precisely the correct _categorical equivalences_ that generalize the notion of [[equivalence of categories]].

## Definition

\begin{definition}\label{TheModelStructure}

The **model structure for quasi-categories** or **Joyal model structure** $sSet_{Joyal}$ on [[sSet]] has

* [[cofibrations]] are the [[monomorphisms]],

* [[weak equivalences]] are those maps that are taken by the [[rigidification of quasi-categories|rigidification]] functor $\mathfrak{C}$ (the [[left adjoint]] to the [[homotopy coherent nerve]]) to a [[Dwyer-Kan equivalence]] (a [[weak equivalence]] in the [[Dwyer-Kan-Bergner model structure on sSet-enriched categories]]).

\end{definition}

## Properties

### As a Cisinski model structure

The model structure for quasi-categories is the [[Cisinski model structure]] on [[sSet]] whose class of weak equivalences is the localizer generated by the [[spine]] inclusions $\{Sp^n \hookrightarrow \Delta^n\}$. See ([Ara](#Ara)).

### General properties
 {#GeneralProperties}

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

The model structure for quasi-categories is

* a [[left proper model category]],

* a [[combinatorial model category]].

=--

\begin{lemma}
\label{CartesianProductPreservesWeakCategoricalEquivalences}
  The image under the [[Cartesian product]]-functor of two [[weak categorical equivalences]] (Def. \ref{TheModelStructure}) is again a weak categorical equivalence.
\end{lemma}
[[Lurie (2009), Cor. 2.2.5.4](#Lurie09)]

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

The model structure for quasi-categories is a [[monoidal model category]] with respect to [[cartesian product]] (hence a [[cartesian closed model category]]) and thus is naturally an [[enriched model category]] over itself, hence is $sSet_{Joyal}$-enriched (reflecting the fact that it tends to present an [[(infinity,2)-category]]). It is however _not_ $sSet_{Quillen}$-enriched and thus not a "[[simplicial model category]]" with respect to this enrichment.

=--

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

For $p \colon \mathcal{C} \to \mathcal{D}$ a morphism of [[simplicial sets]] such that $\mathcal{D}$ is a [[quasi-category]]. Then $p$ is a fibration in $sSet_{Joyal}$ precisely if both of the following conditions hold:

1. it is an *[[inner fibration]]*;

1. it is an *[[isofibration]]*:

   in that for every [[equivalence in an (∞,1)-category|equivalence]] in $\mathcal{D}$ and a lift of its [[domain]] through $p$, there is also a lift of the whole equivalence through $p$ to an equivalence in $\mathcal{C}$.

=--

This is due to [[Joyal]]. [[Lurie (2009), cor. 2.4.6.5](#Lurie09)]

So every [[fibration]] in $sSet_{Joyal}$ is an [[inner fibration]], but the converse is in general false. A notable exception are the fibrations to the point:

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

The [[fibrant objects]] in $sSet_{Joyal}$ are precisely those that are [[inner fibration|inner fibrant]] over the point, hence those simplicial sets which are [[quasi-categories]].

=--

[[Lurie (2009), theorem 2.4.6.1](#Lurie09)]

### Relation to the model structure for $\infty$-groupoids {#RelToInfGrpds}

The inclusion of [[(∞,1)-category|(∞,1)-categories]] [[∞Grpd]] $\stackrel{i}{\hookrightarrow}$ [[(∞,1)Cat]] has a left and a right [[adjoint (∞,1)-functor]]

$$
  (grpdfy \dashv i \dashv Core)
  \;\;
  \colon
  \;\;
  (\infty,1)Cat
  \stackrel{\overset{grpdfy}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{Core}{\longrightarrow}}}
  \infty Grpd
  \,,
$$

where

* $Core$ is the operation of taking the [[core]], the maximal $\infty$-groupoid inside an $(\infty,1)$-category;

* $grpdfy$ is the operation of _groupoidification_ that freely generates an $\infty$-groupoid on a given $(\infty,1)$-category

(see [[Higher Topos Theory|HTT, around remark 1.2.5.4]])

The adjunction $(grpdfy \dashv i)$ is modeled by the [[Bousfield localization of model categories|left Bousfield localization]]

\[
  \label{BousfieldLocalizationToKanQuillenStructure}
  sSet_{Quillen}
  \underoverset
    {\underset{Id}{\longrightarrow}}
    {\overset{Id}{\longleftarrow}}
    {\;\; \bot \;\;}
  sSet_{Joyal}
  \,.
\]

Notice that the left [[derived functor]] $\mathbb{L} Id \colon (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ$ takes a fibrant object on the left -- a [[quasi-category]] -- then does nothing to it but regarding it now as an object in $sSet_{Quillen}$ and then producing its fibrant replacement there, which is [[Kan fibrant replacement]]. This is indeed the operation of _groupoidification_ .

The other adjunction is given by the following

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

There is a [[Quillen adjunction]]

$$
  (k_! \dashv k^!)
  \;\; \colon
  sSet_{Quillen}
  \stackrel
    {\overset{k^!}{\longleftarrow}}
    {\underset{k_!}{\longrightarrow}}
   sSet_{Joyal}
$$

which arises as [[nerve and realization]] for the [[cosimplicial object]]

$$
  k : \Delta \to sSet : [n] \mapsto \Delta'[n]
  \,,
$$

where $\Delta^'[n] = N(\{0 \stackrel{\simeq}{\to} 1 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n\})$ is the [[nerve]] of the [[groupoid]] freely generated from the linear [[quiver]] $[n]$.

This means that for $X \in SSet$ we have

* $k^!(X)_n  = Hom_{sSet}(\Delta'[n],X)$.

* and $k_!(X)_n = \int^{[k]} X_k \cdot \Delta'[k]$.

=--

This is ([Joyal & Tierney (2007), prop 1.19](#JoyalTierney07))

The following proposition shows that $(k_! \dashv k^!)$ is indeed a model for $(i \dashv Core)$:

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

* For any $X \in sSet$ the canonical morphism $X \to k_!(X)$ is an acyclic cofibration in $sSet_{Quillen}$;

* for $X \in sSet$ a [[quasi-category]], the canonical morphism $k^!(X) \to Core(X)$ is an acyclic fibration in $sSet_{Quillen}$.

=--

This is ([Joyal & Tierney (2007), prop 1.20](#JoyalTierney07))

## History

[[André Joyal]] on the history of the Joyal model structure (also on [MathOverflow](https://mathoverflow.net/questions/425082/when-did-the-joyal-model-structure-on-simplicial-sets-originate/427514#427514)):

> I became interested in quasi-categories (without the name) around 1980 after attending a talk by Jon Beck on the work of Boardman and Vogt. I wondered if category theory could be extended to quasi-categories. In my mind, a crucial test was to show that a quasi-category is a Kan complex if its homotopy category is a groupoid. All my attempts at showing this have failed for about 15 years, until I stopped trying hard! I found a proof after extending to quasi-categories a few basic notions of category theory. This was around 1995. The model structure for quasi-categories was discovered soon after. I did not publish it immediately because I wanted to show that it could be used for proving something new in homotopy theory. I am a bit of a perfectionist (and overly ambitious?). I was hoping to develop a synthesis between category theory and homotopy theory (hence the name quasi-categories). I met Lurie at a conference organised by Carlos Simpson in Nice (in 2001?). I gave a talk on the model structure and Lurie asked for a copy of my notes afterward.

## Related concepts

A similar model for [[(∞,n)-categories]] is discussed at

* [[model structure on cellular sets]]

There are analogues of the Joyal model structure for [[cubical sets]] (with or without connection):

* [[model structure for cubical quasicategories]]

## References

The original construction of the Joyal model structure is in

* [[Andre Joyal]], _Theory of quasi-categories I_

Unfortunately, this never became publicly available, but see the lecture notes:

* {#Joyal08} [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008 ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]])

or the construction of the model structure in Cisinski's book

* [[Denis-Charles Cisinski]], _Higher categories and homotopical algebra_, ([pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

which closely follows Joyal's original construction.

A proof that proceeds via [[homotopy coherent nerve]] and [[simplicially enriched categories]]:

* {#Lurie09} [[Jacob Lurie]], following Thm. 2.2.5.1 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf)]

On the relation to the model structure for [[complete Segal spaces]]:

* {#JoyalTierney07} [[Andre Joyal]], [[Myles Tierney]], _Quasi-categories vs. Segal spaces_, in *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) [[arXiv:math/0607820](http://arxiv.org/abs/math/0607820), [doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

Discussion with an eye towards [[Cisinski model structures]] and the [[model structure on cellular sets]] is in

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

See also

* [[Denis-Charles Cisinski]], _Algèbre homotopique et catégories supérieures_, course notes, 2009, [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/coursinftycat.pdf).

On a model structure for [[(infinity,2)-sheaves]] with values in quasicategories:

* {#Meadows15} [[Nicholas Meadows]], _The Local Joyal Model Structure_ ([arXiv:1507.08723](http://arxiv.org/abs/1507.08723))

On [[transferred model structure|transfer]] of the Joyal model structure to [[reduced simplicial sets]], modelling quasi-categories with a single object:

* [Burke (2021)](model+structure+on+reduced+simplicial+sets#Burke21)

[[!redirects Joyal model structure]]
[[!redirects Joyal model structure on simplicial sets]]
[[!redirects model structure on quasicategories]]
[[!redirects model structure on quasi-categories]]
[[!redirects model structure for quasicategories]]
[[!redirects Joyal model structure for quasi-categories]]

[[!redirects weak categorical equivalence]]
[[!redirects weak categorical equivalences]]


model structure for ring spectra

+-- {: .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]] structue on ([[commutative monoid|commutative]]) [[monoids]] in a [[symmetric monoidal category of spectra]] which serves to [[presentable (infinity,1)-category|present]] the [[homotopy theory]] of [[A-∞ rings]]/[[E-∞ rings]].

## Related concepts

* [[model structure for spectra]]

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

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

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

## References

Constructions modeled on [[symmetric ring spectra]] include

* {#MMSS98b} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]],  _[[Model categories of diagram spectra]]_, Proc. London Math. Soc. 82 (2001), ([KTheory:0320](http://www.math.uiuc.edu/K-theory/0320/))

* {#Shipley04} [[Brooke Shipley]], _A convenient model category for commutative ring spectra_, Contemporary Mathematics, Volume 346, 2004

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

[[!redirects model structures for ring spectra]]

[[!redirects model structure on ring spectra]]
[[!redirects model structures on ring spectra]]


model structure for Segal categories

+-- {: .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#
* table of contents
{:toc}

## Idea

A [[model category]] structure whose fibrant objects are precisely the [[Reedy model structure|Reedy fibrant]] [[Segal categories]]. This is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] [[(∞,1)Cat]] from the point of view of (weakly) [[∞Grpd]]-[[enriched categories]].

## Definition

Write $PreSegalCat \hookrightarrow [\Delta^{op}, sSet]$ for the full [[subcategory]] on those [[bisimplicial sets]] $X$ for which $X_0$ is a discrete simplicial set (the "precategories").

The [[nerve]] functor

$$
  N : Cat \to PreSegalCat
$$

has a [[left adjoint]] ("fundamental category" functor)

$$
  \tau_1 : PreSegalCat \to Cat
  \,.
$$

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

Say a morphism $f : X \to Y$ in $PreSegalCat$ is

* [[full and faithful functor|full and faithful]] if for all $a,b \in X_0$ the induced morphism

  $$
    X(a,b) \to X(f(a),f(b))
  $$

  is a [[weak homotopy equivalence]] of simplicial sets;

  * _essentially surjective_ if $\tau_1(f)$ is [[essentially surjective functor|essentially surjective]].

  * a _categorical equivalence_ if it is both full and faithful as well as essentially surjective.

=--

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

There is an essentially unique completion functor

$$
  compl \colon PreSegalCat \to PreSegalCat
$$

equipped with a [[natural transformation]]

$$
  i \colon id_{PreSegalCat} \to compl
$$

such that for all pre-Segal categories $X$

1. $compl(X)$ is a [[Segal category]];

1. $i_X \colon X \to compl(X)$ is an [[isomorphism]] on the sets of objects;

1. $i_X$ is a categorical equivalence if $X$ is already a Segal category;

1. $compl(i_X)$ is a categorical equivalence.

=--

This is ([HS, def. 2.1, lemma 2.2](#HS)).

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

Say a morphism $f : X \to Y$ in $PreSegalCat$ is

* a cofibration precisely if it is a [[monomorphism]];

* a weak equivalence precisely if its completion $compl(f)$ by prop. \ref{Completion} is a categorical equivalence.

=--

(...)

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

This defines a [[model category]] structure for Segal categories (...)

=--

(...)

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

It follows that a map $X \to Y$ between Segal categories is a weak equivalence precisely if it is a categorical equivalence.

Because by prop. \ref{Completion} we have a commuting square of the form

$$
  \array{
    X &\underoverset{\simeq}{i_X}{\to}& compl(X)
    \\
    \downarrow && \downarrow
    \\
    X &\underoverset{\simeq}{i_Y}{\to}& compl(Y)
  }
$$

where the horizontal morphisms are categorical equivalences, and by prop. \ref{ModelStructureForSegalCategories} these satisfy [[2-out-of-3]].

=--

## Properties

### General

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

Equipped with the classes of maps defined in def. \ref{CofibrationsAndEquivalences}, $PreSegalCat$ is a [[model category]] which is

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

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

* a [[Cisinski model structure]].

=--

Cartesian closure is shown in ([Pellissier](#Pellissier)). The fact that it is a Cisinski model structure follows from the result in ([Bergner](#Bergner)).

### Relation to other model structures

See _[[table - models for (infinity,1)-categories]]_.

## Related concepts

* [[model structure for Segal operads]]

## References

The model structure for Segal categories was introduced in

* [[André Hirschowitz]], [[Carlos Simpson]], _Descente pour les $n$-champs (Descent for $n$-stacks)_ ([arXiv:9807049](http://arxiv.org/abs/math/9807049))
 {#HS}

(even for [[Segal n-categories]]). An alternative proof of the existence of this model structure is given in section 5 of

* [[Julie Bergner]], _Three models for the homotopy theory of homotopy theories_ ([arXiv:math/0504334](http://arxiv.org/abs/math/0504334))

The cartesian closure of the model structure was established in

* Regis Pellissier. Catégories enrichies faibles. Thèse, Université de Nice-Sophia Antipolis (2002), ([arXiv:math/0308246](http://arxiv.org/abs/math/0308246))
 {#Pellissier}

The fact that the fibrant Segal categories in this model structure are precisely the [[Reedy model structure|Reedy fibrant]] Segal categories is due to

* [[Julie Bergner]], _A characterization of fibrant Segal categories_, Proceedings of the AMS (2007) ([arXiv:0603400](http://arxiv.org/abs/math/0603400), [pdf](http://www.math.ucr.edu/~jbergner/ReedyFib.pdf))
 {#Bergner}

Model structures for Segal categories enriched over more general [[(∞,1)-categories]] are discussed in section 2 of

* [[Jacob Lurie]], _[[(infinity,2)-Categories and the Goodwillie Calculus]]_

* [pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf)


model structure for Segal operads

+-- {: .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 Segal operads_  is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] of [[(∞,1)-operads]] regarding these as [[enriched (∞,1)-category|∞Grpd-enriched]] [[operads]].

It is the [[operad|operadic]] analog of the _[[model structure for Segal categories]]_: its [[fibrant objects]] are operadic analogs of [[Segal categories]].

## Definition

Write $\Omega$ for the [[tree category]], the [[site]] for [[dendroidal sets]].

### Segal operads

Write $\eta$ for the tree with a single edge and no vertices.
Write

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

for the category of [[simplicial presheaves]] on the tree category -- _simplicial dendroidal sets_ or _dendroidal simplicial sets_ (see [[model structure for complete dendroidal Segal spaces]] for more on this).

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

A **Segal pre-operad** $X \in [\Omega^{op}, sSet]$ is a simplicial dendroidal set such that $X(\eta)$ is a [[discrete object|discrete]] [[simplicial set]] (a plain set regarded as a simplicially constant simplicial set). Write

$$
  SegalPreOperad \hookrightarrow [\Omega^{op}, sSet]
$$

for the [[full subcategory]] on the Segal pre-operads.

A **Segal operad** is a Segal pre-operad such that for every [[tree]] $T \in \Omega$ the [[powering]]

$$
  X^{\Omega[T]} \to X^{Sp(T)}
  \in
  sSet
$$

of the [[spine]] inclusion $(Sp(T) \hookrightarrow T) \in$ [[dendroidal set|dSet]] into $X$ is an [[weak equivalence|acyclic]] [[Kan fibration]]. Write

$$
  SegalOperad \hookrightarrow SegalPreOperad
$$

for the [[full subcategory]] on the Segal operads.

A **Reedy-fibrant Segal operad** is a Segal operad which is moreover fibrant in the [[generalized Reedy model structure]] $[\Omega^{op}, sSet]_{gReedy}$.

=--

This is ([Cisinski-Moerdijk, def. 7.1, def. 8.1](#CisinskiMoerdijk)).

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

The definition of Segal pre-operads encodes a [[set]] of colors of an [[operad]], together with for each [[tree]] $T$ an [[∞-groupoid]] of operations in the operad of the shape of this tree  --- notably $\infty$-groupoids of $n$-ary operations if the tree is the $n$-corolla, $T = C_n$.

The condition on Segal operads encodes the existence of _composition_ of these operad operations by [[∞-anafunctors]]. See the discussion at _[[Segal category]]_ for more on this.

The Reedy fibrancy condition is mostly a technical convenience.

=--

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

The inclusion def. \ref{InclusionOfSegalPreOperads} has a [[left adjoint|left]] and [[right adjoint|right]] [[adjoint functors]]

$$
  sdSet
  \stackrel{\overset{\gamma_!}{\to}}{\stackrel{\overset{\gamma^*}{\leftarrow}}{\underset{\gamma_*}{\to}}}
  SegalPreOperad
  \,.
$$

=--

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

One way to see the existence of the adjoints is to note that
$SegalPreOperad$ is a [[category of presheaves]] over the [[site]] $S(\Omega)$ which is the [[localization]] of $\Omega \times \Delta$ at morphisms of the form $(-,Id_\eta)$, where $\eta$ is the tree with one edge and no vertex. Write

$$
  \gamma : \Delta \times \Omega \to S(\Omega)
$$

for the [[localization]] functor, then the inclusion of Segal pre-operads is the precomposition with this functor

$$
  \gamma^* : SegalPreOperad \simeq [S(\Omega)^{op}, sSet] \hookrightarrow [\Omega^{op}, sSet]
  \,.
$$

Therefore the left and right adjoint to $\gamma^*$ are given by left and right [[Kan extension]] along $\gamma$.

Explicitly, these adjoints are given as follows.

For $X \in [\Omega^{op}, sSet]$, the Segal pre-operad $\gamma_!(X)$ sends a tree $T$
either to $X(T)$, if $T$ is non-linear, hence if it admits no morphism to $\eta$, or else to the [[pushout]]

$$
  \array{
    X(\eta) &\to& X(T)
    \\
    \downarrow && \downarrow
    \\
    \pi_0 X(\eta) &\to& \gamma_!(X)(T)
  }
$$

in [[sSet]], where the top morphism is $X(T \to \eta)$ for the unique morphism to $\eta$.

In words, $\gamma_!(X)$ is obtained from $X$ precisely by contracting the simplicial set of colors to its set of connected components.

=--

### Special morphisms

We discuss morphisms between Segal pre-operads with special properties, which will appear in the model structure.

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

Say a morphism $f$ in $SegalPreOperad$ is a **normal monomorphism** precisely if $\gamma^*(f)$ is a normal monomorphism (see [[generalized Reedy model structure]]),  which in turn is the case if it is simplicial-degreewise a normal morphisms of [[dendroidal sets]] (see there for details).

Correspondingly, a Segal pro-operad $X$ is called _normal_ if $\emptyset \to X$ is a normal monomorphism.

=--

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

A morphism in $SegalPreOperad$ is called an **acyclic fibration** precisely
if it has the [[right lifting property]] against all normal monomorphisms, def. \ref{NormalMonomorphism}.

=--

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

Say a morphism $f$ in $SegalPreOperad$ is a _Segal weak equivalence_ precisely if $\gamma^*(f)$ is a weak equivalence in the [[model structure for dendroidal complete Segal spaces]] $[\Omega^{op}m, sSet]_{gReedy \atop cSegal}$.

=--

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

Call a morphism in $SegalPreOperad$

* a weak equivalence precisely if it is a Segal weak equivalence, def. \ref{SegalWeakEquivalence};

* a cofibration precisely if it is a normal monomorphism, def. \ref{NormalMonomorphism}.

=--

Theorem \ref{ExistenceOfTheModelStructure} below asserts that this is indeed a model category struture whose fibrant objects are the Segal operads.

## Properties

### Of the various classes of morphisms

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

If $f : X \to Y$ in $[\Omega^{op}, sSet]$ is a normal monomorphism and
$\pi_0 X(\eta) \to \pi_0 Y(\eta)$ is a monomorphism, then $\gamma_!(f)$
is normal in $SegalPreOperad$.

=--

([Cis-Moer, lemma 7.4](#CisinskiMoerdijk)).

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

The class of normal monomorphisms in $SegalPreOperad$ is
[[cofibrantly generated model category|generated]] (under
[[pushout]], [[transfinite composition]] and [[retracts]]) by the set

$$
  \{
     \gamma_!(\partial \Delta[n] \times \Omega[T]  \cup \Delta[n] \times \partial \Omega[T])
     \to
    \gamma_! (\Delta[n], \Omega[T])
  \}_{n \in \Delta, T \in \Omega, {\vert T\vert} \geq 1}
 \cup
  \{
    \emptyset \to \eta
  \}
$$

=--

([Cis-Moer, prop 7.5](#CisinskiMoerdijk)).

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

Let $X \in [\Omega^{op}, sSet]_{gReedy \atop Segal}$ be fibrant. Then $\gamma_* X$ is a Reedy fibrant Segal operad. If $X$ is moreover fibrant in $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$ then the [[unit of an adjunction|counit]] $\gamma^* \gamma* X \to X$ is a weak equivalence in $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$.

=--

([Cis-Moer, prop 8.2](#CisinskiMoerdijk)).

+-- {: .num_lemma #AcyclicFibrationsAreIndeedWeakEquivalences}
###### Lemma

An acyclic fibration in $SegalPreOperad$, def. \ref{PreOperadAcyclicFibration}, is also a weak equivalence in $[\Omega^{op}, sSet]_{gReedy \atop Segal}$.

=--

([Cis-Moer, prop 8.12](#CisinskiMoerdijk)).

### Of the model structure itself

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

The structures in  def. \ref{WeakEquivalencesAndFibrations}
make the category $SegalPreOperad$ a [[model category]] which is

* [[cofibrantly generated model category|cofibrantly generated]];

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

=--

This is ([Cis-Moer, theorem 8.13](#CisinskiMoerdijk)).

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

The existence of the cofibrantly generated model structure follows with _[Smith's theorem](combinatorial+model+category#SmithTheorem)_: by the discussion there it is sufficient to notice that

1. the Segal equivalences are an accessibly embedded accessible full subcategory of the arrow category;

1. the acyclic cofibrations are closed under pushout and retract;

   (both of these because these morphisms come from the [[combinatorial model category]] $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$)

1. the morphisms with right lifting against the normal monomorphisms are weak equivalences, by lemma \ref{AcyclicFibrationsAreIndeedWeakEquivalences}.

=--

### Relation to other model structures

We discuss the relation to various other model structures for operads. For an overview see _[[table - models for (infinity,1)-operads]]_.

#### To dendroidal complete Segal spaces

(...) [[model structure for dendroidal complete Segal spaces]]

## References

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal Segal spaces and infinity-operads_ ([arXiv:1010.4956](http://arxiv.org/abs/1010.4956))
 {#CisinskiMoerdijk}

[[!redirects Segal operad]]
[[!redirects Segal operads]]


model structure for weak complicial sets

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of [[weak complicial set]] is a model for (the [[omega-nerve]] of) the notion of [[weak ∞-category]]. The **model structure for weak $\omega$-categories** is a [[model category]] structure on the category of [[stratified simplicial set]]s, such that cofibrant-fibrant objects are precisely the [[weak complicial set]]s. This model structure may therefore be understood as a presentation of the [[(∞,1)-category]] of weak $\omega$-categories.

## References

Section 6 of

* [[Dominic Verity]], _Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory_ ([arXiv:math/0604414](http://arxiv.org/abs/math.CT/0604414)).

[[!redirects model structure for weak ∞-categories]]