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\section*{(infinity,1)-category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-topos theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{quasicategories}{Quasi-categories}\dotfill \pageref*{quasicategories} \linebreak
\noindent\hyperlink{__and_simplicially_enriched_categories}{$Top$-, $Kan$- and simplicially enriched categories}\dotfill \pageref*{__and_simplicially_enriched_categories} \linebreak
\noindent\hyperlink{homotopical_categories}{Homotopical categories}\dotfill \pageref*{homotopical_categories} \linebreak
\noindent\hyperlink{model_categories}{Model categories}\dotfill \pageref*{model_categories} \linebreak
\noindent\hyperlink{segal_categories_and_complete_segal_spaces}{Segal categories and complete Segal spaces}\dotfill \pageref*{segal_categories_and_complete_segal_spaces} \linebreak
\noindent\hyperlink{categories}{$A_\infty$-categories}\dotfill \pageref*{categories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{category_theory_2}{$(\infty,1)$-Category theory}\dotfill \pageref*{category_theory_2} \linebreak
\noindent\hyperlink{the_collection_of_all_categories}{The collection of all $(\infty,1)$-categories}\dotfill \pageref*{the_collection_of_all_categories} \linebreak
\noindent\hyperlink{model_category_presentations}{Model category presentations}\dotfill \pageref*{model_category_presentations} \linebreak
\noindent\hyperlink{a_modelindependent_approach}{A model-independent approach}\dotfill \pageref*{a_modelindependent_approach} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{LectureNotes}{Surveys and lecture notes}\dotfill \pageref*{LectureNotes} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 \emph{[[category]]} and \emph{[[∞-groupoid]]}.

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

\begin{itemize}%
\item an [[internal category in an (∞,1)-category|internal category]] in [[∞-groupoids]]/basic [[homotopy theory]] (as such usually modeled as a [[complete Segal space]]).


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



\end{itemize}
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:

\begin{itemize}%
\item 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]]


\item and know everything about higher [[equivalences]].



\end{itemize}
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]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{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.

\hypertarget{quasicategories}{}\subsubsection*{{Quasi-categories}}\label{quasicategories}

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 \emph{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 \emph{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 \emph{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

\begin{itemize}%
\item Boardman and Vogt called a \emph{weak Kan complex} ;


\item [[Andre Joyal]] called a [[quasi-category]];


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



\end{itemize}
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

\begin{itemize}%
\item the [[(infinity,1)-category of (infinity,1)-functors|quasi-category of (∞,1)-functors]] between two [[quasi-categories]] $C$ and $D$;


\item the quasi-category of all [[∞-groupoids]];


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



\end{itemize}
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.

\hypertarget{__and_simplicially_enriched_categories}{}\subsubsection*{{$Top$-, $Kan$- and simplicially enriched categories}}\label{__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

\begin{itemize}%
\item the [[model structure on topological spaces|standard model structure]] on the [[nice category of spaces|nice category]] of compactly generated weakly Hausdorff [[topological spaces]];


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



\end{itemize}
In fact, this is also equivalent to

\begin{itemize}%
\item the [[model structure on simplicial sets|standard model structure]] on the category [[SimpSet|SSet]] of [[simplicial sets]].

\end{itemize}
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

\begin{itemize}%
\item 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);


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

\begin{itemize}%
\item 0-cells are the points of $X$;


\item 1-cells are the paths in $X$;


\item 2-cells are the triangles in $X$;


\item etc.



\end{itemize}


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

For more see

\begin{itemize}%
\item [[relation between quasi-categories and simplicial categories]]

\end{itemize}
\hypertarget{homotopical_categories}{}\subsubsection*{{Homotopical categories}}\label{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 \emph{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 CW{\tt \symbol{94}}(-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

\hypertarget{model_categories}{}\subsubsection*{{Model categories}}\label{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.

\begin{itemize}%
\item A [[simplicial model category]] $\mathbf{A}$ is, in particular, a [[simplicially enriched category]].


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


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



\end{itemize}
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]] $A$.

\hypertarget{segal_categories_and_complete_segal_spaces}{}\subsubsection*{{Segal categories and complete Segal spaces}}\label{segal_categories_and_complete_segal_spaces}

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

\begin{itemize}%
\item [[Segal categories]];


\item [[complete Segal spaces]].



\end{itemize}
Segal categories can be thought of as categories which are \emph{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]].

\hypertarget{categories}{}\subsubsection*{{$A_\infty$-categories}}\label{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.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{category_theory_2}{}\subsubsection*{{$(\infty,1)$-Category theory}}\label{category_theory_2}

A crucial point about the notion of \emph{$(\infty,1)$-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 \emph{[[(∞,1)-category theory]]}.

\hypertarget{the_collection_of_all_categories}{}\subsubsection*{{The collection of all $(\infty,1)$-categories}}\label{the_collection_of_all_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]].

\hypertarget{model_category_presentations}{}\subsubsection*{{Model category presentations}}\label{model_category_presentations}

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

See \emph{[[table - models for (∞,1)-categories]]}.

\hypertarget{a_modelindependent_approach}{}\subsection*{{A model-independent approach}}\label{a_modelindependent_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 \emph{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.

\begin{enumerate}%
\item 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.


\item Define a \emph{relative quasicategory} to be a quasicategory equipped with a full 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).


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


\item 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 \href{http://nyjm.albany.edu/j/2016/22-4.html}{MazelGee16, Figure 1}.


\item Using the previous observation, expand the diagram in the introduction of \href{http://arxiv.org/abs/1112.0040}{BarwickSchommerPries} (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 $(Cat_{sSet})_{Bergner}$.


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


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


\item It follows from the main theorem of \href{http://arxiv.org/abs/math/0409598}{Toen} 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 $i$.


\item 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 \emph{any} vertex, and propose that to work ``model-independently'' is to work \emph{within} $Cat_{(\infty,1)}$.



\end{enumerate}
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.

\href{http://ncatlab.org/nlab/files/relcats-modelcats-qcats-inftycats.jpg}{This diagram}, 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 \href{https://nforum.ncatlab.org/discussion/7029/a-diagram-relating-different-models-of-inftycategories/#Item_0}{here}.)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[table - models for (∞,1)-categories]]


\item [[0-category]], [[(0,1)-category]]


\item [[category]]


\item [[2-category]], [[(2,1)-category]]


\item [[3-category]]


\item [[n-category]]


\item [[(∞,0)-category]]


\item [[(n,1)-category]]


\item \textbf{(∞,1)-category}, [[internal (∞,1)-category]], [[∞-groupoid]]

\begin{itemize}%
\item [[locally cartesian closed (∞,1)-category]]


\item [[(∞,1)-topos]]


\item [[semiadditive (∞,1)-category]], [[additive (∞,1)-category]]


\item [[stable (∞,1)-category]]


\item [[monoidal (∞,1)-category]]


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


\item [[disjunctive (∞,1)-category]]



\end{itemize}

\item [[(∞,2)-category]]


\item [[(∞,n)-category]]


\item [[(n,r)-category]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{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:

\begin{itemize}%
\item [[André Joyal]], \emph{The theory of quasicategories and its applications} lectures at \href{http://www.crm.es/HigherCategories/}{Simplicial Methods in Higher Categories}, (\href{http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf}{pdf})

\end{itemize}
Further notes (where the term ``[[logos]]'' is used instead of [[quasi-category]]) are in

\begin{itemize}%
\item [[André Joyal]], \emph{Notes on Logoi}, 2008 (\href{http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf}{pdf})

\end{itemize}
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

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]} .

\end{itemize}
An brief exposition from the point of view of [[algebraic topology]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{What is\ldots{} an $\infty$-category?}, \emph{Notices of the AMS}, September 2008 (\href{http://www.ams.org/notices/200808/tx080800949p.pdf}{pdf})

\end{itemize}
A useful comparison of the four [[model category]] structures on

\begin{itemize}%
\item [[quasi-categories]];


\item [[simplicially enriched categories]];


\item [[Segal categories]];


\item [[complete Segal spaces]].



\end{itemize}
is in

\begin{itemize}%
\item [[Julie Bergner]], \emph{A survey of $(\infty,1)$-categories} (\href{http://arxiv.org/abs/math/0610239}{arXiv})

\end{itemize}
More discussion of the other two models can be found at

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}

\end{itemize}
and in the references listed at \emph{[[(∞,n)-category]]}.

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

\begin{itemize}%
\item [[Dan Dugger]], [[David Spivak]], \emph{Rigidification of quasi-categories} (\href{http://arxiv.org/abs/0910.0814}{arXiv:0910.0814})


\item [[Dan Dugger]], [[David Spivak]], \emph{Mapping spaces in quasi-categories} (\href{http://arxiv.org/abs/0911.0469}{arXiv:0911.0469})



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

\begin{itemize}%
\item [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], \emph{[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]} , volume 113 of Mathematical Surveys and Monographs

\end{itemize}
A model by [[stratified spaces]] is in

\begin{itemize}%
\item [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], \emph{A stratified homotopy hypothesis} (\href{http://arxiv.org/abs/1502.01713}{arXiv:1502.01713})

\end{itemize}
A more model-independent abstract formulation is discussed in

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]], \emph{Infinity category theory from scratch}, 2016 (\href{http://www.math.jhu.edu/~eriehl/scratch.pdf}{pdf})

\end{itemize}
And is being developed in a book in progress, regularly being updated:

\begin{itemize}%
\item [[Emily Riehl]] and [[Dominic Verity]], \emph{Elements of $\infty$-Category Theory}, (2019) (\href{http://www.math.jhu.edu/~eriehl/elements.pdf}{pdf})

\end{itemize}
For discussion in [[homotopy type theory]] see \emph{[[internal category in homotopy type theory]]} and see

\begin{itemize}%
\item [[Emily Riehl]], [[Michael Shulman]], \emph{A type theory for synthetic $\infty$-categories} (\href{https://arxiv.org/abs/1705.07442}{arXiv:1705.07442})


\item [[Emily Riehl]], \emph{The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories}, talk at \href{http://www.math.ias.edu/vvmc2018}{Vladimir Voevodsky Memorial Conference 2018} (\href{http://www.math.jhu.edu/~eriehl/Voevodsky.pdf}{pdf})



\end{itemize}
\hypertarget{LectureNotes}{}\subsubsection*{{Surveys and lecture notes}}\label{LectureNotes}

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

\begin{itemize}%
\item Omar Antol\'i{}n Camarena, \emph{A whirlwind tour of the world of $(\infty,1)$-categories}, 2013 (\href{http://arxiv.org/abs/1303.4669}{arXiv:1303.4669})

\end{itemize}
A foundational set of lecture notes is developing in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Higher category theory and homotopical algebra} (\href{http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf}{pdf})

\end{itemize}
A survey with an eye towards [[higher algebra]] is in

\begin{itemize}%
\item [[Moritz Groth]], \emph{A short course on $\infty$-categories} (\href{http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf}{pdf})

\end{itemize}
A survey on various notions of Homotopical category:

\begin{itemize}%
\item [[Emily Riehl]], \emph{Homotopical categories: from model categories to (∞,1)-categories} (\href{https://arxiv.org/abs/1904.00886}{arXiv:1904.00886})

\end{itemize}
Lecture notes are in

\begin{itemize}%
\item [[Dylan Wilson]], \emph{Lectures on higher categories} (\href{https://sites.google.com/a/uw.edu/dwilson/notes}{pdf})


\item [[Emily Riehl]], \emph{[[Categorical Homotopy Theory]]}



\end{itemize}
See also

\begin{itemize}%
\item [[Zhen Lin Low]], \emph{[[Notes on homotopical algebra]]}

\end{itemize}
[[!redirects (∞,1)-category]] [[!redirects (∞,1)-categories]] [[!redirects (infinity,1)-categories]] [[!redirects infinity comma one category]] [[!redirects (?\%2C1)-category]]



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