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#### Model category theory
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[[!include model category theory - contents]]
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#### Enriched category theory
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[[!include enriched category theory contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[model category|model structure]] $sPSh(C)$ on [[sSet]]-[[enriched functor category|enriched presheaves]] is supposed to be a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]] on an [[sSet-site]] $C$.

It generalizes the [[model structure on simplicial presheaves]] which is the special case obtained when $C$ happens to be just an ordinary [[category]].

This means that in as far as the [[model structure on simplicial presheaves]] models [[∞-stacks]], the model structure on [[sSet]]-[[enriched category|enriched categories]] model [[derived stack]]s.

## Definition

The construction of the model structure on [[sSet]]-[[enriched category|enriched categories]] closely follows the discussion of the [[model structure on simplicial presheaves]], only that everything now takes place in [[enriched category theory]].

+-- {: .num_defn}
###### Notation and conventions

Regard the [[closed monoidal  category]]  [[sSet]] as a [[simplicially enriched category]] in the canonical way.

For $C$ any $sSet$-category, write $sk_1 C$ for its underlying ordinary [[category]].

Write $[C^{op}, sSet]$ for the [[enriched functor category]].

=--

Hence the ordinary category $sk_1 [C^{op}, sSet]$ has as objects [[enriched functor]]s $C^{op} \to sSet$ and a morphism $f : A \to B$ in $sk_1 [C^{op}, sSet]$ is a [[natural transformation]]  given by a collection of morphisms $f_c : A(c) \to B(c)$ in [[sSet]], for each object $c \in C$.

+-- {: .num_defn}
###### Definition
**(global model structure)**

Let $C$ be a [[simplicially enriched category]].

The **global projective model structure** $sk_1[C^{op},sSet]_{proj}$ on $sk_1[C^{op}, sSet]$

* fibrations and weak equivalences are those transformations that are objectwise fibrations and weak equivalences in the standard [[model structure on simplicial sets]].

=--

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

The global projective model structure on $sk_1 [C^{op}, sSet]$ makes the $sSet$-category $[C^{op}, sSet]$ a [[combinatorial simplicial model category]].

=--

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

For $C$ an [[sSet-site]], the **local projective model structure** on $sk_1 [C^{op}, sSet]$ is the [[Bousfield localization of model categories|left Bousfield localization]] of $sk_1 [C^{op}, sSet]_{proj}$ at...

=--

This appears on ([ToënVezzosi, page 14](#ToenVezzosi)).

## Properties

### Over an unenriched site

It seems that the claim is that, indeed, in the special case that $C$ happens to be an ordinary category, the model structure on $sPSh(C)_{proj}^{l loc}$ reproduces the projective [[local model structure on simplicial presheaves]].

### Presentation of $(\infty,1)$-toposes

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

For an [[sSet-site]] $C$ regarded as an [[(∞,1)-site]], the local model structure on $[C^{op}, sSet]$ is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]] on $C$, in that there is an [[equivalence of (∞,1)-categories]]

$$
  ([C^{op}, sSet]_{loc})^\circ \simeq Sh_{(\infty,1)}(C)
  \,.
$$

=--

This is [Lurie, prop. 6.5.2.14, remark 6.5.2.15](#Lurie).

## Examples

### Derived geometry

Where a [[topos]] or [[(∞,1)-topos]] over an ordinary [[site]] encodes [[higher geometry]], over a genuine [[sSet-site]] one speaks of [[derived geometry]]. An [[∞-stack]] on such a higher site is also called a [[derived stack]].

Therefore the model structure on $sSet$-presheaves serves to model contexts of derived geometry. For instance over the [[etale (∞,1)-site]].

(...)

## Related concepts

* [[model topos]]

* [[enriched Reedy category]]

## References

The theory of model structures on $sSet$-enriched presheaf categories was developed in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Topos Theory_ ([arXiv:math/0207028](http://arxiv.org/abs/math/0207028))
{#ToenVezzosi}

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Segal topoi and stacks over Segal categories_ Proceedings of the Program _Stacks, Intersection theory and Non-abelian Hodge Theory_ , MSRI, Berkeley, January-May
2002 ([arXiv:0212330](http://arxiv.org/abs/math/0212330))

The relation to intrinsically defined [[(∞,1)-topos theory]] is around remark 6.5.2.15 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
{#Lurie}

[[!redirects model structure on SimpSet-enriched presheaves]]
[[!redirects model structure on sSet-enriched presheaves]]
[[!redirects model structure on SSet-enriched presheaves]]
[[!redirects model structure on sSet-presheaves]]
[[!redirects model structure on SSet-presheaves]]
[[!redirects model structure on simplicially enriched presheaves]]


model structure on sSet-operads

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#### Higher algebra
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#Contents#
* table of contents
{:toc}

## Idea

The [[operad|operadic]] generalization of the [[model structure on sSet-categories]]: a [[presentable (infinity,1)-category|presentation]] of [[(∞,1)Operad]].

## Definition

All operads considered here are multi-coloured [[symmetric operads]] ([[symmetric multicategories]]).

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

Call a morphism of [[simplicial operads]] $f : P \to Q$

* _fully faithful_ if it is a [[weak homotopy equivalence]] on all component simplicial sets.

* _essentially surjective_ if the induced morphism $\pi_0(f)$ on [[homotopy categories]] is an [[essentially surjective functor]].

* a _weak equivalence_ if it is fully faithful and essentially surjective.

* a _local fibration_ if it is componentwise a [[Kan fibration]].

* a _fibration_ if it is a local fibration and the underlying [[functor]] $\pi_0(f) : Ho(j^* P) \to Ho(j^* Q)$ on the [[homotopy categories]] of the underlying [[simplicial categories]] is an [[isofibration]].

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

This defines on $sSet Operad$ the structure of a [[model category]] which is

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

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

=--

This is ([Cisinski-Moerdijk, theorem 1.14](#CisinskiMoerdijk)).

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

For $C \in $ [[Set]], let $sSet Operad_C \hookrightarrow sSet Operad$
be the full subcategory on operads with $C$ as their set of colours.

Then $sSet Operad_C \simeq (Operad_C)^{\Delta^{op}}$ is the category of [[simplicial objects]] in $C$-coloured [[symmetric operads]], and restricted to this the above model category structure is corresponding the [[model structure on simplicial algebras]].

=--

See ([Cisinski-Moerdijk, remark 1.9](#CisinskiMoerdijk)).

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

Restricted along the inclusion

$$
  j_! : sSet Cat \hookrightarrow sSet Operad
$$

the above model structure restricts to the [[model structure on sSet-categories]] by [[Julie Bergner]].

=--

## Properties

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

A morphism in $sSet Operad$ is an acyclic fibration precisely
if it is componentwise an acyclic [[Kan fibration]].

=--

## Related concepts

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

## References

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets and simplicial operads_ ([arXiv:1109.1004](http://arxiv.org/abs/1109.1004))
 {#CisinskiMoerdijk}

[[!redirects model structure on simplicial operads]]


model structure on strict omega-categories

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## Idea

The _model structure on strict $\omega$-categories_ is a [[model category]] structure that presentes the [[(∞,1)-category]] of [[strict ∞-categories]].

It resticts to the [[model structure on strict ∞-groupoids]].

These structures also go by the name [[canonical model structure]] or [[folk model structure]].

## Properties

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

Every object is fibrant. The acyclic fibrations are precisely the functors that are [[k-surjective functor]]s for all $k \in \mathbb{N}$.

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

The [[transferred model structure]] on [[Str∞Grpd]] along the [[forgetful functor]]

$$
  U : Str \omega Grpd \to Str \omega Cat
$$

exists and coincides with the [[model structure on strict ∞-groupoids]] defined in ([BrownGolasinski](#BrownGolasinski)).

=--

This is proven in ([AraMetayer](#AraMetayer)).

## Related concepts

* [[model structure for weak complicial sets]]

## References

The [[model structure on strict ∞-groupoids]] was introduced in

* [[Ronnie Brown]], [[Marek Golasinski]]; _A model structure for the homotopy theory of crossed complexes_ ([numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1989__30_1_61_0))
{#BrownGolasinski}

The model structure on strict $\omega$-categories was discussed in

* [[Yves Lafont]], [[Francois Métayer]], [[Krzysztof Worytkiewicz]], _A folk model structure on omega-cat_ ([arXiv](http://arxiv.org/abs/0712.0617)) Adv. Math. 224 (2010), no. 3, 1183–1231.

Dicussion of cofibrant [[resolution]] in this model structure by [[polygraph]]s/[[computad]] is in

* [[Francois Métayer]], _Cofibrant complexes are free_ ([arXiv](http://arxiv.org/abs/math.CT/0701746))

* [[Francois Métayer]], _Resolutions by polygraphs_ ([tac](http://www.tac.mta.ca/tac/volumes/11/7/11-07.pdf))

The relation between the model structure on strict $\omega$-categories and that on strict $\omega$-groupoids is established in

* Dimitri Ara, [[Francois Metayer]], _The Brown-Golasinski model structure on strict $\infty$-groupoids revisited_ ([arXiv](http://arxiv.org/abs/1010.2599)) Homology Homotopy Appl. 13 (2011), no. 1, 121–142.
{#AraMetayer}

[[!redirects model structure on strict ∞-categories]]


model structure on strict omega-groupoids

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#Contents#
* table of contents
{:toc}

## Idea

There is a [[cofibrantly generated model category|cofibrantly generated]] ("[[folk model structure|folk]]") [[model category|model structure]] on the category of [[strict ∞-groupoid]]s, equivalently that of [[crossed complex]]es.

It is the [[transferred model structure]] along the [[forgetful functor]] of the [[model structure on strict ∞-categories]].

## References

The model structure was given in

* [[Ronnie Brown]], [[Marek Golasinski]]; _A model structure for the homotopy theory of crossed complexes_ ([numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1989__30_1_61_0)) Cahiers Topologie Géom. Différentielle 30 (1989) 61-82.
{#BrownGolasinski}

The relation to the model structure on strict $\omega$-categories was established in

* Dimitri Ara, [[Francois Metayer]], _The Brown-Golasinski model structure on $\infty$-groupoids revisited_ ([pdf](http://hal.archives-ouvertes.fr/docs/00/52/58/42/PDF/folkgr.pdf)) Homology, Homotopy Appl. 13 (2011), no. 1, 121–142.
{#AraMetayer}

[[!redirects model structure on strict ∞-groupoids]]
[[!redirects model structure on strict ∞-groupoids]]


model structure on symmetric spectra

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#Contents#
* table of contents
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## Idea

A [[model structure on spectra]] for [[symmetric spectra]]

The [[category]] of _symmetric spectra_ is a [[presentable (∞,1)-category|presentation]] of the [[symmetric monoidal (∞,1)-category]] [[stable (infinity,1)-category of spectra|of spectra]], with the special property that it implements the [[smash product of spectra]] such as to yield itself a [[symmetric monoidal category|symmetric]] [[monoidal model category|monoidal]] [[model category of spectra]]: the _[[model structure on symmetric spectra]]_. This implies in particular that with respect to this [[symmetric smash product of spectra]] an [[E-∞ ring]] is presented simply as a plain [[commutative monoid]] [[internalization|in]] symmetric spectra.

## Properties

### Relation to model structures on sequential spectra

There is a [[Quillen equivalence]] to the [[Bousfield-Friedlander model structure]] on [[sequential spectra]] ([HoveyShipleySmith 00, section 4.3](#HoveyShipleySmith00), [Mandell-May-Schwede-Shipley 01, theorem 0.1](#MandellMaySchwedeShipley01)).

### Relation to model structure on $\mathcal{S}$-modules

There is also a [[Quillen equivalence]] to the [[model structure on S-modules]] ([Schwede 01](#Schwede01))

## Related concepts

[[model structure on spectra]]

* [[sequential spectrum]], [[model structure on sequential spectra]]

with [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], **model structure on symmetric spectra**

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* [[S-module]], [[model structure on S-modules]]

* [[excisive functor]], [[model structure on excisive functors]]

* [[Model categories of diagram spectra]]

## References

The projective and injective model structure on symmetric spectra are due to

* {#HoveyShipleySmith00} [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], _Symmetric spectra_, J. Amer. Math. Soc. 13 (2000), 149-208 ([arXiv:math/9801077](http://arxiv.org/abs/math/9801077))

The "S-model structure" (also called "flat model structure" in [Schwede 12, part III](#Schwede12)) is due to

* [[Brooke Shipley]], _A Convenient Model Category for Commutative Ring Spectra_, 2003 ([web](https://www.semanticscholar.org/paper/A-Convenient-Model-Category-for-Commutative-Ring-Shipley/ae906091dab8b64213f94b8ecc8047c37be20144))

See also

* {#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]

Another proof, beyond ([Hovey-Shipley-Smith 00](#HoveyShipleySmith00)), of the Quillen equivalence to the [[Bousfield-Friedlander model structure]] is due to

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], theorem 0.1 of _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

The Quillen equivalence to the [[model structure on S-modules]] is due to

* {#Schwede01} [[Stefan Schwede]], _S-modules and symmetric spectra_, Math. Ann. 319, 517–532 (2001) ([pdf](http://www.math.uni-bonn.de/people/schwede/comparison.pdf))

Comprehensive review is in

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

Generalization to a [[model structure for parameterized spectra]]:

* {#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]])

[[!redirects model structures on symmetric spectra]]

[[!redirects model structure on symmetric spectrum objects]]
[[!redirects model structures on symmetric spectrum objects]]