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\section*{quasi-category}

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

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

[[!include quasi-category theory contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_simplicially_enriched_categories}{Relation to simplicially enriched categories}\dotfill \pageref*{relation_to_simplicially_enriched_categories} \linebreak
\noindent\hyperlink{higher_associahedra_in_quasicategories}{Higher associahedra in quasi-categories}\dotfill \pageref*{higher_associahedra_in_quasicategories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{constructions_in_quasicategories}{Constructions in quasi-categories}\dotfill \pageref*{constructions_in_quasicategories} \linebreak
\noindent\hyperlink{RelatedConcepts}{Related concepts}\dotfill \pageref*{RelatedConcepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{quasi-category} is a [[geometric definition of higher categories|geometric model]] for [[(∞,1)-category]].

In analogy to how a [[Kan complex]] is a model in terms of [[simplicial set]]s of an [[∞-groupoid]] -- also called an [[(∞,0)-category]] -- a [[quasi-category]] is a model in terms of [[simplicial set]]s of an [[(∞,1)-category]].

\begin{uremark}
In older literature, such as [[The Joy of Cats]], the term ``quasicategory'' was sometimes used for a ``very large'' category whose objects are [[large categories]] or otherwise built out of [[proper classes]], but nowadays this usage is fairly archaic. See also [[metacategory]].

\end{uremark}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{quasi-category} or \textbf{[[weak Kan complex]]} is a [[simplicial set]] $C$ satisfying the following equivalent conditions

\begin{itemize}%
\item all \emph{inner} [[horn]]s in $C$ have fillers. This means that the lifting condition given at [[Kan complex]] is imposed only for horns $\Lambda^i[n]$ with $0 \lt i \lt n$.


\item the morphism of simplicial sets

\begin{displaymath}
sSet(\Delta[2],C) \to sSet(\Lambda^1[2],C)
\end{displaymath}
(induced from the inner [[horn]] inclusion $\Lambda^1[2] \to \Delta[2]$) is an acyclic [[Kan fibration]].



\end{itemize}
\end{defn}
The equivalence of these two definitions is due to [[Andre Joyal]] and recalled as [[Higher Topos Theory|HTT, corollary 2.3.2.2]]. Quasi-categories are the [[fibrant objects]] in the [[model structure for quasi-categories]].

\begin{remark}
\label{}\hypertarget{}{}
The second condition says manifestly that a quasi-category is a simplicial set in which composition of any two composable edges is defined up to a contractible space of choices. This is the [[coherence law]] on composition.

\end{remark}
\begin{udefn}
An \textbf{[[algebraic quasi-category]]} is a quasi-category equipped with a \emph{choice} of inner horn fillers.

\end{udefn}
While quasi-categories provide a [[geometric definition of higher categories]], algebraic quasi-categories provide an [[algebraic definition of higher categories]]. For more details on this see [[model structure on algebraic fibrant objects]].

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

\hypertarget{relation_to_simplicially_enriched_categories}{}\subsubsection*{{Relation to simplicially enriched categories}}\label{relation_to_simplicially_enriched_categories}

The [[homotopy coherent nerve]] relates quasi-categories with another model for $(\infty,1)$-categories: [[simplicially enriched categories]].

See [[relation between quasi-categories and simplicial categories]] for more.

\hypertarget{higher_associahedra_in_quasicategories}{}\subsubsection*{{Higher associahedra in quasi-categories}}\label{higher_associahedra_in_quasicategories}

While the geometric definition of [[(∞,1)-category]] in terms of quasi-categories elegantly captures all the higher categorical data automatically, it may be of interest in applications to explicitly extract the associators and higher associators encoded by this structure, that would show up in any [[algebraic definition of higher categories|algebraic definition of the same categorical structure]], such as [[algebraic quasi-categories]].

For a discussion of this see

\begin{itemize}%
\item [[Emily Riehl]], \emph{Associativity data in an $(\infty,1)$-category} (\href{http://math.uchicago.edu/~eriehl/associativity.pdf}{pdf} \href{http://golem.ph.utexas.edu/category/2009/10/associativity_data_in_an_1cate.html}{blog})

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The two basic examples for quasi-categories are

\begin{itemize}%
\item Every [[Kan complex]] is, in particular, a quasi-category.


\item The [[nerve]] of a [[category]] is a quasi-category.



\end{itemize}
Since the nerve of a category is a [[Kan complex]] iff the category is a [[groupoid]], quasi-categories are a minimal common generalization of Kan complexes and nerves of categories.

By the [[homotopy hypothesis]]-theorem every Kan complex arises, up to equivalence, as the [[fundamental ∞-groupoid]] of a [[topological space]].

Analogously, every [[directed topological space]] $X$ has naturally a [[fundamental (∞,1)-category]] given by a quasi-category whose $k$-cells are maps $\Delta^k_{Top} \to X$ that map the 1-[[simplicial skeleton|skeleton]] of the topological simplex in an order-preserving way to directed paths in $X$.

The [[directed homotopy theory]] that would state that this or a similar construction exhausts all quasicategories up to equivalence, does not quite exist yet.

\hypertarget{constructions_in_quasicategories}{}\subsection*{{Constructions in quasi-categories}}\label{constructions_in_quasicategories}

The point of quasi-categories is that they are supposed to provide a fully [[homotopy theory|homotopy-theoretic]] refinement of the ordinary notion of [[category]]. In particular, all the familiar constructions of [[category theory]] have natural analogs in the context of quasi-categories. See for instance

\begin{itemize}%
\item [[hom-object in a quasi-category]]


\item [[equivalence in a quasi-category]]


\item [[equivalence of quasi-categories]]


\item [[join of quasi-categories]]


\item [[over quasi-category]]


\item [[terminal object in a quasi-category]]


\item [[monomorphism in an (∞,1)-category]]


\item [[limit in quasi-categories]]


\item [[sub-quasi-category]]


\item [[opposite quasi-category]]


\item [[fibrations of quasi-categories]]

\begin{itemize}%
\item [[inner Kan fibration]]


\item [[Cartesian fibration]]


\item [[left Kan fibration]]/[[right Kan fibration]]


\item [[minimal Joyal fibration]]



\end{itemize}


\end{itemize}
\hypertarget{RelatedConcepts}{}\subsection*{{Related concepts}}\label{RelatedConcepts}

One may try to further weaken the filler conditions in order to describe [[(∞,n)-categories]] for $n \gt 1$. One approach along these lines is the theory of [[weak complicial sets]].

Or one may change the shape category to pass from [[simplicial sets]] to [[cellular sets]]. A quasi-category-like definition of [[(∞,n)-categories]] on these -- \emph{[[n-quasicategories]]} -- is discussed at \emph{[[model structure on cellular sets]]}.

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

Quasi-categories were originally defined in

\begin{itemize}%
\item [[Michael Boardman]], [[Rainer Vogt]], \emph{Homotopy invariant algebraic structures on topological spaces}, Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, 1973.

\end{itemize}
They occured as \emph{[[weak Kan complexes]]} in

\begin{itemize}%
\item [[Rainer Vogt]], \emph{Homotopy limits and colimits}, Math. Z., 134, (1973), 11--52.

\end{itemize}
Vogt's main theorem involved a category of [[homotopy coherent diagram|homotopy coherent diagrams]] defined on a topologically enriched category and showed it was equivalent to a quotient category of the category of (commutative) diagrams on the same category.

[[Jean-Marc Cordier]] in

\begin{itemize}%
\item [[J.-M. Cordier]], \emph{Sur la notion de diagramme homotopiquement coh\'e{}rent}, Cahiers de Top. G\'e{}om. Diff., 23, (1982), 93 --112,

\end{itemize}
defined the [[homotopy coherent nerve]] of any [[simplicially enriched category]]. This generalised the [[nerve]] of an ordinary category. In

\begin{itemize}%
\item [[J.-M. Cordier]] and [[Tim Porter]], \emph{Vogt's theorem on categories of homotopy coherent diagrams}, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65--90,

\end{itemize}
it was shown that this homotopy coherent nerve was a quasi-category if the simplicial enrichment was by Kan complexes.

A systematic study of SSet-enriched categories in this context is in

\begin{itemize}%
\item J-M Cordier, [[Tim Porter]] \emph{Homotopy coherent category theory} Trans. Amer. Math. Soc. 349 (1997), no. 1, 1-54. (\href{http://www.ams.org/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf}{pdf})

\end{itemize}
The importance of quasi-categories as a basis for [[category theory]] has been particularly emphasized in work by [[André Joyal]]

\begin{itemize}%
\item [[André Joyal]], \emph{Quasi-categories and Kan complexes}, J. Pure Appl. Algebra, 175 (2002), 207-222.

\end{itemize}
For several years Joyal has been preparing a textbook on the subject. This still doesn't quite exist, but an extensive writeup 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}
and more recently, with more details

\begin{itemize}%
\item [[André Joyal]], \emph{Notes on quasi-categories} (\href{http://www.math.uchicago.edu/~may/IMA/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 in

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

\end{itemize}
An overview of the material there is contained in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Cat\'e{}gories sup\'e{}rieures et th\'e{}orie des topos}, S\'e{}minaire Bourbaki, 21.3.2015, \href{http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf}{pdf}.

\end{itemize}
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}
Survey includes

\begin{itemize}%
\item [[Emily Riehl]], \emph{Categorical homotopy theory}, Lecture notes (\href{http://www.math.jhu.edu/~eriehl/cathtpy.pdf}{pdf})


\item [[Charles Rezk]], \emph{Stuff about quasicategories}, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017,(\href{http://math.uiuc.edu/~rezk/595-fal16/quasicats.pdf}{pdf})


\item [[Moritz Groth]], \emph{A short course on ∞-categories} (\href{https://arxiv.org/abs/1007.2925}{arXiv:1007.2925})



\end{itemize}
An in-depth study of adjunctions between quasi-categories and the monadicity theorem is given in

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]] \emph{The 2-category theory of quasi-categories} (\href{http://arxiv.org/abs/1306.5144}{arXiv}), \emph{Homotopy coherent adjunctions and the formal theory of monads} (\href{http://arxiv.org/abs/1310.8279}{arXiv})

\end{itemize}
[[!redirects quasi-categories]] [[!redirects quasicategory]] [[!redirects quasicategories]]

[[!redirects inner Kan complex]] [[!redirects inner Kan complexes]]



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