(infinity,1)-category

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#### $(\infty,1)$-Category theory
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#### Homotopy theory
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#### Higher category theory
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#### $(\infty,1)$-topos theory
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#Contents#
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## Idea

According to the general pattern on [[(n,r)-category]], an $(\infty,1)$-category is a (weak) [[∞-category]] in which all $n$-morphisms for $n \geq 2$ are [[equivalences]]. This is the joint generalization of the notion of _[[category]]_ and _[[∞-groupoid]]_.

More precisely, this is the notion of _[[category]]_ up to [[coherence|coherent]] [[homotopy]]:
an $(\infty,1)$-category is equivalently

* an [[internal category in an (∞,1)-category|internal category]] in [[∞-groupoids]]/basic [[homotopy theory]] (as such usually modeled as a [[complete Segal space]]).

* a category [[enriched (infinity,1)-category
|homotopy enriched]] over [[∞Grpd]] (as such usually modeled as a [[Segal category]]).

Among all [[(n,r)-category|(n,r)-categories]], $(\infty,1)$-categories are special in that they are the simplest structures that at the same time:

* admit a [[higher category theory|higher version]] of [[category theory]] ([[limit]]s, [[adjunction]]s, [[Grothendieck construction]], etc, [[sheaf and topos theory]], etc.) : [[(infinity,1)-category theory]]

* and know everything about higher [[equivalences]].

Notably for understanding the collections of all [[(n,r)-category|(n,r)-categories]] for arbitrary $n$ and $r$, which in general is an $(n+1,r+1)$-category, the knowledge of the underlying $(n,1)$- (and hence $(\infty,1)$-)category already captures much of the information of interest: it allows to decide if two given $(n,r)$-categories are equivalent and allows to obtain new $(n,r)$-categories from existing ones by universal constructions.

The collection of all $(\infty,1)$-categories forms the [[(∞,2)-category]] [[(∞,1)Cat]].

## Definitions

There are a number of different ways to make the idea of an $(\infty,1)$-category precise, including [[quasi-categories]], [[simplicial set|simplicially]] [[enriched categories]], [[topologically enriched categories]], [[Segal categories]], [[complete Segal spaces]], and $A_\infty$-[[A-infinity categories|categories]] (most of which can be done either simplicially or topologically).  Additionally, any notion of [[∞-category]] can be specialized to a notion of $(\infty,1)$-category by simply requiring all $n$-cells for $n\gt 1$ to be invertible.

Unlike the case for general notions of $n$-category, almost all the definitions of $(\infty,1)$-category are known to form [[model categories]] that are [[Quillen equivalence|Quillen equivalent]].  See also [[n-category]] for a summary of the state of the art about definitions of $n$-category and comparisons between them.

### Quasi-categories

We start with the definition of "$(\infty,1)$-category" that was promoted by [[Andre Joyal]] as a good model for the theory. This goes back to Boardman-Vogt in the 1970s and was further developed, by [[Jean-Marc Cordier]] and [[Tim Porter]] in the early 1980s.

This is a [[geometric definition of higher category]] which conceives an $(\infty,1)$-category as a [[simplicial set]] with extra [[stuff, structure, property|property]]. It is a straightforward generalization of the definition of [[∞-groupoid]] as a [[Kan complex]], and, in fact, one alternative term used early on was 'weak Kan complex';  see below.

Recall that a [[Kan complex]] is a [[simplicial set]] in which every [[horn]] $\Lambda^k[n]$, $0 \leq k \leq n$ has a _filler_. This condition may be read in words as: every collection of adjacent $n$-cells has a composite $n$-cell, even if the orientations of the cells don't match. This implicitly encodes the _invertibility_ of every cell: if the orientation does not match, we can invert the cell and then compose.

From this perspective one observes, by looking closely at the combinatorics, that the invertibility of the 1-cells in the simplicial set is enforced particularly by the condition that the [[horn|outer horns]] $\Lambda^0[n]$ and $\Lambda^n[n]$ have fillers.

Therefore in a [[simplicial set]] in which _only the inner horns_ $\Lambda^k[n]$ for $0 \lt k \lt n$ have fillers all cells are required to have a kind of inverse, except the 1-cells. (They may have inverses, too, but are not required to).

This is evidently a realization of the idea of an [[(n,r)-category]] with $n = \infty$ and $r = 1$.

Such a simplicial set with fillers for all inner horns

* Boardman and Vogt called a _weak Kan complex_ ;

* [[Andre Joyal]] called a [[quasi-category]];

* [[Jacob Lurie]] called an $\infty$-category.

Here we follow Joyal and say [[quasi-category]] when we mean concretely the simplicial sets with extra property. We use the more general term
"$(\infty,1)$-category" for this or any of its equivalent models, discussed below, in order to distinguish from the term [[∞-category]] or [[∞-category]] that is more traditionally understood to generically mean an $\infty$-category with no conditions on invertibility
(in terms of [[(n,r)-category]]: an $(\infty,\infty)$-category).

With [[quasi-category|quasi-categories]] being just [[simplicial sets]] with extra property, there are evident and simple definitions of

* the [[(infinity,1)-category of (infinity,1)-functors|quasi-category of (∞,1)-functors]] between two [[quasi-categories]] $C$ and $D$;

* the quasi-category of all [[∞-groupoids]];

* the [[(infinity,1)-category of (infinity,1)-categories|quasi-category of all quasi-categories]].

Similarly, [[Andre Joyal]] and [[Jacob Lurie]] have shown that all other constructions in [[category theory]] have good generalizations to quasi-categories, which usually have conceptually simple formulations: see [[Higher Topos Theory]] for more.

### $Top$-, $Kan$- and simplicially enriched categories

Despite the conceptual simplicity of [[quasi-category|quasi-categories]],
for computations and in particular for obtaining examples,
it is often useful to pass to a slightly different model.

Recall that we said at the beginning that an $(\infty,1)$-category is supposed to be like an  [[enriched category]] which is enriched over the category of [[∞-groupoids]].
This turns out to make sense literally
if one takes care to remember that $\infty$-groupoids
themselves form a higher category.

As discussed at  [[homotopy hypothesis]] there is a Quillen equivalence of the [[model category|model categories]] of

* the [[model structure on topological spaces|standard model structure]] on the [[nice category of spaces|nice category]] of compactly generated weakly Hausdorff [[topological spaces]];

* the [[model structure on simplicial sets|standard model structure]] on the category of [[Kan complex]]es.

In fact, this is also equivalent to

* the [[model structure on simplicial sets|standard model structure]] on the category [[SimpSet|SSet]] of [[simplicial sets]].

If we take the notion of [[Kan complex]] to be the most manifest incarnation of the idea "[[∞-groupoid]]", then under these equivalences one may think of

* a [[simplicial set]] as representing the [[Kan complex]] which is obtained from it by "freely throwing in the missing inverses" of cells (technically: as representing its fibrant replacement);

* a [[topological space]] $X$ as representing the [[Kan complex]] $\Pi(X)$, whose

  * 0-cells are the points of $X$;

  * 1-cells are the paths in $X$;

  * 2-cells are the triangles in $X$;

  * etc.

With this interpretation understood (i.e. with these model structures understood), [[SimpSet|SSet]]-enriched categories do model $(\infty,1)$-categories.

For more see

* [[relation between quasi-categories and simplicial categories]]

### Homotopical categories

A [[homotopical category]] is a category $C$
equipped with a class $W$ of [[category with weak equivalences|weak equivalences]].
Every homotopical category $(C,W)$ has a _quasi-localisation_ $C[W(-1)]$
which is a [[quasi-category]]. The simplicial set $C[W(-1)]$ is
obtained from the [[nerve]] of $C$ by freely gluing a [[homotopy]] inverse to each [[morphism]] in $W$, and then, by adding simplices to turn it into a quasi-category (this last step is called a fibrant completion).

The [[quasi-category]] $C[W(-1)]$ is equivalent to the [[Dwyer-Kan localisation]] of $C$ with respect to $W$, via the equivalence between [[quasi-category|quasi-categories]] and [[simplicially enriched category|simplicial categories]] mentioned above.

Conversely, every quasi-category is equivalent to
the quasi-localisation of a homotopical category.
This gives a representation of all $(\infty,1)$-categories
in terms of [[homotopical category|homotopical categories]].
It follows that many aspects of the theory of $(\infty,1)$-categories
can be expressed in terms of [[category theory]].

When the homotopical category (C,W) is obtained from a Quillen
model structure (by forgetting the cofibrations and the fibrations)
the quasi-category C[W^(-1)] has finite limits and colimits.
Conversely, I conjecture that every quasi-category
with finite limits and colimits is equivalent to the quasi-localisation
of a model category. In fact, every locally presentable quasi-category
is a quasi-localisation of a combinatorial model by a result of Lurie.
More can be said: the underlying category can taken to be
a category of presheaves by a result of Daniel Dugger.

http://arxiv.org/abs/math/0007070

### Model categories

A specific notion of [[homotopical category]] is that of a [[model category]]. $(\infty,1)$-categories obtained as the Dwyer-Kan simplicial localizations of model categories have for instance finite $(\infty,1)$-[[limit]]s and $(\infty,1)$-[[colimit]]s. The [[presentable (∞,1)-category|locally presentable (∞,1)-categories]] are precisely those presented this way by [[combinatorial model category|combinatorial model categeories]].

At the very beginning, a [[model category]] was understood as a "model for the category [[Top]] of topological spaces," or more precisely [[homotopy types]]: some [[category]] with extra [[stuff, structure, property|structure and properties]] which allows one to perform all operations familiar of the [[homotopy theory]] of [[topological spaces]].

As mentioned above, from the point of view of [[(∞,1)-categories]], [[Top]] may naturally be regarded an as [[(∞,1)-category]] and is in fact the archetypical example, analogous to how [[Set]] is the archetypical example of an ordinary [[category]].

This indicates that, more generally, a [[model category]] should actually be a means to model (i.e. encode) in 1-categorical terms an $(\infty,1)$-category, and of course this is true since indeed any [[category with weak equivalences]] presents an $(\infty,1)$-category via Dwyer-Kan simplicial localization.  In the case of a model category, however, or at least a [[simplicial model category]], this $(\infty,1)$-category has a different, simpler construction.

* A [[simplicial model category]] $\mathbf{A}$ is, in particular, a [[simplicially enriched category]].

* the full [[SSet]]-[[subcategory]] $\mathbf{A}^\circ$ on the fibrant-cofibrant objects of $\mathbf{A}$ happens to be [[Kan complex]]-[[enriched category|enriched]];

* the [[homotopy coherent nerve]] $N(\mathbf{A}^\circ)$ of $\mathbf{A}^\circ$ is the [[quasi-category]] _presented_ by $A$.

Up to equivalence, this gives the same $(\infty,1)$-category as the Dwyer-Kan hammock localization.  With the relation between [[simplicially enriched categories]] and [[quasi-category|quasi-categories]] via [[homotopy coherent nerve]] understood, we shall here often not distinguish between $\mathbf{A}^\circ$ and $N(\mathbf{A}^\circ)$ as the $(\infty,1)$-category [[presentable (infinity,1)-category|presented]] by a [[model category]] $A$.

### Segal categories and complete Segal spaces

Other models for $(\infty,1)$-categories are

 * [[Segal categories]];

 * [[complete Segal spaces]].

Segal categories can be thought of as categories which are
_weakly_ enriched in topological spaces/simplicial sets/Kan complexes,
where the definition of "weak" makes use of the notion of [[homotopy]]
and [[homotopy limit]] in [[Top]] or [[SimpSet|SSet]].

Complete Segal spaces are like [[internal categories in an (∞,1)-category]].

This construction principle in particular lends itself
to iteration and hence to an inductive definition of
[[(∞,n)-category]] via [[Segal n-categories]] and [[n-fold complete Segal spaces]].

### $A_\infty$-categories

An $A_\infty$-[[A-infinity category|category]] can also be thought of as a category "weakly enriched" in spaces (i.e. $\infty$-groupoids), except that in contrast to the Segal approaches the "weakness" is specified [[algebraic definition of higher category|algebraically]] and parametrized by an [[operad]].  This approach can be generalized to the [[Trimble n-category|Trimble]] definition of $n$-category or $(\infty,n)$-category.

## Properties

### $(\infty,1)$-Category theory

A crucial point about the notion of _$(\infty,1)$-category_ is that it supports all the standard constructions and theorems of [[category theory]], if only the consistent replacements are made ([[isomorphism]] becomes [[equivalence]], etc.).

See _[[(∞,1)-category theory]]_.

### The collection of all $(\infty,1)$-categories

The collection of all $(\infty,1)$-categories
forms an [[(∞,2)-category]] called [[(∞,1)Cat]].

Often it is useful to regard that as a (large) $(\infty,1)$-category itself, by discarding the non-invertible [[natural transformations]].

### Model category presentations

There is a wealth of different presentations of $(\infty,1)$-categories.

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

## A model-independent approach

In practice, it can be useful to be able to treat all "presentations of $(\infty,1)$-categories" on the same equal footing (e.g. relative categories and topologically-enriched categories).  While truly model-independent foundations of $(\infty,1)$-category theory do not (yet) exist, this can be accomplished _within_ any model of $(\infty,1)$-categories, which we proceed to describe.  As quasicategories are by far the most well-developed, we use them as an ambient framework.  We also take care to make as few choices (even "contractible" ones) as possible.  However, we do not explicitly mention set-theoretic issues, though these are easily handled using Grothendieck universes.

1. Consider the $Kan$-enriched category $\underline{QCat}$ of quasicategories; for quasicategories $C$ and $D$, the Kan complex of morphisms between them is $\underline{hom}_{\underline{QCat}} = \iota(\underline{hom}_{sSet}(C,D))$, the largest Kan complex contained in their internal hom simplicial set.

1. Define a _relative quasicategory_ to be a quasicategory equipped with a wide sub-quasicategory of "weak equivalences" containing all equivalences.  For relative quasicategories $(C,W_C)$ and $(D,W_D)$, write $\underline{hom}_{\underline{RelQCat}}((C,W_C),(D,W_D)) \subset \underline{hom}_{\underline{QCat}}(C,D)$ for the sub-Kan complex consisting of those maps which take $W_C$ into $W_D$.  Note that using this definition, this is actually the inclusion of a disjoint union of connected components among Kan complexes (in the strictest possible sense).

1. There is an evident inclusion $min : \underline{QCat} \to \underline{RelQCat}$, which takes a quasicategory $C$ to the relative quasicategory $(C,C^\simeq)$.

1. Although a Quillen equivalence $M_1 \rightleftarrows M_2$ between model categories determines an equivalence of homotopy categories, note that neither adjoint functor need preserve weak equivalences.  On the other hand, the restrictions $M_1^c \hookrightarrow M_1 \rightarrow M_2$ and $M_1 \leftarrow M_2 \hookleftarrow M_2^f$ (to the cofibrant objects of $M_1$ and the fibrant objects of $M_2$) do preserve weak equivalences, and these determine a hexagonal diagram of weak equivalences between relative categories (in the Barwick--Kan model structure), as in [MazelGee16, Figure 1](http://nyjm.albany.edu/j/2016/22-4.html).

1. Using the previous observation, expand the diagram in the introduction of [BarwickSchommerPries](http://arxiv.org/abs/1112.0040) (relating a great many Quillen equivalent model categories presenting "the homotopy theory of $(\infty,1)$-categories") into a diagram of weak equivalences between relative categories.  As relative categories are particular examples of relative quasicategories, this defines a functor $F : K \to \underline{RelQCat}$ among fibrant objects of $(Cat_{sSet})_{Bergner}$.

1. Now, apply the right Quillen equivalence $N^{hc} : (Cat_{sSet})_{Bergner} \to sSet_{Joyal}$ (the homotopy-coherent nerve) to this cospan $\underline{QCat} \xrightarrow{min} \underline{RelQCat} \leftarrow K$.

1. The morphism $N^{hc}(min)$ of quasicategories admits a contractible Kan complex worth of quasicategorical left adjoints, any of which presents the _localization_ of relative quasicategories.  Choose one, and denote this quasicategorical adjunction by $L : N^{hc} (\underline{RelQCat}) \rightleftarrows N^{hc} ( \underline{QCat}) : min$.

1. It follows from the main theorem of [Toen](http://arxiv.org/abs/math/0409598) that the composite map $L \circ N^{hc}(F) : N(K) \cong N^{hc}(K) \to N^{hc}(\underline{RelQCat}) \to N^{hc}(\underline{QCat})$ is "essentially contractible" in the quasicategorical sense.  More precisely, for any cofibration into an acyclic object $i : N(K) \to K' \approx pt$ in $sSet_{Joyal}$, there exists a contractible Kan complex worth of extensions of $L \circ N^{hc}(F)$ over $i$.

1. Define $(The(\infty,1)Cats) \subset N^{hc}(\underline{QCat})$ to be the maximal sub-Kan complex generated by the image of $L \circ N^{hc}(F)$.  We write $Cat_{(\infty,1)} \in (The(\infty,1)Cats)$ for _any_ vertex, and propose that to work "model-independently" is to work _within_ $Cat_{(\infty,1)}$.

This sequence of maneuvers balances twin aims.  On the one hand, Toen's theorem asserts that after choosing a basepoint, this Kan complex is a model for $B(\mathbb{Z}/2)$.  Thus, any sort of object which might be considered as "a presentation of an $(\infty,1)$-category" canonically determines an object of $Cat_{(\infty,1)}$ (where "canonical" must still be taken in the quasicategorical sense).  On the other hand, it is completely independent of which vertex of $(The(\infty,1)Cats)$ we choose.

[This diagram](http://ncatlab.org/nlab/files/relcats-modelcats-qcats-inftycats.jpg), taking place in $N^{hc}(\underline{QCat})$, elaborates on certain salient aspects of the passage from models of $(\infty,1)$-categories to a model-independent approach.  (For a small amount of explanation of this diagram, see [here](https://nforum.ncatlab.org/discussion/7029/a-diagram-relating-different-models-of-inftycategories/#Item_0).)

## Related concepts

* [[table - models for (∞,1)-categories]]

* [[0-category]], [[(0,1)-category]]

* [[category]]

* [[2-category]], [[(2,1)-category]]

* [[3-category]]

* [[n-category]]

* [[(∞,0)-category]]

* [[(n,1)-category]]

* **(∞,1)-category**, [[internal (∞,1)-category]], [[∞-groupoid]]

  * [[locally cartesian closed (∞,1)-category]]

  * [[(∞,1)-topos]]

  * [[semiadditive (∞,1)-category]], [[additive (∞,1)-category]]

  * [[stable (∞,1)-category]]

  * [[monoidal (∞,1)-category]]

  * [[locally presentable (∞,1)-category]], [[accessible (∞,1)-category]], [[compactly generated (∞,1)-category]]

  * [[disjunctive (∞,1)-category]]

  * [[model (∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* [[(n,r)-category]]

## References

### General

For several years [[Andre Joyal]] -- who was one of the first to promote the idea that for studying [[higher category theory]] it is good to first study $(\infty,1)$-categories in terms of [[quasi-category|quasi-categories]] -- has been preparing a textbook on the subject. This still doesn't quite exist, but an extensive write-up of lecture notes does:

* {#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)_, Quadern **45** 2, Centre de Recerca Matemàtica, Barcelona 2008 ( [[JoyalTheoryOfQuasiCategories.pdf:file]])

Further notes (where the term "[[logos]]" is used instead of [[quasi-category]]):

* {#Joyal08} [[André Joyal]], _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf), [[JoyalOnLogoi2008.pdf:file]])

Meanwhile [[Jacob Lurie]], building on Joyal's work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\infty,1)$-categories in terms of the models [[quasi-category]] and [[simplicially enriched category]] is

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

An brief exposition from the point of view of [[algebraic topology]] is in

* [[Jacob Lurie]], _What is... an $\infty$-category?_, _Notices of the AMS_, September 2008 ([pdf](http://www.ams.org/notices/200808/tx080800949p.pdf))

A useful comparison of the four [[model category]] structures on

 * [[quasi-categories]];

 * [[simplicially enriched categories]];

 * [[Segal categories]];

 * [[complete Segal spaces]].

is in

* [[Julie Bergner]], _A survey of $(\infty,1)$-categories_,  In: [[John Baez]], [[Peter May]] (eds.),  _[[Towards Higher Categories]]_, The IMA Volumes in Mathematics and its Applications, vol 152, Springer 2007 ([arXiv:math/0610239](http://arxiv.org/abs/math/0610239), [doi:10.1007/978-1-4419-1524-5_2](https://doi.org/10.1007/978-1-4419-1524-5_2))

* [[Julia Bergner]], _Equivalence of models for equivariant $(\infty,1)$-categories_, Glasgow Mathematical Journal, Volume 59, Issue 1 (2016) ([arXiv:1408.0038](https://arxiv.org/abs/1408.0038), [doi:10.1017/S0017089516000136](https://doi.org/10.1017/S0017089516000136))

More discussion of the other two models can be found at

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

and in the references listed at _[[(∞,n)-category]]_.

The relation between [[quasi-category|quasi-categories]] and [[simplicially enriched categories]] was discussed in detail in

* [[Dan Dugger]], [[David Spivak]], _Rigidification of quasi-categories_ ([arXiv:0910.0814](http://arxiv.org/abs/0910.0814))

* [[Dan Dugger]], [[David Spivak]], _Mapping spaces in quasi-categories_, Algebraic & Geometric Topology 11 (2011) 263–325 ([arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263))

The presentation of $(\infty,1)$-categories by [[homotopical categories]] and [[model categories]] is discussed in

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]],  [[Jeff Smith]], _[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]_ , volume 113 of Mathematical Surveys and Monographs

A model by [[stratified spaces]] is in

* {#AyalaFrancisRozenblyum15} [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], _A stratified homotopy hypothesis_ ([arXiv:1502.01713](http://arxiv.org/abs/1502.01713))

A more model-independent abstract formulation is discussed in

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Infinity category theory from scratch_, Higher Structures **4** 1 (2020) [[arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf), [lectures](https://www.epfl.ch/labs/hessbellwald-lab/seminar/ytm2015/)]

* {#RiehlVerity22} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$

For discussion in [[homotopy type theory]] see _[[internal category in homotopy type theory]]_ and see

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], _A type theory for synthetic $\infty$-categories_ ([arXiv:1705.07442](https://arxiv.org/abs/1705.07442))

* {#Riehl18} [[Emily Riehl]], _The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories_, talk at [Vladimir Voevodsky Memorial Conference 2018](http://www.math.ias.edu/vvmc2018) ([pdf](http://www.math.jhu.edu/~eriehl/Voevodsky.pdf))

### Surveys and lecture notes
 {#LectureNotes}

An introduction to [[higher category theory]] through $(\infty,1)$-categories:

* Omar Antolín Camarena, _A whirlwind tour of the world of $(\infty,1)$-categories_, 2013 ([arXiv:1303.4669](http://arxiv.org/abs/1303.4669))

Elementary exposition with an eye towards [[homotopy type theory]]:

* {#Riehl22} [[Emily Riehl]], *$\infty$-Category theory for undergraduates*, [talk](Center+for+Quantum+and+Topological+Systems#RiehlDec2022) at *[[CQTS]]* (Dec. 2022) [[web](Center+for+Quantum+and+Topological+Systems#RiehlDec2022), video: [YT](https://www.youtube.com/watch?v=7g2rkiFsbXo)]

* {#Riehl23} [[Emily Riehl]], *[Could $\infty$-category theory be taught to undergraduates?](https://www.ams.org/journals/notices/202305/noti2692/noti2692.html?adat=May%202023&trk=2692&galt=feature&cat=feature&pdfissue=202305&pdffile=rnoti-p727.pdf)*, Notices of the AMS (May 2023) [[published pdf](https://www.ams.org/journals/notices/202305/rnoti-p727.pdf), [arxiv:2302.07855](https://arxiv.org/abs/2302.07855)]

A foundational set of lecture notes:

* [[Denis-Charles Cisinski]], _[[Higher Categories and Homotopical Algebra]]_,  Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

A survey with an eye towards [[higher algebra]] is in

* [[Moritz Groth]], _A short course on $\infty$-categories_ ([pdf](https://arxiv.org/pdf/1007.2925))

A survey on various notions of [[homotopical categories]]:

* [[Emily Riehl]], _Homotopical categories: from model categories to (∞,1)-categories_, in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds,) *[[Stable categories and structured ring spectra]]*,  MSRI Book Series, Cambridge University Press ([arXiv:1904.00886](https://arxiv.org/abs/1904.00886))

Also:

* [[Markus Land]], *Introduction to Infinity-Categories*, Birkhäuser 2021 ([doi:10.1007/978-3-030-61524-6](https://link.springer.com/book/10.1007/978-3-030-61524-6))

Lecture notes:

* [[Dylan Wilson]], _Lectures on higher categories_ ([pdf](https://sites.google.com/a/uw.edu/dwilson/notes))

* [[Emily Riehl]], _[[Categorical Homotopy Theory]]_

See also

* [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_

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(infinity,1)-category of (infinity,1)-categories

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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The collection of all [[(∞,1)-categories]] forms naturally the [[(∞,2)-category]] [[(∞,1)Cat]].
But for many purposes it is quite sufficient to regard only invertible [[natural transformations]] between [[(∞,1)-functor]], which means that one needs just the maximal [[(∞,1)-category]] inside that $(\infty,2)$-category of all $(\infty,1)$-categories.

Given that an $(\infty,1)$-category is a context for abstract [[homotopy theory]], the $(\infty,1)$-category of $(\infty,1)$-categories is also called the the _homotopy theory of homotopy theories_ ([Rezk 98](#Rezk98), [Bergner 07](#Bergner07)).

(Another, complementary, truncation is to the [[homotopy 2-category of (∞,1)-categories]].)

## Definition

### Intrinsic definition

The full [[SSet]]-[[enriched category theory|enriched]]-[[subcategory]] of [[SSet]] on those simplicial sets which are [[quasi-categories]] is -- by the properties discussed at [[(∞,1)-category of (∞,1)-functors]] --  itself a [[quasi-category]]-[[enriched category]]. This is the [[(∞,2)-category]] of [[(∞,1)-categories]].

The [[sSet]]-[[subcategory]] of that obtained by picking of each [[hom-object]] the [[core]], i.e. the maximal [[∞-groupoid]]/[[Kan complex]] yields an [[∞-groupoid]]/[[Kan complex]]-[[enriched category]]. This is the **$(\infty,1)$-category of $(\infty,1)$-categories** in its incarnation as a [[simplicially enriched category]]. Forming its [[homotopy coherent nerve]] produces the **quasi-category of quasi-categories** .

### Models

The [[Andre Joyal|Joyal]]-[[model structure for quasi-categories]] is an $sSet_{Joyal}$-[[enriched model category]] and hence its full [[SSet]]-[[subcategory]] on cofibrant-fibrant objects is the $(\infty,2)$-category of $(\infty,1)$-categories.

An $SSet_{Quillen}$-[[enriched model category]] (i.e. enriched over the ordinary [[model structure on simplicial sets]]) whose full subcategory of fibrant-cofibrant objects is the $(\infty,1)$-category $(\infty,1)Cat$ is the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]] (over the terminal set). Its underlying plain model category is [[Quillen equivalence|Quillen equivalent]] to the Joyal-model structure, but it is indeed $sSet_{Quillen}$-enriched.

Other model structures that present the $(\infty,1)$-category of all $(\infty,1)$-categories are

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

* [[model structure on simplicially enriched categories]]

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

### The nerve into simplicial spaces

The nerve functor

$$
  N : (\infty,1)Cat_1 \to PSh(\Delta, \infty Gpd) : C \mapsto n \mapsto Core(C^{[n]})
$$

is fully faithful. Thus, the $(\infty,1)$-category of $(\infty,1)$-categories can be identified with the $(\infty,1)$-category of [[category object in an (infinity,1)-category#GrpdInCatIsChoiceOfGroupoidObjects|internal categories in $\infty Gpd$]]

This is closely related to the complete Segal space model.

$N$ is, in fact, the embedding of a [[reflective sub-(infinity,1)-category]]. The $(\infty,1)$-categories can be identified with the subcategory of $PSh(\Delta, \infty Gpd)$ of local objects with respect to the spine inclusions $Sp^n \subseteq \Delta^n$ and with the map $J \to 1$, where $J$ is the indiscrete simplicial space on two discrete objects.

Alternatively, the map $J \to 1$ can be replaced with the projection from the simplicial discrete space formed from the union of two 2-simplices expressing the idea of a morphism with a left and right inverse $fg \simeq 1$ and $gh \simeq 1$.

## Applications

* of particular interest is the $(\infty,1)$-subcategory $(\infty,1)PresCat_1 \hookrightarrow (\infty,1)Cat_1$ of [[presentable (infinity,1)-category|presentable (∞,1)-categories]].

## Related concepts

* [[homotopy 2-category of (∞,1)-categories]]

## References

In terms of [[complete Segal spaces]]:

* {#Rezk98} [[Charles Rezk]], _A model for the homotopy theory of homotopy theory_, Trans. Amer. Math. Soc. 353 (2001), 973-1007 ([arXiv:math/9811037](https://arxiv.org/abs/math/9811037), [doi:10.1090/S0002-9947-00-02653-2](https://doi.org/10.1090/S0002-9947-00-02653-2))

* {#Bergner07} [[Julia Bergner]], _Three models for the homotopy theory of homotopy theories_, Topology Volume 46, Issue 4, September 2007, Pages 397-436 ([arXiv:math/0504334](https://arxiv.org/abs/math/0504334), [doi:10.1016/j.top.2007.03.002](https://doi.org/10.1016/j.top.2007.03.002))

In terms of [[quasi-categories]]:

* [[Jacob Lurie]], Chapter 3 of: _[[Higher Topos Theory]]_ (2009)

[[!redirects (∞,1)-category of (∞,1)-categories]]

[[!redirects homotopy theory of homotopy theories]]
[[!redirects homotopy theory of homotopy theory]]

[[!redirects homotopy theories of homotopy theories]]
[[!redirects homotopy theories of homotopy theory]]


(infinity,1)-category of (infinity,1)-functors

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the notion of [[functor category]] from [[category theory]] to [[(∞,1)-category|(∞,1)]]-[[higher category theory]].

## Definition

Let $C$ and $D$ be [[(∞,1)-categories]], taken in their incarnation as [[quasi-category|quasi-categories]]. Then

$$
  Func(C,D) := sSet(C,D)
$$

is the [[simplicial set]] of morphisms of simplicial sets between $C$ and $D$ (in the standard [[sSet]]-[[enriched category|enrichment]] of $SSet$):

$$
  sSet(C,D) := [C,D] := ([n] \mapsto Hom_{sSet}(\Delta[n]\times C,D))
  \,.
$$

The objects in $Fun(C,D)$ are the [[(∞,1)-functors]] from $C$ to $D$, the morphisms are the corresponding [[natural transformations]] or [[homotopy|homotopies]], etc.

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

The simplicial set $Fun(C,D)$ is indeed a [[quasi-category]].

In fact, for $C$ and $D$ any simplicial sets, $Fun(C,D)$ is a [[quasi-category]] if $D$ is a [[quasi-category]].

=--

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

Using that [[sSet]] is a [[closed monoidal category]] the [[horn]] filling conditions

$$
  \array{
    \Lambda[n]_i &\to& [C,D]
    \\
    \downarrow & \nearrow
    \\
    \Delta[n]
  }
$$

are equivalent to

$$
  \array{
    C \times \Lambda[n]_i &\to& D
    \\
    \downarrow & \nearrow
    \\
    C \times \Delta[n]
  }
  \,.
$$

Here the vertical map is [[fibrations of quasi-categories|inner anodyne]] for inner horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$, and hence the lift exists whenever $D$ has all inner horn fillers, hence when $D$ is a [[quasi-category]].

=--

For the definition of $(\infty,1)$-functors in other models for $(\infty,1)$-categories see [[(∞,1)-functor]].

## Properties

### Model category presentation
 {#ModelCategoryPresentation}

The projective and injective [[global model structure on functors]] (as well as the [[Reedy model structure]] if $C$ is a [[Reedy category]]) [[presentable (infinity,1)-category|presents]] $(\infty,1)$-categories of $(\infty,1)$-functors, at least when there exists [[combinatorial simplicial model category|combinatorial simplicial]]-[[structure]] model on the [[codomain]] [[model category]].

Let

* $C$ be a [[small category|small]] [[sSet]]-[[enriched category]];

* $A$ a [[combinatorial simplicial model category]] and

  $A^\circ$ its [[full subcategory|full]] [[sSet-enriched category|sSet-]][[subcategory]] of [[bifibrant objects]];

* $[C,A]$ the [[sSet]]-[[enriched functor category]] equipped with the projective[^1] [[global model structure on functors]], and

  $[C,A]^\circ$ its full [[sSet]]-[[subcategory]]  on [[bifibrant objects]].

* $N \colon sSet\text{-}Cat \to sSet$ the [[homotopy coherent nerve]].

[^1]: [Lurie's Remark 4.2.4.5](#Lurie) claims that Prop. \ref{PresentationByModelStructuresOnFunctors} also works also for the injective structure, but this is not so clear, see also [MO:q/440965](https://mathoverflow.net/q/440965/381).

[[right adjoints preserve limits|Since]]$\;N$ is a [[right adjoint]] it [[preserved limit|preserves]] [[products]] so that we obtain a morphism

\[
  N(C) \times N\big([C,A]\big)
   \xrightarrow{\; \sim \;}
  N\big(C \times [C,A]\big)
   \xrightarrow{\; N(ev) \;}
  N(A)
\]

induced from the [[internal hom]]-[[adjunct]] of $Id \colon [C,A] \to [C,A]$.

Noticing that the [[bifibrant objects]] of $[C,A]$ are [[enriched functors]] that, in particular, take values in [[bifibrant objects]] of $A$, this restricts to a morphism of the form

$$
  N(C) \times N\big([C,A]^\circ\big)
  \xrightarrow{\;\; N(ev) \;\;}
  N\big(A^\circ\big)
  \,,
$$

which, by the [[internal hom]]-[[adjunction]], corresponds to a morphism

$$
  N\big([C,A]^\circ\big)
  \xrightarrow{\phantom{--}}
  sSet\big(
    N(C)
    ,\,
    N(A^\circ)
  \big)
  \,.
$$

Here $A^\circ$ is [[Kan complex]]-[[enriched category|enriched]], by the [[axioms]] of an [[classical model structure on simplicial sets|$sSet_{Quillen}$-]][[enriched model category]], and so $N(A^\circ)$ is a [[quasi-category]]. Therefore we may write this as

$$
  \cdots
  \;=\;
  Func\big(
    N(C)
    ,\,
    N(A^\circ)
  \big)
  \,.
$$

\begin{proposition}
\label{PresentationByModelStructuresOnFunctors}
This canonical morphism

\[
  \label{TheNerveComparisonEquivalence}
  N\big([C,A]^\circ\big)
    \xrightarrow{\phantom{---}}
  Func\big(
    N(C)
    ,\,
    N(A^\circ)
  \big)
\]

is an [[equivalence of (infinity,1)-categories|equivalence of $\infty$-categories]] in that it is a [[weak equivalence]] in the [[model structure for quasi-categories]].
\end{proposition}
This is [Lurie (2009), Prop. 4.2.4.4](#Lurie).

\begin{proof}
The strategy is to show that the objects on both sides are both [[exponential objects]] in the [[homotopy category]] of [[Joyal model structure|$sSet_{Joyal}$]], which, by the [uniqueness of adjoints](adjoint+functor#UniquenessOfAdjoints), implies that they are [[isomorphism|isomorphic]] in the homotopy category, which finally is equivalent to the statement to be proven.

That $Func\big(N(C), N(A^\circ)\big) \simeq \big(N(A^\circ)\big)^{N(C)}$
is an [[exponential object]] in the [[homotopy category]] is pretty immediate.

That the left hand is an isomorphic exponential follows from [Lurie 09, corollary  A.3.4.12](#Lurie), which asserts that for $C$ and $D$ [[sSet-enriched categories]] with $C$ [[cofibrant object|cofibrant]] and $A$ as above, we have that [[composition]] with the [[evaluation map]] induces a [[bijection]]

$$
  Hom_{Ho(sSet Cat)}\big(D, [C,A]^\circ\big)
  \xrightarrow{\simeq}
  Hom_{Ho(sSet Cat)}\big(C \times D, A^\circ\big)
  \,.
$$

Since $Ho(sSet Cat_{Bergner}) \simeq Ho(sSet_{Joyal})$ this
identifies also $N\big([C,A]^\circ\big)$ with the exponential object in question.
\end{proof}

> The following proof is fresh, still needs double-checking.

\begin{corollary}
\label{CompatibilityOfModelPresentationWithPrecomposition}
  In the above situation, consider an [[sSet]]-[[enriched functor]] $f \,\colon\, C' \longrightarrow C$ between any [[small category|small]] [[sSet-enriched categories]]. Then under the identification of Prop. \ref{PresentationByModelStructuresOnFunctors} the two $\infty$-functors given by

1. [[homotopy coherent nerve]] of the [[derived functor]] $\mathbb{R}f^\ast$ of precomposition with $f$

1. the precomposition with the [[homotopy coherent nerve]] of $f$

are related by a [[natural equivalence]] of $\infty$-functors:

\begin{tikzcd}[sep=30pt]
  N\big(
    [C,A]^\circ
  \big)
  \ar[dd, "{ N\big( \mathbb{R}f^\ast \big) }"{description}]
  \ar[rr, "{ \sim }"]
  &&
  \mathrm{Func}\big(
    N(C)
    ,\,
    N(A^\circ)
  \big)
  \ar[dd, "{ N(f)^\ast }"{description}]
  \\
  \\
  N\big(
    [C',A]^\circ
  \big)
  \ar[uurr, Rightarrow, shorten=50pt, "{ \sim }"{sloped}]
  \ar[rr, "{ \sim }"]
  &&
  \mathrm{Func}\big(
    N(C')
    ,\,
    N(A^\circ)
  \big)
\end{tikzcd}

\end{corollary}
\begin{proof}
\label{ProofOfCompatibilityOfModelPresentationWithPrecomposition}
Consider the following diagram of [[simplicial sets]] (the outer ones being [[quasi-categories]]):

\begin{tikzcd}[
  row sep=30pt,
  column sep=20pt
]
  N\big(
    [C,A]^\circ
  \big)
  \ar[dr, hook]
  \ar[rrrr]
  \ar[ddddd, "{ N(\mathbb{R}f^\ast) }"{description} ]
  &[-10pt]
  &&
  &[-35pt]
  \mathrm{Func}\big(
    N(C)
    ,\,
    N(A^\circ)
  \big)
  \ar[dl, hook]
  \ar[ddddd, "{ N(f)^\ast }"{description}]
  \\
  &
  N\big(
    [C,A]
  \big)
  \ar[rr]
  \ar[dd, "{ N(f^\ast) }"{description} ]
  &&
  \mathrm{Func}\big(
    N(C)
    ,\,
    N(A)
  \big)
  \ar[dd, "{ N(f)^\ast }"{description}]
  \\
  \\
  &
  N\big(
    [C',A]
  \big)
  \ar[rr]
  \ar[d, "{ N(Q) }"{description}]
  \ar[dr, "{\mathrm{id}}"{description}, "{\ }"{name=t, swap}]
  &&
  \mathrm{Func}\big(
    N(C')
    ,\,
    N(A)
  \big)
  \\
  {}
  &
  N\big(
    [C',A]
  \big)
  \ar[r, "{ \mathrm{id} }"{description} ]
  \ar[to=t, Rightarrow]
  &
  N\big(
    [C',A]
  \big)
  \ar[ur]
  &&
  \\[-10pt]
  N\big(
    [C',A]^\circ
  \big)
  \ar[ur, hook]
  \ar[rrrr]
  &&&&
  \mathrm{Func}\big(
    N(C')
    ,\,
    N(A^\circ)
  \big)
  \ar[uul, hook]
\end{tikzcd}

Here $Q$ denotes any [[functorial factorization|functorial]] [[cofibrant replacement]] (which exists, by [this Example](functorial+factorization#InCombinatorialModelCategories), since $[C',A]$ is a [[combinatorial model category]] by the above discussion) and the double arrow denotes  (the image under the hc-nerve of) the [[natural transformation]] with components the corresponding resolution equivalences $Q(\text{-}) \xrightarrow{\;\sim\;} (\text{-})$ (which are components of a [[natural transformation]], by the nature of [[functorial factorization]]).

The left square commutes by the construction of right derived functors of [[right Quillen functors]] (eg. [this Prop.](Introduction+to+Homotopy+Theory#ComputationOfLeftRightDerivedFunctorsViaResolutions)) and the middle square is the [[naturality square]] of the comparison map discussed [above](#ModelCategoryPresentation). The remaining outer squares just exhibit the restriction to [[bifibrant objects]], as discussed above.

The total diagram is of the claimed from. It just remains to see that the 2-morphism filling it is really an equivalence, but this follows by Prop. \ref{EquivalencesDetectedOnObjects}.
\end{proof}

\linebreak

### Limits and colimits {#Limits}

For $C$ an ordinary [[category]] that admits small [[limit]]s and [[colimit]]s, and for $K$ a [[small category]], the [[functor category]] $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See [[limits and colimits by example]]. The analogous statement is true for $(\infty,1)$-categories of $(\infty,1)$-functors

\begin{proposition}
Let $K$ and $C$ be [[quasi-categories]], such that $C$ has all [[limit in a quasi-category|colimits]] indexed by $K$.

Let $D$ be a small quasi-category. Then

* The $(\infty,1)$-category $Func(D,C)$ has all $K$-indexed colimits;

* A morphism $K^\triangleright \to Func(D,C)$ is a colimiting cocone precisely if for each object $d \in D$ the induced morphism $K^\triangleright \to C$ is a colimiting cocone.

\end{proposition}

This is ([Lurie, corollary 5.1.2.3](#Lurie)).

### Equivalences

\begin{proposition}
\label{EquivalencesDetectedOnObjects}
**(equivalences of $\infty$-functors detected on objects)**
\linebreak
A morphism $\alpha$ in $Func(D,C)$ (that is, a [[natural transformation]]) is an [[equivalence in an (infinity,1)-category|equivalence]] if and only if each component $\alpha_d$ is an equivalence in $C$.
\end{proposition}

This is due to [Joyal (2008), Chapter 5, Theorem C ([p. 125](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=125))](#Joyal08).

## Examples

* Between ordinary categories, it reproduces the ordinary [[category of functors]].

* Since the standard [[model structure on simplicial sets]] presents [[∞Grpd]]

  $$
    (sSet_{Quillen})^\circ \simeq \infty Grpd
  $$

  the [[model structure on simplicial presheaves]] (more precisely and more generally the [[model structure on sSet-enriched presheaves]]) on the [[opposite (∞,1)-category]] $C^{op}$ [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-presheaves]] on $C$:

  $$
    N([C^{op},sSet_{Quillen}]^\circ) \simeq Func(C^{op},\infty Grpd)
    = PSh_{(\infty,1)}(C)
    \,.
  $$

## Related concepts

* [[functor category]]

* [[2-functor 2-category]]

## References

The intrinsic definition is in section 1.2.7 of

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

The discussion of [[model category]] models is in A.3.4.

The theorem about equivalences (Prop. \ref{EquivalencesDetectedOnObjects}) is due to:

* {#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]]]

[[!redirects (∞,1)-category of (∞,1)-functors]]

[[!redirects (∞,1)-categories of (∞,1)-functors]]

[[!redirects (infinity,1)-functor category]]
[[!redirects (infinity,1)-functor categories]]

[[!redirects (infinity,1)-functor (infinity,1)-category]]
[[!redirects (infinity,1)-functor (infinity,1)-categories]]

[[!redirects functor (infinity,1)-category]]
[[!redirects (∞,1)-functor categories]]
[[!redirects functor (∞,1)-categories]]
[[!redirects (infinity,1)-functor category]]
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[[!redirects (∞,1)-functor categories]]
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[[!redirects functor (∞,1)-category]]

[[!redirects (∞,1)-functor (∞,1)-category]]
[[!redirects (∞,1)-functor (∞,1)-categories]]

[[!redirects infinity1-category of infinity1-functors]]


(infinity,1)-category of (infinity,1)-presheaves

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

#Contents#
* table of contents
{:toc}

## Definition ##

For [[∞Grpd]] the [[(∞,1)-category]] of [[∞-groupoids]], and for $S$ a [[(∞,1)-category]] (or in fact any [[simplicial set]]), an **$(\infty,1)$-presheaf** on $S$ is an $(\infty,1)$-functor

$$
  F : S^{op} \to \infty Grpd
  \,.
$$

The **$(\infty,1)$-category of $(\infty,1)$-presheaves** is the [[(∞,1)-category of (∞,1)-functors]]

$$
  PSh_{(\infty,1)}(S) := Func(S^{op}, \infinity Grpd)
  \,.
$$

## Properties

### Models {#Models}

A [[model category|model]] for an $(\infty,1)$-presheaf categories is the [[model structure on simplicial presheaves]]. See also the discussion at [[models for ∞-stack (∞,1)-toposes]].

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

For $C$ a [[simplicially enriched category]] with [[Kan complex]]es as hom-objects, write $[C^{op}, sSet_{Quillen}]_{proj}$ and $[C^{op}, sSet_{Quillen}]_{inj}$ for the projective or injective, respectively, global [[model structure on simplicial presheaves]]. Write $(-)^\circ$ for the full [[sSet]]-[[enriched category|enriched]] [[subcategory]] on fibrant-cofibrant objects, and $N(-)$ for the [[homotopy coherent nerve]] that sends a Kan-complex enriched category to a [[quasi-category]].

Then there is an [[equivalence of quasi-categories]]

$$
  PSh(N(C))
  \simeq
  N ([C^{op}, sSet_{Quillen}]_{proj})^\circ
  \,.
$$

Similarly for the injective model structure.

=--

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

This is a special case of the more general statement that the [[model structure on functors]] models an [[(∞,1)-category of (∞,1)-functors]]. See there for more details.

=--

Notice that the result in particular means that any $(\infty,1)$-presheaf -- an "$\infty$-[[pseudofunctor]]" -- may be _straightened_ or _rectified_ to a genuine [[sSet]]-[[enriched functor]], that respects horizontal compositions strictly.

### Limits and colimits

In an ordinary [[category of presheaves]], [[limit]]s and colimits are computed objectwise, as described at [[limits and colimits by example]]. The analogous statement is true for [[limit in a quasi-category|(∞,1)-limits]] and colimits in an $(\infty,1)$-category of $(\infty,1)$-presheaves.

This is a special case of the general existence of limits and colimits in an [[(∞,1)-category of (∞,1)-functors]]. See there for more details.

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

For $C$ a small $(\infty,1)$-category, the $(\infty,1)$-category $PSh(C)$ admits all small limits and colimits.

=--

See around [[Higher Topos Theory|HTT, cor. 5.1.2.4]].

### As the free completion under colimits {#FreeColimCompletion}

An ordinary [[category of presheaves]] on a small category $C$ is the [[free cocompletion]] of $C$, the free completion under forming colimits.

The analogous result holds for $(\infty,1)$-category of $(\infty,1)$-presheaves.

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

Let $C$ be a small [[quasi-category]] and $j \colon S \to PSh(C)$ the [[(∞,1)-Yoneda embedding]].

The identity [[(∞,1)-functor]] $Id \,\colon\, PSh(C) \to PSh(C)$ is the left [[(∞,1)-Kan extension]] of $j$ along itself.

=--

This is [[Higher Topos Theory|HTT, lemma 5.1.5.3]].

For $D$ a [[quasi-category]] with all small [[limit in a quasi-category|colimits]], write  $Func^L(PSh(C),D) \subset Func(PSh(C),D)$ for the full [[sub-quasi-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that [[preserved colimit|preserve]] small [[limit in a quasi-category|colimits]].

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

Pre-composition with the Yoneda embedding $j \colon C \to PSh(C)$ induces an [[equivalence of quasi-categories]]

$$
  Func^L(PSh(C),D) \to Func(C,D)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, theorem 5.1.5.6]].

In terms of the model given by the [[model structure on simplicial presheaves]], this is statement made in [Dugger (2001)](#Dugger2001), which gives that article its name.

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

Let $A$ and $B$ be [[model categories]], $D$ a plain [[category]] and

$$
  \array{
    D &\overset{r}{\longrightarrow}& A
    \\
    &  \mathllap{{}_{\gamma}}\searrow
    \\
    && B
  }
$$

two plain [[functors]]. Say that a **model-category theoretic factorization** of $\gamma$ through $A$ is

1. a [[Quillen adjunction]] $(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B$

1. a [[natural transformation|natural]] weak equivalence $\eta  : L \circ r \to \gamma$

   $$
     \array{
       D &&\stackrel{r}{\to}&& A
       \\
       & \mathllap{{}_\gamma}\searrow
         &{}^\eta\swArrow
       & \swarrow_{\mathrlap{L}}
       \\
       && B
     }
     \,.
   $$

Let the [[category]] of such factorizations have morphisms $\big((L \dashv R), \eta \big) \to \big((L' \dashv R'\big), \eta' )$ given by [[natural transformation]]s $L \to L'$ such that for all all objects $d \in D$ the diagrams

$$
  \array{
     L\circ r(d) &&\longrightarrow&& L'\circ r(d)
     \\
     & {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}}
     \\
     &&
     \gamma()
  }
$$

commutes.

=--

Notice that the [[(∞,1)-category]] presented by a [[model category]] -- at least by a [[combinatorial model category]] -- has all [[limit in a quasi-category|(∞,1)-categorical colimits]], and that the Quillen left adjoint functor $L$ presents, via its [[derived functor]], a left [[adjoint (∞,1)-functor]] that preserves $(\infty,1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\infty,1)$-functors into $(\infty,1)$-categories that have all colimits.

+-- {: .un_theorem}
###### Theorem
**(model category presentation of free $(\infty,1)$-cocompletion)**

For $C$ a [[small category]], the projective global [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{proj}$ on $C$ is universal with respect to such factorizations of functors out of $C$:

every functor $C \to B$ to any [[model category]] $B$ has a factorization through $[C^{op}, sSet]_{proj}$ as above, and the category of such factorizations is [[contractible]].

=--

This is [Dugger (2001), theorem 1.1](#Dugger2001),
where the proof appears on page 30.

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

To produce the factorization $[C^{op},sSet] \to B$ given the functor $\gamma$, first
notice that the ordinary [[Yoneda extension]] $[C^{op},Set] \to B$ would be given by the left [[Kan extension]] given by the [[coend]] formula

$$
  F \mapsto \int^{c \in C} \gamma(c) \cdot F(c)
  \,,
$$

where the dot in the integrand is the [[copower|tensoring]] of cocomplete category $B$ over [[Set]]. To refine this to a [[Quillen adjunction|left Quillen functor]] $L : [C^{op},sSet] \to B$, _choose_ a [[cosimplicial object|cosimplicial]] [[resolution]]

$$
  \Gamma \;\colon\; C \to [\Delta,B]
$$

of $\gamma$. Then set

$$
  L \;\colon\;
  F \mapsto
  \int^{c \in C}
  \int^{[n] \in \Delta}
  \Gamma^n(c) \cdot F_n(c)
  \,.
$$

The [[right adjoint]] $R \colon B \to [C^{op},sSet]$ of this functor is given by

$$
  R(X) \colon c \mapsto Hom_B(\Gamma^\bullet(c), X)
  \,.
$$

For $(L \dashv R) \colon [C^{op}, sSet]_{proj} \stackrel{\to}{\leftarrow} B$ to be a [[Quillen adjunction]], it is  sufficient to check that $R$ preserves [[fibrations]] and [[acyclic fibrations]]. By definition of the projective model structure this means that for every (acyclic) fibration $b_1 \to b_2$ in $B$ we have for every object $c \in C$ that that

$$
  Hom_C\big(\Gamma^\bullet(c), b_1 \to b_2\big)
$$

is an (acyclic) fibration of simplicial sets. But this is one of the standard properties of [[cosimplicial object|cosimplicial [[resolutions]].

Finally, to find the natural weak equivalence $\eta \colon L \circ j \simeq \gamma$, write $j : C \to [C^{op},sSet]$ for the [[Yoneda embedding]] and notice that by [[Yoneda reduction]] it follows that for $x \in C$ we have

$$
  L(j(x))
  \;=\;
  \int^{c \in C} \int^{[n] \in \Delta}
  \Gamma^n(c) \cdot C(c,x)
  \;=\;
  \Gamma^0(x)
$$
(where equality signs denote [[isomorphisms]]).

By the very definition of cosimplicial resolutions, there is a natural weak equivalence $\Gamma(x) \stackrel{\simeq}{\to}$. We can take this to be the component of $\eta$.

=--

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

The [[(∞,1)-Yoneda embedding]] $j \colon C \to PSh(C)$ generates $PSh(C)$ under small colimits:

a full [[sub-quasi-category|(∞,1)-subcategory]] of $PSh(C)$ that contains all representables and is closed under forming $(\infty,1)$-colimits is already equivalent to $PSh(C)$.

=--

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

This is [[Higher Topos Theory|HTT, corollary 5.1.5.8]].

=--

### Functorality
 {#Functoriality}

The two descriptions of $PSh(S)$ as the functor category $Func(S^{op}, \infty Grpd)$ and as the free cocompletion $S \to PSh(S)$ ([above](#FreeColimCompletion)) both extend to contravariant and covariant [[(infinity,1)-functors|functors]] on [[(infinity,1)Cat|$(\infty, 1)Cat$]].

These two constructions are related.

$Func(-, \infty Grpd) \colon (\infty, 1)Cat^{op} \to (\infty, 1)\widehat{Cat}$ takes values in the subcategory of [[presentable (∞,1)-categories]] and functors [[preserved colimit|preserving]] [[small limit|small]] [[(infinity,1)-limit|limits]] and [[(infinity,1)-colimit|colimits]], and thus having [[left adjoint|left]] and [[right adjoint|right]] [[adjoint (infinity,1)-functor|adjoints]].

Let $Pr^L(\infty,1)Cat$ be the (∞,1)category of [[presentable (∞,1)-categories]] and functors that are left adjoints,
and similarly for $Pr^R(\infty,1)Cat$.

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

The [[adjoint (infinity,1)-functor#category_of_adjunctions|local left adjoint]]
to $Func(-, \infty Grpd) \colon (\infty, 1)Cat^{op} \to Pr^R(\infty,1)Cat$
is naturally equivalent to the free cocompletion functor $P \colon (\infty, 1)Cat \to Pr^L(\infty,1)Cat$
by a natural equivalence whose components at a small (∞,1)-category $S$ are homotopic to the [[identity functor]] on $PSh(S)$.

=--

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

Consider the (∞,1)-category of possibly large (∞,1)-categories that have small colimits, and functors preserving small colimits. By [[Higher Topos Theory|HTT, 5.3.6.10]], the inclusion into the full (∞,1)-category of (∞,1)-categories has a left adjoint $P$, which is characterized by the natural equivalence
$$
  Func^L\big(P(S), -\big) \simeq Func(S, -)
$$
given by pre-composition with the [[(infinity,1)-Yoneda embedding|Yoneda embedding]].

Let $Func(-, \infty Grpd)^{ladj}$ denote the [local left adjoint](adjoint+infinity1-functor#LadjAndRadj) to $Func(-, \infty Grpd)$.
Then for $C \in Pr^L(\infty,1)Cat$
and $S \in (\infty,1)Cat$, there are natural equivalences
$$
  \begin{aligned}
  Func^L\big(Func(S^{op}, \infty Grpd)^{ladj}, C\big)
  &\simeq Func^R\big(C^{radj}, Func(S^{op}, \infty Grpd)\big)^{op}
  \\ &\simeq Func\big(S^{op}, Func^R(C^{radj}, \infty Grpd)\big)^{op}
  \\&\simeq Func\big(S, Func^L(\infty Grpd, C)\big)
  \\&\simeq Func(S, C)
  \end{aligned}
$$
The second equivalence holds because, for presentable $C$, the property of being in $Func^R$ is characterized by being accessible and preserving small limits, which can be determined pointwise.

The first and third equivalences are general properties of [local left adjoints](adjoint+infinity1-functor#LadjAndRadj) and opposite functor categories.

The last equivalence is because $Func^L(\infty Grpd, C) \simeq Func(1, C)$.

Since both constructions corepresent the same functor $Func(S, -)$, the yoneda lemma ensures there is a natural equivalence $ P(S) \simeq Func(S^{op}, \infty Grpd)^{ladj} $.

Now consider the chain of equivalences

$$
  Func(S, C)
  \leftarrow Func^L\big(Func(S^{op}, \infty Grpd)^{ladj}, C\big)
  \to Func^L\big(P(S), C\big)
  \to Func(S, C)
  \mathrlap{\,,}
$$

where the first map is inverse to composition with the Yoneda embedding, the second map is the equivalence constructed previously, and the third map is composition with the Yoneda embedding.

Be careful to note that the first equivalence has not yet been shown to be natural in $S$.

The overall composite is essentially uniquely determined by an automorphism of $\alpha_S$ of $S$. It remains to be shown that  $\alpha_S$ is an identity.

When $S = \Delta^n$, $S$ has no nontrivial automorphisms, so
$\alpha_{\Delta^n}$ is the identity.

For a functor $\phi : \Delta^n \to S$. The naturality of the above equivalence gives a commutative square

\begin{tikzcd}
  P(\Delta^n) \arrow[r] \arrow[d] &
  \mathrm{Func}(\Delta^{op}, \infty \mathrm{Grpd})^{ladj} \arrow[d]
  \\ P(S) \arrow[r] &
  \mathrm{Func}(S^{op}, \infty \mathrm{Grpd})^{ladj}
\end{tikzcd}

[[Higher Topos Theory|HTT, 5.2.6.3]] asserts $P(\phi)$ is left adjoint to $\Func(\phi, \infty Grpd)$, so the left and right arrows are homotopic.

We've already seen the top arrow is homotopic to the identity, and so we infer $\alpha_S \phi \simeq \phi$. Since the $\Delta^n$ generate $(\infty,1)Cat$, $\alpha_S$ is thus homotopic to the identity.

=--

Note that a natural automorphism of either functor extending $PSh$ would induce a natural automorphism of the identity functor on $(\infty,1)Cat$. By [[Higher Topos Theory|HTT, 5.2.9.1]], the space of such automorphisms is contractible.

### Relation to slicing
  {#WithOvercategories}

The following analog of the corresponding result for
1-[[categories of presheaves]] holds for $(\infty,1)$-presheaves. See [[functors and comma categories]].

+-- {: .num_prop #SlicingCommutesWithFormingPresheaves}
###### Proposition
**(slicing commutes with passing to presheaves)**

Let $\mathcal{C}$ be a [[small (∞,1)-category]] and
$p \colon \mathcal{K} \to \mathcal{C}$ a [[diagram]].

Write $\mathcal{C}_{/p}$  and $PSh_\infty(\mathcal{C})/_{y p}$
for the corresponding [[over quasi-category|over categories]], where
$y \colon \mathcal{C} \to PSh_\infty(\mathcal{C})$ is the [[(∞,1)-Yoneda embedding]].

Then we have an [[equivalence of quasi-categories|equivalence of (∞,1)-categories]]

$$
  PSh_\infty(\mathcal{C}_{/p})
    \stackrel{\simeq}{\to}
  PSh_\infty(\mathcal{C})_{/y p}
  \,.
$$

=--

This appears as [[Higher Topos Theory|HTT, 5.1.6.12]].

## $(\infty,1)$-subcategories of $(\infty)$-presheaf categories

### Locally presentable $(\infty,1)$-categories

A [[reflective (∞,1)-subcategory]] of an $(\infty,1)$-category of $(\infty,1)$-presheaves is called a [[presentable (∞,1)-category]].

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

If that [[left adjoint|left]] [[adjoint (∞,1)-functor]] to the embedding of the [[reflective (∞,1)-subcategory]] furthermore preserves finite [[limit]]s, then the subcategory is an [[(∞,1)-category of (∞,1)-sheaves]]: an [[(∞,1)-topos]]

## Related concepts

[[!include locally presentable categories - table]]

## References ##

* {#Dugger01} [[Dan Dugger]], *[[Universal homotopy theories]]*, Advances in Mathematics **164** (2001) 144-176 [[arXiv:math/0007070](https://arxiv.org/abs/math/0007070), [doi:10.1006/aima.2001.2014](https://doi.org/10.1006/aima.2001.2014)]

* {#Lurie09} [[Jacob Lurie]], section 5.1 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html)]

[[!redirects (infinity,1)-categories of (infinity,1)-presheaves]]

[[!redirects (∞,1)-category of (∞,1)-presheaves]]
[[!redirects (∞,1)-categories of (∞,1)-presheaves]]

[[!redirects free (∞,1)-cocompletion]]
[[!redirects free (infinity,1)-cocompletion]]

[[!redirects (∞,1)-presheaf (∞,1)-topos]]
[[!redirects (∞,1)-presheaf (∞,1)-toposes]]

[[!redirects (infinity,1)-presheaf (infinity,1)-category]]
[[!redirects (∞,1)-presheaf (∞,1)-category]]
[[!redirects (infinity,1)-presheaf (infinity,1)-categories]]
[[!redirects (∞,1)-presheaf (∞,1)-categories]]

[[!redirects infinity-category of infinity-presheaves]]
[[!redirects infinity-categories of infinity-presheaves]]

[[!redirects infinity1-category of infinity1-presheaves]]
[[!redirects infinity1-categories of infinity1-presheaves]]


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

#Contents#
* table of contents
{:toc}

## Idea

The notion of $(\infty,1)$-category of $(\infty,1)$-sheaves is the generalization of the notion of [[category of sheaves]] from [[category theory]] to the [[higher category theory]] of [[(∞,1)-categories]].

## Definition
 {#Definition}

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

An **$(\infty,1)$-category of $(\infty,1)$-sheaves** is a [[reflective sub-(∞,1)-category]]

$$
  Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  PSh(C)
$$

of an [[(∞,1)-category of (∞,1)-presheaves]] such that the following equivalent conditions hold

* $L$ is a [[topological localization]];

* there is the structure of an [[(∞,1)-site]] on $C$ such that the objects of $Sh(C)$ are precisely those [[(∞,1)-presheaves]] $A$ that are [[local objects]] with respect to the _covering_ [[monomorphism in an (∞,1)-category|monomorphisms]] $p : U \to j(c)$ in $PSh(C)$ in that

  \[
   \label{DescentCondition}
    A(c) \simeq PSh(j(c),A) \stackrel{PSh(p,A)}{\to} PSh(U,A)
  \]

  is an [[(∞,1)-equivalence]] in [[∞Grpd]].

=--

This is [[Higher Topos Theory|HTT, def. 6.2.2.6]].

An $(\infty,1)$-category of $(\infty,1)$-sheaves is an **[[(∞,1)-topos]]**.

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

Equivalence (eq:DescentCondition) is the _[[descent]]_ condition and the presheaves satisfying it are the **[[(∞,1)-sheaves]]** .

Typically $U$ here is the [[Cech nerve]]

$$
  C(\{U_i\}) = \lim_{\to_{[n]}} U_{i_0, \cdots U_{i_n}}
$$

of a [[covering]] family $\{U_i \to c\}$
(where the colimit is the [[limit in a quasi-category|(∞,1)-categorical colimit]] or [[homotopy colimit]]) so that the above [[descent]] condition becomes

$$
  A(c) \simeq PSh(\lim_\to U_\cdots, A)
 \simeq
  \lim_{\leftarrow} A(U_\cdots)
 =
 \lim_{\leftarrow}
 \left(
    \cdots
    \stackrel{\to}{\stackrel{\to}{\to}}
    \prod_{i,j} A(U_i) \times_{A(c)} A(U_j)
    \stackrel{\to}{\to}\prod_i A(U_i)
 \right)
 \,.
$$

=--

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

Sometimes [[(∞,1)-sheaves]] are called **[[∞-stacks]]**, though sometimes the latter term is reserved for [[hypercomplete]] $(\infty,1)$-sheaves and at other times again it refers to [[(∞,2)-sheaves]].

The [[(n,1)-category|(n,1)-categorical]] counting is:

* [[sheaf]] = 0-stack = 0-[[truncated]] $(\infty,1)$-sheaf

* $(2,1)$-sheaf = [[stack]] = 1-truncated $(\infty,1)$-sheaf

* $(3,1)$-sheaf = 2-stack = 2-truncated $(\infty,1)$-sheaf

* etc.

* $(\infty,1)$-sheaf = [[∞-stack]]  (or = [[hypercomplete]] $(\infty,1)$-sheaf).

=--

## Properties

### Localizations and Grothendieck topology {#LocAndTopology}

We reproduce the proof that the two characterization in def. \ref{CategoryOfSheaves} above are indeed equivalent.

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

For $C$ an [[(∞,1)-site]], the full [[sub-(∞,1)-category]] of $PSh(C)$ on [[local objects]] with respect to the covering monomorphisms in $PSh(C)$ is indeed a [[topological localization]], and hence $Sh(C)$ is indeed an exact [[reflective sub-(∞,1)-category]] of $PSh(C)$ and hence an [[(∞,1)-topos]].

=--

This is [[Higher Topos Theory|HTT, Prop. 6.2.2.7]]

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

We must prove that the [[(∞,1)-sheafification]] functor $L \colon PSh(C)\to Sh(C)$ preserves [[finite (∞,1)-limits]]. To do so we give an explicit construction of $L$. Given a presheaf $F\in PSh(C)$, define a new presheaf $F^+$ by the formula

$$F^+(c)={\lim_{\rightarrow}}_U {\lim_{\leftarrow}}_{u\in U} F(u)$$

where the colimit is taken over all covering sieves $U$ of $c$; this is called the _[[plus construction]]_. It defines a functor $PSh(C)\to PSh(C)$ and there is an obvious morphism $F\to F^+$ natural in $F$.

It is clear that the construction $F\mapsto F^+$ preserves
[[finite (∞,1)-limits]], since [[filtered (∞,1)-colimits]] do, and it is easy to see that the map $F\to F^+$ becomes an [[equivalence in an (∞,1)-category|equivalence]] in $Sh(C)$. Given an [[ordinal]] $\alpha$, let $F^{(\alpha)}$ be the $\alpha$-iteration of the [[plus construction]] applied to the presheaf $F$. Then the functor $F\mapsto F^{(\alpha)}$ preserves finite limits and the canonical map $F\to F^{(\alpha)}$ becomes an equivalence in $Sh(C)$. In particular, if $F^{(\alpha)}$ is a sheaf, then $F^{(\alpha)}\simeq L(F)$. Thus, it suffices to show that there exists an ordinal $\alpha$ such that, for every $F\in PSh(C)$, $F^{(\alpha)}$ is a sheaf.

Fix $c\in C$ and a covering sieve $U$ of $C$. Given a presheaf $G\in PSh(C/c)$, we define an auxiliary presheaf $Match(U,G)\in PSh(C/c)$ by the formula

$$Match(U,G)(f: d\to c)={\lim_{\leftarrow}}_{u\in f^\ast U}G(u).$$

Restriction maps induce a morphism $\theta_G: G\to Match(U,G)$. Since we clearly have $G(u)\stackrel{\sim}{\to} Match(U,G)(u)$ for $u\in U$, the functor $Match(U,-)$ is _idempotent_ in the sense that $Match(U,\theta_G)$ and $\theta_{Match(U,G)}$ are (equivalent) equivalences.

By definition, $F\in PSh(C)$ is a sheaf if and only if $F(c)\stackrel{\sim}{\to} Match(U,F|_{C/c})(c)$ for every $c\in C$ and every covering sieve $U$ of $c$. Our goal is therefore to find an ordinal $\alpha$ (depending only on the (∞,1)-site $C$) such that, for every $F\in PSh(C)$, the map

$$F^{(\alpha)}(c) \to \Match(U,F^{(\alpha)}|_{C/c})(c)$$

is an equivalence.

The morphism $G\to G^+$ in $PSh(C/c)$ factors as

$$G\to Match(U,G)\to G^+.$$

Applying $Match(U,-)$ to this factorization, we get a commutative diagram

$$
  \array{
     G &\to& Match(U,G) &\to& G^+
     \\
     \downarrow^{\mathrlap{\theta_G}} && \downarrow^{\mathrlap{\theta_{Match(U,G)}}} && \downarrow^{\mathrlap{\theta_{G^+}}}
     \\
     Match(U,G) &\to& Match(U,Match(U,G)) &\to& Match(U,G^+)
  }
$$

in which the map $\theta_{Match(U,G)}$ is an equivalence since $Match(U,-)$ is idempotent. By cofinality, the colimit of the maps $\theta_{G^{(n)}}$ as $n\to\infty$ is an equivalence. Applying this to $G=F|_{C/c}$, we get

$$ F^{(\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(n)}|_{C/c})(c).$$

This _almost_ means that $F^{(\omega)}$ is a sheaf. The problem is that the filtered colimit on the right-hand side need not commute with the limit appearing in the definition of $Match(U,-)$, that is, the canonical map

$$ {\lim_{\rightarrow}}_{\alpha \lt \omega} Match(U,F^{(\alpha)}|_{C/c})(c) \to \Match(U,F^{(\omega)}|_{C/c})(c) $$

need not be an equivalence. To solve this problem, we choose a cardinal $\kappa$ such that for every $c\in C$ and every covering sieve $U$ of $c$, the functor $Match(U,(-)|_{C/c})(c):Psh(C)\to \infty Grpd$ preserves $\kappa$-filtered colimits. This is possible because $C$ is small and each of these functors, being the composition of the restriction functor $PSh(C)\to PSh(U)$ and the limit functor $PSh(U)\to\infty Grpd$, has a [[left adjoint|left]] [[adjoint (∞,1)-functor]] and is therefore accessible (see [[HTT|HTT Prop. 5.4.7.7]]). Then the above map with $\omega$ replaced by $\kappa$ is an equivalence. For every ordinal $\alpha\lt\kappa$, applying the above to $F^{(\alpha)}$ shows that

$$ F^{(\alpha+\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(\alpha+n)}|_{C/c})(c),$$

Since $\kappa$ is a limit ordinal, we deduce that $F^{(\kappa)}$ is a sheaf by cofinality.

=--

And conversely:

+-- {: .num_prop}
###### Proposition
**(equivalence of site structures and categories of sheaves)**

For $C$ a [[small (∞,1)-category]], there is a bijective correspondence between structure of an [[(∞,1)-site]] on $C$ and equivalence classes of [[topological localization]]s of $PSh(C)$.

=--

This is [[Higher Topos Theory|HTT, prop. 6.2.2.9]].

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

For $C$ a small [[(∞,1)-site]] and $Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)$ the corressponding reflective inclusion of [[(∞,1)-sheaves]] into [[(∞,1)-presheaves]] on $C$ we have that the image under $L$ of a sub-$(\infty,1)$-functor  $p : U \to j(c)$ of a [[representable functor|representable]] $j(c)$ is covering precisely if $L(p)$ is an equivalence.

=--

This is [[Higher Topos Theory|HTT, lemma 6.2.2.8]].

+-- {: .proof}
###### Proof of the Lemma

Since $Sh(C)$ is the reflectuive localization of $PSh(C)$ at covering monomorphisms, it is clear that if $p : U \to j(c)$ is covering, then $L(p)$ is an equivalence.

To see the converse, form the 0-truncation of $L i$ and conclude as for ordinary sheaves on the [[homotopy category of an (infinity,1)-category|homotopy catgegory]] of $C$.

...

=--

+-- {: .proof}
###### Proof of the Proposition

We have seen in (...) that for every structure of an $(\infty,1)$-site on $C$ we obtain a topological localization of the presheaf category, and that this is an injective map from site structures to equivalence classes of sheaf categories. It remains to show that it is also a surjective map, i.e. that every [[topological localization]] of $PSh(C)$ comes from the structure of an [[(∞,1)-site]] on $C$.

So consider $S \subset Mor(PSh(C))$ a strongly saturated class of morphisms which s topological (closed under pullbacks). Write $S_0 \subset S$ for the subcalss of those that are [[monomorphism in an (infinity,1)-category|monomorphisms]] of the form $U \to j(c)$.

Observe that then $S$ is indeed generated by (is the smallest strongly saturated class containing) $S_0$: since by the [[co-Yoneda lemma]] every object $X \in PSh(C)$ is a colimit $x \simeq {\lim_\to}_k j(\Xi_k)$ over representables. It follows that every monomorphism $f : Y \to X$ is a colimit (in $Func(\Delta[1],PSh(C))$) of those of the form $U \to j(c)$: for consider the pullback diagram

$$
  \array{
     f^* ({\lim_\to}_k \Xi_k) &\to& Y
     \\
     \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}}
     \\
     {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X
  }
  \;\;\;\;\;
  \simeq
  \;\;\;\;\;
  \array{
     ({\lim_\to}_k f^* \Xi_k) &\to& Y
     \\
     \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}}
     \\
     {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X
  }
$$

where the equivalence is due to the fact that we have [[universal colimits]] in $PSh(C)$. This realizes $f$ as a colimit over morphisms of the form $f^* j(\Xi_k) \to j(\Xi_k)$ that are each a pullback of a monomorphism. Since monomorphisms are stable under pullback (see [[monomorphism in an (∞,1)-category]] for details), all these component morphisms are themselves monomorphisms.

So every monomorphism in $S$ is generated from $S_0$, but by the assumption that $S$ is topological, it is itself entirely generated from monomorphisms, hence is generated from $S_0$.

So far this establishes that evry topological localization of $PSh(C)$ is a localization at a collection of sieves/ subfunctors $U \to j(c)$ of representables. It remains to show that this collection of subfunctors is indeed an Grothendieck topology and hence exhibits on $C$ the structure of an [[(∞,1)-site]]. We check the necessary three axioms:

1. _equivalences cover_ -- The equivalences $j(c) \stackrel{\simeq}{\to} j(c)$ belong to $S$ and are monomorphisms, hence belong to $S_0$.

1. _pullback of a cover is covering_ - Since monomorphisms are stable under pullback, we haave for every $p : U \to j(c)$ in $S$ and every $j(f) : j(d) \to j(c)$ that also the pullback $f^* p$

   $$
     \array{
        f^* U &\to& U
        \\
        \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}}
        \\
        j(d) &\stackrel{f}{\to}& j(c)
     }
   $$

   is a monomorphism and in $S$, hence in $S_0$.

1. _if restriction of a sieve to a cover is covering, then the sieve is covering_  -- Let $p : U \to j(c)$ be an arbitrary monomorphism and $f : X \to j(d)$ in $S_0$. Write $X \simeq {\lim_\to}_k \Xi_k$ and consider the pullback

   $$
     \array{
       {\lim_\to}_k p^* \Xi_k &\stackrel{p^* f}{\to}&  U
       \\
       \downarrow^{{\lim_\to}_k f_k^* p} && \downarrow^{\mathrlap{p}}
       \\
       {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c)
     }
     \,,
   $$

   where again we made use of the [[universal colimits]] in $PSh(C)$. Now notice that

   1. $f$ is in $S$ by assumption;

   1. $p^* f$ is by pullback stability of $S$;

   1. each of the $f_k p$ is in $S$ by assumption, hence ${\lim_k f_k^* p}$
      is by the fact that $S$ is strongly saturated.

   1. so by commutativity $p \circ p^*f$ is in $S$.

   1. finally by 2-out-of-3 this means that $p$ is in $S$.

=--

### Over paracompact topological spaces {#OverParacompactSpaces}

We discuss how $(\infty,1)$-sheaves over a [[paracompact topological space]] are equivalent to topological spaces [[overcategory|over]] $X$. This is the analogue of the 1-categorical statement that [[sheaves]] on $X$ are equivalent to [[etale space]]s over $X$: an etale space over $X$ is one whose [[fiber]]s are [[discrete topological space]], hence 0-[[truncated]] spaces. The [[n-category]] analogy has [[homotopy n-type]]s as fibers.

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

For $Y \to X$ a [[morphism]] in [[Top]], and $U \in Op(X)$ an [[open subset]] of $X$, write

$$
  Sing_X(Y,U) := Hom_X(U \times \Delta^\bullet, X)
$$

for the [[simplicial set]] (in fact a [[Kan complex]]) of [[continuous map]]s

$$
  \array{
    U \times \Delta^k && \to && Y
    \\
    & \searrow && \swarrow
    \\
    && X
  }
$$

from $U$ times the topological $k$-[[simplex]] $\Delta^k$ into $Y$, that are [[section]]s of $Y \to X$.

=--

This is a relative version of the [[singular simplicial complex]] functor.

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

Let $(X, \mathcal{B})$ be a [[topological space]] equipped with a [[base for the topology]] $\mathcal{B}$.

There is a [[model category]] structure on the [[over category]] $Top/X$ with weak equivalences and fibration precisely those morphisms $Y \to Z$ over $X$ such that for each $U \in \mathcal{B}$ the induced morphism $Sing_X(Y,U) \to Sing_X(Z,U)$ is a weak equivalence or fibration, respectively, in the standard [[model structure on simplicial sets]].

=--

This is [[Higher Topos Theory|HTT, prop 7.1.2.1]].

Write $(Top/X)^\circ$ for the [[(∞,1)-category]] [[locally presentable (∞,1)-category|presented]] by this model structure.

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

Let $X$ be a [[paracompact topological space]] and write as usual $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ for the $(\infty,1)$-category of $(\infty,1)$-sheaves on the [[category of open subsets]] of $X$; equipped with the canonical structure of a [[site]].

Let $\mathcal{B}$ be the set of **$F_\sigma$-open subsets** of $X$. This are those [[open subset]]s that are countable unions of [[closed subset]]s, equivalently the 0-sets of [[continuous function]]s $X \to [0,1]$.

Let $Top/X^\circ$ be the corresponding $(\infty,1)$-categoty according to the [above proposition](#OverTopModelStructure). Then $Sing_X(-,-)$ constitutes  an [[equivalence of (∞,1)-categories]]

$$
  Top/X^\circ \simeq Sh_{(\infty,1)}(X)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, corollary 7.1.4.4]].

### Difference to more general $(\infty,1)$-toposes {#DiffToOthers}

The [[(∞,1)-topos]]es that are $(\infty,1)$-categories of sheaves, i.e. that arise by [[topological localization]] from an [[(∞,1)-category of (∞,1)-presheaves]], enjoy  a number of special properties over other classes of $(\infty,1)$-toposes, such as notably [[hypercomplete (∞,1)-topos]]es.

The following lists these properties. ([[Higher Topos Theory|HTT, section 6.5.4]].)

#### Universal property

The construction of [[(∞,1)-sheaf]] [[(∞,1)-topos]]es on a given [[locale]] is singled out over the construction of other kinds of $(\infty,1)$-toposes (such as [[hypercomplete (∞,1)-topos]]es) by the following universal property:

forming $(\infty,1)$-sheaves is, roughly, [[right adjoint]] to the functor $\tau_{\leq -1}$ that sends each $(\infty,1)$-topos to its underlying [[locale]] of [[subobject]]s of the [[terminal object]].

See [[Higher Topos Theory|HTT, item 1) of section 6.5.4]].

For $X,Y$ two $(\infty,1)$-toposes, write $Geom(X,Y) \subset Func(X,Y)$ for the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that are [[geometric morphism]]s.

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

For $C$ an [[essentially small (∞,1)-category|small]] [[(n,1)-category]] with [[finite (∞,1)-limits]] and equipped with the structure of an [[(∞,1)-site]] and for $Y$ an [[(∞,1)-topos]], the [[truncated|truncation functor]]

$$
  \tau_{\leq n-1} :
  Geom(Y, Sh(C)) \to Geom(\tau_{\leq n-1} Y, \tau_{\leq n-1} Sh(C))
$$

is an [[equivalence in a quasi-category|equivalence]] (of [[(∞,1)-categories]]).

=--

This is [[Higher Topos Theory|HTT, lemma 6.4.5.6]].

See also [[n-localic (∞,1)-topos]].

#### Compact generation

+-- {: .num_prop}

###### Proposition

Let $X$ be a [[coherent topological space]] and let $Op(X)$ be its [[category of open subsets]] with the standard structure of an [[(∞,1)-site]].

Then $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ is _compactly generated_ in that it is generated by [[filtered colimit]]s of [[compact object]]s.

Moreover, the compact objects of $Sh_{(\infty,1)}(X)$ are those that are [[stalk]]wise compact objects in [[∞Grpd]] and [[locally constant ∞-stack|locally constant]] along a suitable [[stratification]] of $X$.

=--

This is [[Higher Topos Theory|HTT, prop. 6.5.4.4]].

This statement is false for the [[hypercompletion]] of $Sh_{(\infty,1)}(X)$, in general.

#### Nonabelian cohomology {#NonabelianCohomology}

For $X$ a [[topological space]], let

$$
  (LConst \dashv \Gamma) :  Sh_{(\infty,1)}(X)
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
$$

be the [[global section]]s terminal [[geometric morphism]].

For $A \in \infty Grpd$, the ([[nonabelian cohomology|nonabelian]]) [[cohomology]] of $X$ with coefficients in $A$ is usually defined in [[∞Grpd]] as

$$
  H(X,A) := \pi_0 Func(Sing X, A)
  \,,
$$

where $Sing X$ is the [[fundamental ∞-groupoid]] of $X$. On the other hand, if we send $A$ into $Sh_{(\infty,1)}(X)$ via $LConst$, the there is the _intrinsic_ [[cohomology]] of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$

$$
  H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A)
  \,.
$$

Noticing that $X$ is in fact the [[terminal object]] of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that [[global section]]s functor, this is equivalently

$$
  \cdots \simeq \pi_0 \Gamma LConst A
  \,.
$$

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

If $X$ is a [[paracompact space]], then these two definitions of [[nonabelian cohomology]] of $X$ with [[constant ∞-stack|constant coefficients]] $A \in \infty Grpd$ agree:

$$
  H(X,A) := \pi_0 \infty Grpd(Sing X,A)  \simeq Sh_{(\infty,1)}(X)(X,LConst A)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, theorem 7.1.0.1]].

## Models

The topological localizations of an [[(∞,1)-category of (∞,1)-presheaves]] are [[presentable (∞,1)-category|presented]] by the [[Bousfield localization of model categories|left Bousfield localization]] of the global [[model structure on simplicial presheaves]] at the set of [[Cech cover]]s.

The [[hypercompletion|hypercomplete]] $(\infty,1)$-sheaf toposes are [[presentable (infinity,1)-category|presented]] by the local Joyal-Jardine [[model structure on simplicial presheaves]].

Detailed discussion of this [[model category]] presentation is at

* [[models for infinity-stack (infinity,1)-toposes|models for (infinity,1)-category of (infinity,1)-sheaves]] .

## Related concepts

* [[geometric homotopy type theory]]

## References

The study of simplicial presheaves apparently goes back to

* K. Brown, [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]

which considers locally [[Kan complex|Kan]] [[simplicial presheaf|simplicial presheaves]] as a [[category of fibrant objects]].

This was later conceived in terms of a [[model structure on simplicial presheaves]] and on simplicial sheaves by Joyal and Jardine. Toën summarizes the situation and emphasizes the interpretation in terms of [[∞-stacks]] living in $(\infty,1)$-categories for instance in

B. Toën, _Higher and derived stacks: a global overview_   ([arXiv](http://arxiv.org/abs/math/0604504)) .

This concerns mostly [[hypercomplete]] $(\infty,1)$-sheaves, though.

The full picture in terms of Grothendieck-[[(∞,1)-topos]]es of [[(∞,1)-sheaves]] is the topic of

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

  * localization $(\infty,1)$-functors ($(\infty,1)$-sheafification for the present purpose) are discussed in section 5.2.7;

  * local objects ($(\infty,1)$-sheaves for the present purpose) and [[local isomorphism]]s are discussed in section 5.5.4;

  * the definition of $(\infty,1)$-topoi of $(\infty,1)$-sheaves is then definition 6.1.0.4 in section 6.1;

  * the characterization of $(\infty,1)$-sheaves in terms of [[descent]] is in section 6.1.3

  * the relation between the [[model structure on simplicial presheaves|Brown?Joyal?Jardine model]] and the general story is discussed at length in section 6.5.4

An overview is in

* [[Denis-Charles Cisinski]], _Catégories supérieures et théorie des topos_, Séminaire Bourbaki, 21.3.2015, [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf).

[[!redirects (infinity,1)-category of (infinity,1)-sheaves]]
[[!redirects (∞,1)-category of (∞,1)-sheaves]]
[[!redirects (infinity,1)-categories of (infinity,1)-sheaves]]
[[!redirects (∞,1)-categories of (∞,1)-sheaves]]

[[!redirects (∞,1)-sheaf (∞,1)-topos]]
[[!redirects (∞,1)-sheaf (∞,1)-toposes]]
[[!redirects (∞,1)-sheaf (∞,1)-topoi]]

[[!redirects ∞-stack (∞,1)-topos]]
[[!redirects ∞-stack (∞,1)-toposes]]
[[!redirects ∞-stack (∞,1)-topoi]]

[[!redirects infinity-stack (infinity,1)-topos]]
[[!redirects infinity-stack (infinity,1)-toposes]]
[[!redirects infinity-stack (infinity,1)-topoi]]

[[!redirects (∞,1)-sheaf (∞,1)-category]]
[[!redirects (∞,1)-sheaf (∞,1)-categories]]

[[!redirects (infinity,1)-sheaf (infinity,1)-category]]
[[!redirects (infinity,1)-sheaf (infinity,1)-categories]]

[[!redirects (∞,1)-topos of (∞,1)-sheaves]]
[[!redirects (∞,1)-toposes of (∞,1)-sheaves]]
[[!redirects (infinity,1)-topos of (infinity,1)-sheaves]]
[[!redirects (infinity,1)-toposes of (infinity,1)-sheaves]]

[[!redirects (∞,1)-sheaf ∞-topos]]
[[!redirects (∞,1)-sheaf ∞-toposes]]


(infinity,1)-category of cartesian sections

# Idea #

Let $p : E \to X$ be an [[(∞,1)-functor]] of [[(infinity,1)-category|(∞,1)-categories]]. A _cartesian section_ of $p$ is a [[section]] $\sigma : X \to E$ that sends all 1-morphisms in $X$ to [[Cartesian morphism]]s in $E$.

#Definition#

...

#Remarks#

If $p : E \to X$ is a [[Cartesian fibration]] classified by an [[(∞,1)-functor]] $F : X \to (\infty,1)Cat^{op}$ then $\Gamma_X^{cart}(E)$ is equivalent to the [[limit in a quasi-category|limit]] of $F$

$$
  \Gamma_X^{cart}(E) \simeq lim F
  \,.
$$

See the discussion at [[limit in a quasi-category]] for details.

#References#

In corollary 3.3.3.2 of

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

the collection of cartesian sections of $p : E \to X$ appears as $Maps_X^\flat(X^#, E^{cart})$.

Here

* the [[simplicial set]] $Maps_X^\flat(\cdots)$ is the simplicial set underlying the [[internal hom]] of [[marked simplicial set]]s over $X$ (beginning of section 3.1.3);

* $X^#$ is the [[simplicial set]] $X$ with _all_ cells marked (beginning of section 3.1)

* and  $E^{cart}$ is $E$ with precisely all [[Cartesian morphism]]s marked (def. 3.1.1.9).

[[!redirects (∞,1)-category of cartesian sections]]
[[!redirects (infinity,1)-category of cartesian section]]


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

#Contents#
* table of contents
{:toc}

## Idea

An [[(∞,1)-category]] presented by a [[model structure on chain complexes]].

Its [[homotopy category of an (∞,1)-category]] is the [[derived category]] of the underlying [[abelian category]].

## Related concepts

* [[stable (∞,1)-category]], [[triangulated category]]

* [[truncation of a chain complex]], [[suspension of a chain complex]]

[[!redirects (∞,1)-category of chain complexes]]

[[!redirects (∞,1)-categories of chain complexes]]
[[!redirects (infinity,1)-categories of chain complexes]]

[[!redirects (∞,1)-category of unbounded chain complexes]]
[[!redirects (infinity,1)-category of unbounded chain complexes]]
[[!redirects (∞,1)-categories of unbounded chain complexes]]
[[!redirects (infinity,1)-categories of unbounded chain complexes]]

[[!redirects stable (∞,1)-category of chain complexes]]
[[!redirects stable (∞,1)-categories of chain complexes]]

[[!redirects stable (infinity,1)-category of chain complexes]]
[[!redirects stable (infinity,1)-categories of chain complexes]]


(infinity,1)-category theory

+-- {: .rightHandSide}
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

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

## Idea

In the pattern of [[(n,r)-categories]] the notion of [[(∞,1)-category]] is special in that it is precisely the structure that

* admits [[homotopy theory]] in that it encodes the notion of higher coherent equivalence;

* admits [[category theory]] in terms of [[universal construction]]s: [[limit]]s, [[adjunction]]s, [[Grothendieck construction]]s, etc.

This entry surveys the _category theory of $(\infty,1)$-categories_ .

## Basic notions {#BasicCatNotions}

* [[nLab:homotopy category of an (∞,1)-category]]

* presentation by a [[nLab:category with weak equivalences]]:
  [[nLab:Dwyer-Kan localization]]

* [[nLab:(∞,1)-functor]]

* [[nLab:(∞,1)-category of (∞,1)-categories]]

* the archetypical $(\infty,1)$-category: [[nLab:∞Grpd]]

* [[(∞,1)-category of (∞,1)-presheaves]]

* [[opposite quasi-category]]

* [[nLab:terminal object in a quasi-category]]

## Universal constructions {#UniversalConstructions}

The [[universal construction]]s of [[category theory]] generalize, with unique existence of universal morphisms replaced by the requirement of a _[[contractible space]]_ of universal morphisms.

* [[limit in a quasi-category]]

* [[adjoint (∞,1)-functor]]

* [[(∞,1)-Grothendieck construction]]

## Related concepts

* [[(∞,1)-category of (∞,1)-categories]]

* [[homotopy 2-category of (∞,1)-categories]]

* [[synthetic (infinity,1)-category theory]]

[[!include table of category theories]]

## References

See the references at _[[(infinity,1)-category|$(\infty,1)$-category]]_.

[[!redirects (∞,1)-category theory]]


(infinity,1)-functor

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[[!include quasi-category theory contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

An _$(\infty,1)$-functor_ is a [[homomorphism]] between [[(∞,1)-categories]]. It generalizes

* the notion of [[functor]] between [[categories]];

* the notion of [[pseudofunctor]] from a [[category]] to a [[(2,1)-category]];

* the notion of [[2-functor]] between [[(2,1)-categories]];

* the notion of [[n-functor]] between [[(n,r)-category|(n,1)-categories]].

An $(\infty,1)$-functor is _functorial_ (respects [[composition]]) only up to [[coherent]] higher [[homotopies]]. It may be thought of as a **[[homotopy coherent functor]]** or **strongly homotopy functor**.

The collection of all $(\infty,1)$-functors between two $(\infty,1)$-categories form an [[(∞,1)-category of (∞,1)-functors]].

## Definition

The details of the definition depend on the model chosen for
[[(∞,1)-categories]].

1. [[quasi-category]]

1. [[simplicially enriched category]]

1. [[Segal category]]

1. [[complete Segal space]]

### In terms of quasi-categories

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

For $C$ and $D$ [[quasi-category|quasi-categories]], an **$(\infty,1)$-functor** $F : C \to D$ is simply a morphism of the underlying [[simplicial sets]].

A **natural transformation** $\eta : F \to G$ between two such $(\infty,1)$-functors is a simplicial [[homotopy]]

$$
  \array{
    C
    \\
    {}^{\mathllap{i_0}}\downarrow & \searrow^{\mathrlap{F}}
    \\
    C \times \Delta[1] &\stackrel{\eta}{\to}& D
    \\
    {}^{\mathllap{i_1}}\uparrow & \nearrow_{G}
    \\
    C
  }
  \,.
$$

A **[[modification]]** $\rho$ between natural transformations is an order 2 simplicial homotopy

$$
  \rho : C \times \Delta[2] \to D
  \,.
$$

Generally a **$k$-[[transfor]]** $\phi$ of $(\infty,1)$-functors is a simplicial homotopy of order $k$ between the corresponding quasi-categories

$$
  \phi : C \times \Delta[k] \to D
  \,.
$$

In total, the [[(∞,1)-category of (∞,1)-functors]] between given quasi-categories $C$ and $D$ is the simplicial function complex

$$
  (\infty,1)Cat(C,D) :=
  sSet(C,D)
  :=
  \int^{k \in \Delta} \Delta[k] \cdot Hom_{sSet}(C \times \Delta[k], D)
$$

as computed by the canonical [[sSet]]-[[enriched category|enrichment]] of $sSet$ itself.

=--

This serves to define the [[(∞,1)-category of (∞,1)-functors]].

## Examples

### $\infty$-Pseudo-functors / homotopy presheaves

Let $C$ be an ordinary [[category]]. The above definition in particular serves to generalize the notion of a [[pseudofunctor]] (functor up to homotopy)

$$
  F : C^{op} \to Grpd
$$

with values in the [[2-category]] [[Grpd]] as it appears in the theory of [[stack]]s/[[2-sheaves]]:

let $KanCplx \subset sSet$ be the [[full subcategory]] of [[sSet]] on the [[Kan complex]]es. This is naturally a [[simplicially enriched category]]. Write $N(\mathbf{KanCplx})$ for the [[homotopy coherent nerve]] of this simplicially enriched category. This is the [[quasi-category]]-incarnaton of [[∞Grpd]].

Write $N(C^{op})$ for the ordinary [[nerve]] of the ordinary category $C^{op}$ (passing to the [[opposite category]] is just a convention here, with no effect on the substance of the statement). Then an **$\infty$-pseudofunctor** or **[[(∞,1)-presheaf]]** or **homotopy presheaf** on $C$ is a morphism of simplicial sets

$$
  F : N(C^{op}) \to N(\mathbf{KanCplx})
  \,.
$$

One sees easily in low degrees that this does look like the a [[pseudofunctor]] there:

1. the 1-cells of $N(C)$ are just the [[morphism]]s in $C$, so that on 1-cells we have that $F$ is an assignment

   $$
     F : (x \stackrel{f}{\leftarrow} y) \mapsto (F(x) \stackrel{F(f)}{\to} F(y)
   $$

   of morphisms in $C$ to morphisms in $KanCplx$, as befits a functor;

1. the 2-cells of $N(C)$ are pairs of composable morphisms, so that on 2-cells we have that $F$ is an assignment

   $$
     F :
     \left(
       \array{
          && y
          \\
          & {}^{\mathllap{g}}\swarrow
          & & \nwarrow^{\mathrlap{f}}
          \\
          x &&\stackrel{g \circ f}{\leftarrow}&& z
       }
     \right)
     \;\;
     \mapsto
     \;\;
     \left(
       \array{
          && F(y)
          \\
          & {}^{\mathllap{F(g)}}\nearrow
          & \Downarrow^{\mathrlap{F(f,g)}} &
          \searrow^{\mathrlap{F(f)}}
          \\
          F(x) &&\stackrel{F(g \circ f)}{\rightarrow}&& F(z)
       }
     \right)
   $$

   which means that $F$ does not necessarily respect the [[composition]] of morphisms, but instead does introduce [[homotopies]] $F(f,g)$ for very pairs of composable morphisms, which measure how $F(g)\circ F(f)$ differs from $F(g \circ f)$. These are precisely the homotopies that one sees also in an ordinary [[pseudofunctor]]. But for our $(\infty,1)$-functor there are now also higher and higher homotopies:

1. the 3-cells of $N(C)$ are triples of composable morphisms $(f,g,h)$ in $C$. They are sent by $F$ to a tetrahedron that consists of a homotopy-of-homotopies from the $F(f,g) \cdot F( h , g\circ f )$ to $F(g, h) \cdot F(f , h \circ g)$;

1. and so on.

For more see [[(∞,1)-presheaf]].

## Properties

It turns out that every $(\infty,1)$-functor $C \to \infty Grpd$ can be **rectified** to an _ordinary ([[sSet]]-[[enriched functor|enriched]]) functor_ with values in [[Kan complex]]es.

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

For $C = N(\mathbf{C})$ a quasi-category given as the [[homotopy coherent nerve]] of a Kan-complex enriched category $\mathbf{C}$ (which may for instance be just an ordinary 1-category), write

$$
  [\mathbf{C}^{op}, \mathbf{sSet}]
$$

for the [[sSet]]-[[enriched category]] of _ordinary_ ($sSet$-[[enriched functor|enriched]]) functors (respecting composition strictly).

Then: every $(\infty,1)$-functor $N(\mathbf{C}^{op}) \to \infty Grpd$ is equivalent to a strictly composition respecting functor of this sort. Precisely: write $[\mathbf{C}^{op}, \mathbf{sSet}]^\circ$ for the full $\mathbf{sSet}$-enriched subcategory on those strict functors that are fibrant and cofibrant in a [[model structure on simplicial presheaves]] on $\mathbf{C}$. Then we have an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-categories]]

$$
  Hom_{(\infty,1)Cat}(N(\mathbf{C}^{op}), \infty Grpd)
  \simeq
  N([\mathbf{C}^{op}, \mathbf{sSet}]^\circ)
  \,.
$$

=--

More on this is at [[(∞,1)-category of (∞,1)-presheaves]].

## Related concepts

* [[function]]

* [[functor]]

* [[2-functor]] / [[pseudofunctor]] / [[(2,1)-functor]]

  * [[monoidal functor]]

* [[n-functor]]

* **(∞,1)-functor**

  * [[monoidal (∞,1)-functor]]

  * [[adjoint (∞,1)-functor]], [[(∞,1)-monad]]

* [[(∞,n)-functor]]

[[!include properties of functors -- contents]]

## References

section 1.2.7 in

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

discusses morphisms of [[quasi-category|quasi-categories]].

[[!redirects (infinity,1)-functors]]
[[!redirects (∞,1)-functor]]
[[!redirects (∞,1)-functors]]


(infinity,1)-geometric morphism

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

#Contents#
* table of contents
{:toc}

## Idea

By [[categorification]] of the notion of [[geometric morphism]], an _$(\infty,1)$-geometric morphism_ is a pair of [[adjoint (∞,1)-functor]]s between [[(∞,1)-topos]]es where the leftadjoint is left-exact.

## Definition

For $\mathbf{H}$ and $\mathbf{K}$ two [[(∞,1)-topos]]es, a **$(\infty,1)$-geometric morphism** $f : \mathbf{H} \to \mathbf{K}$ is

* an [[(∞,1)-functor]] $f_* : \mathbf{H} \to \mathbf{K}$

  (called the [[direct image]] of the geometric morphism)

* which has a left [[adjoint (∞,1)-functor]] $\mathbf{H} \leftarrow \mathbf{K} : f^*$;

  (called the [[inverse image]] of the geometric morphism)

* such that $f^*$ is a left [[exact (∞,1)-functor]].

The (non-full) [[sub-(∞,1)-category]] of [[(∞,1)Cat]] on [[(∞,1)-toposes]] and $(\infty,1)$-geometric morphisms between them is  [[(∞,1)Toposes]].

## Examples

* [[terminal geometric morphism]]

* [[locally infinity-connected (infinity,1)-topos|locally $\infty$-connected $\infty$-topos]]

* [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]]

## Related concepts

* [[geometric morphism]]

* [[terminal geometric morphism]]

## References

section 6.3.1 in

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

[[!redirects (∞,1)-geometric morphism]]
[[!redirects (∞,1)-geometric morphisms]]

[[!redirects (infinity,1)-geometric morphisms]]