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\section*{higher category theory}

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

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

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{basic_concepts}{Basic concepts}\dotfill \pageref*{basic_concepts} \linebreak
\noindent\hyperlink{basic_constructions}{Basic constructions}\dotfill \pageref*{basic_constructions} \linebreak
\noindent\hyperlink{higher_presheaves}{Higher presheaves}\dotfill \pageref*{higher_presheaves} \linebreak
\noindent\hyperlink{higher_universal_constructions}{Higher universal constructions}\dotfill \pageref*{higher_universal_constructions} \linebreak
\noindent\hyperlink{basic_theorems}{Basic theorems}\dotfill \pageref*{basic_theorems} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{extended_cobordisms}{Extended cobordisms}\dotfill \pageref*{extended_cobordisms} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{1categorical_models}{1-categorical models}\dotfill \pageref*{1categorical_models} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Higher category theory} is the generalization of [[category theory]] to a context where there are not only [[morphism]]s between [[object]]s, but generally [[k-morphism]]s between $(k-1)$-morphisms, for all $k \in \mathbb{N}$.

Higher category theory studies the generalization of [[∞-groupoid]]s -- and hence, via the [[homotopy hypothesis]], of [[topological space]]s -- to that of [[directed space]]s and their \emph{combinatorial or algebraic models} . It is to the theory of [[∞-groupoid]]s as [[category theory]] is to the theory of [[groupoids]] (and hence of [[groups]]).

These [[geometric definition of higher category|combinatorial]] or [[algebraic definition of higher category|algebraic]] models are known as [[n-category|n-categories]] or, when $n \to \infty$, as [[∞-category|∞-categories]] or [[∞-category|∞-categories]], or, in more detail, as [[(n,r)-category|(n,r)-categories]]:

\begin{itemize}%
\item the natural number $n$ denotes the maximal dimension of \emph{non-trivial} cells in the model,


\item while the natural number $r$ denotes the maximal dimension of the \emph{directed} cells.



\end{itemize}
So an ordinary [[topological space]] or [[∞-groupoid]] is an [[(∞,0)-category]]: it has cells of arbitrary dimension and all of them are reversible.

In contrast to that, a [[geometric definition of higher category|combinatorial]] or [[algebraic definition of higher category|algebraic]] model for a [[directed space]] in which the 1-dimensional paths may not all be reversible is an [[(∞,1)-category]]: it still has cells of arbitrary dimension, but only those of dimension greater than 1 are guaranteed to be reversible.

Often it is convenient in practice to consider the case where the possible dimension $n$ of non-trivial cells is finite. This is familiar from how a [[topological space]] that happens to have vanishing [[homotopy group]]s in dimension above some $n$ -- a [[homotopy n-type]] -- is modeled by an [[n-groupoid]]. A fully directed version of this is an [[n-category]], which is short for [[(n,n)-category]]: non-trivial cells up to and including dimension $n$, and all of them allowed to be non-reversible. Actually, it is possible to go on to an $(n,n+1)$-category, or $(n+1)$-[[n-poset|poset]]; you can either consider than the $n$-cells are ordered, or else consider that there are irreversible $(n+1)$-cells which are indistinguishable. (Reversible indistinguishable $(n+1)$-cells are all identities and so might as well not exist.)

For low $n$ very explicit [[algebraic definition of higher category|algebraic models]] for $n$-categories are available but in their full generality become quickly rather untractable as $n$ increases: the series starts with [[bicategory]], [[tricategory]] and [[tetracategory]]. While bicategories have found plenty of applications, already the axioms of tricategories are rather involved and their theory mainly serves to produce the statement that there is a good [[semi-strict infinity-category|semi-strictifications]] of tricategories called [[Gray-category|Gray-categories]].

Indeed, there are many \emph{strictified} models for higher categories: combinatorial or algebraic models that sacrifice full generality for a better concrete control. Notably there is a useful combinatorial/algebraic model for [[strict ∞-category|strict ∞-categories]] which, while falling short, already goes a long way towards describing general higher categorical structures. In fact, by [[Simpson's conjecture]] every [[∞-category]] is equivalent to one that looks like a [[strict ∞-category]] except for possibly having weak unit laws.

The challenge of describing fully general [[∞-category|∞-categories]] is to achieve a combinatorial or algebraic control of all the higher composition rules of higher cells. One can distinguish roughly two orthogonal approaches to dealing with the problem:

in the [[algebraic definition of higher category]] an algebraic machinery is set up that allows to concretely handle the explicit \emph{choices} of composites of cells. Such machinery usually involves [[operad]]ic tools in one way or other. The most sophisticated definitions of this kind are the closely related [[Batanin ∞-category]] and [[Trimble n-category|Trimble ∞-category]].

On the other hand, in the [[geometric definition of higher category]] a combinatorial machinery is set up that allows to guarantee \emph{existence} of composites of cells. In the [[simplicial model for weak omega-categories|simplicial models for weak ∞-categories]] higher categories are characterized as [[simplicial set]]s with the extra [[stuff, structure, property|property]] that certain composites exist. The issue here is to characterize these existence laws correctly.

The basic example for such ``existence laws'' is the \emph{Kan-filler condition} that characterizes simplicial sets that are [[Kan complex]]es, the models for [[(∞,0)-category|(∞,0)-categories]]. More general higher categories are obtained by relaxing the Kan condition in just the right way. For instance by simply restricting the Kan-condition to just a certain sub-set of all cells yields the definition of simplicial sets that are called [[quasi-category|quasi-categories]]. These model [[(∞,1)-category|(∞,1)-categories]].

The right further relaxation of the (weak) Kan filler condition is more involved. An approach to capture this has been given by [[Dominic Verity]]`s definition of simplicial sets that are called [[complicial set]]s and [[weak complicial set]]s.

One expects that every algebraic definition of higher categories admits a construction called a [[nerve]] that maps it to a [[simplicial set]] that would constitute the corresponding geometric model.

Another approach to handle the geometric definition of higher categories is a recursive one that uses $n$-fold simplicial sets. This is based on the notion of [[complete Segal space]], which is essentially a variation of the concept of [[quasi-category]]. Its advantage is that its definition may be applied recursively to yield the notion of [[n-fold complete Segal space]]s. These model [[(∞,n)-category|(∞,n)-categories]] for finite $n$.

Finally, a large supply of further models exists for [[(∞,1)-category|(∞,1)-categories]] in terms of [[enriched category theory]]. [[simplicial model category|Simplicially enriched model categories]] are a highly-developed toolkit for handling [[presentable (infinity,1)-category|presentable (∞,1)-categories]]. [[pretriangulated dg-category|Pretriangulated dg-enriched categories]] and [[A-infinity category|A-∞ categories]] are a comparably highly developed toolkit for handling [[stable (∞,1)-category|stable (∞,1)-categories]].

\hypertarget{basic_concepts}{}\subsection*{{Basic concepts}}\label{basic_concepts}

The basic concept on which higher category theory is built is the notion of \textbf{[[k-morphism]]} for all $k \in \mathbb{N}$, equipped with a notion of composition, such that \textbf{[[coherence law]]s} are satisfied.

This is what it's all about.

\hypertarget{basic_constructions}{}\subsection*{{Basic constructions}}\label{basic_constructions}

\hypertarget{higher_presheaves}{}\subsubsection*{{Higher presheaves}}\label{higher_presheaves}

\begin{itemize}%
\item [[higher topos theory]]

\end{itemize}
\hypertarget{higher_universal_constructions}{}\subsubsection*{{Higher universal constructions}}\label{higher_universal_constructions}

\begin{itemize}%
\item [[2-limit]]


\item [[adjoint (∞,1)-functor|(∞,1)-adjunction]]


\item [[(∞,1)-Kan extension]]

\begin{itemize}%
\item [[limit in a quasi-category|(∞,1)-limit]]

\end{itemize}

\item [[(∞,1)-Grothendieck construction]]



\end{itemize}
\hypertarget{basic_theorems}{}\subsection*{{Basic theorems}}\label{basic_theorems}

\begin{itemize}%
\item [[homotopy hypothesis]]-theorem


\item [[delooping hypothesis]]-theorem


\item [[periodic table]]


\item [[stabilization hypothesis]]-theorem


\item [[michaelshulman:exactness hypothesis]]


\item [[holographic principle of higher category theory|holographic principle]]



\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

See

\begin{itemize}%
\item [[applications of (higher) category theory]].

\end{itemize}
\hypertarget{extended_cobordisms}{}\subsubsection*{{Extended cobordisms}}\label{extended_cobordisms}

One major application of higher category theory and one of the driving forces in developing it has been [[FQFT|extended functorial quantum field theory]]. This has recently led to what may become one of the central theorems of higher category theory, the proof of the [[cobordism hypothesis]]. This roughly characterizes the [[(∞,n)-category of cobordisms]] $Bord_n$ as the free [[(∞,n)-category]] with duals on a single generator.

\hypertarget{models}{}\subsection*{{Models}}\label{models}

There are many different \emph{models} for bringing the abstract notion of higher category onto paper.

\begin{itemize}%
\item [[(n × k)-category]]


\item [[n-fold category]]


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

\begin{itemize}%
\item [[Theta-space]]


\item [[∞-category]]/[[∞-category]]


\item [[(∞,n)-category]]

\begin{itemize}%
\item [[n-fold complete Segal space]]

\end{itemize}

\item [[(∞,2)-category]]


\item [[(∞,1)-category]]

\begin{itemize}%
\item [[quasi-category]]

\begin{itemize}%
\item [[algebraic quasi-category]]

\end{itemize}

\item [[simplicially enriched category]]


\item [[complete Segal space]]


\item [[model category]]


\item [[internal category in homotopy type theory]]



\end{itemize}

\item [[(∞,0)-category]]/[[∞-groupoid]]

\begin{itemize}%
\item [[Kan complex]]

\begin{itemize}%
\item [[algebraic Kan complex]]


\item [[simplicial T-complex]]



\end{itemize}


\end{itemize}

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

\begin{itemize}%
\item [[2-category]], [[(2,1)-category]]


\item [[1-category]]


\item [[0-category]]


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


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



\end{itemize}

\item [[n-poset]] = (n$-$1,n)-category

\begin{itemize}%
\item [[2-poset]]

\end{itemize}

\item [[n-groupoid]] = (n,0)-category

\begin{itemize}%
\item [[2-groupoid]], [[3-groupoid]]

\end{itemize}


\end{itemize}

\item [[categorification]]/[[decategorification]]


\item [[geometric definition of higher category]]

\begin{itemize}%
\item [[Kan complex]]


\item [[quasi-category]]


\item [[simplicial model for weak ∞-categories]]

\begin{itemize}%
\item [[complicial set]]


\item [[weak complicial set]]



\end{itemize}


\end{itemize}

\item [[algebraic definition of higher category]]

\begin{itemize}%
\item [[bicategory]]


\item [[bigroupoid]]


\item [[tricategory]]


\item [[tetracategory]]


\item [[strict ∞-category]]


\item [[Batanin ∞-category]]


\item [[Trimble n-category|Trimble ∞-category]]


\item [[Grothendieck-Maltsiniotis ∞-categories]]



\end{itemize}

\item [[stable homotopy theory]]

\begin{itemize}%
\item [[symmetric monoidal category]]


\item [[symmetric monoidal (∞,1)-category]]


\item [[stable (∞,1)-category]]

\begin{itemize}%
\item [[dg-category]]


\item [[A-∞ category]]


\item [[triangulated category]]



\end{itemize}


\end{itemize}


\end{itemize}
\hypertarget{1categorical_models}{}\subsubsection*{{1-categorical models}}\label{1categorical_models}

\begin{itemize}%
\item [[homotopical category]]


\item [[model category|model category theory]]


\item [[enriched category theory]]



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

\begin{itemize}%
\item [[opetopic type theory]]

\end{itemize}
[[!include table of category theories]]

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

For a very gentle introduction to higher category theory, try \href{http://math.ucr.edu/home/baez/week73.html#tale}{The Tale of -Categories}, which begins in ``week73'' of This Week's Finds and goes on from there \ldots{}; keep clicking the links.

For a slightly more formal but still pathetically easy introduction, try:

\begin{itemize}%
\item [[John Baez]], \href{http://arxiv.org/abs/q-alg/9705009}{An Introduction to n-Categories}, in

\end{itemize}
, eds.

E. Moggi and G. Rosolini, Springer Lecture Notes in Computer Science vol. 1290, Springer, Berlin, 1997.

For a free introductory text on $n$-categories that's \emph{full of pictures}, try this:

\begin{itemize}%
\item [[Eugenia Cheng]] and [[Aaron Lauda]], \href{http://www.cheng.staff.shef.ac.uk/guidebook/}{Higher-Dimensional Categories: An Illustrated Guidebook}.

\end{itemize}
[[Tom Leinster]] has written about ``comparative $\infty$-categoriology'' (to \href{http://golem.ph.utexas.edu/category/2008/01/comparative_smootheology.html}{borrow} a term):

\begin{itemize}%
\item Tom Leinster, \emph{A Survey of Definitions of n-Category} (\href{http://arxiv.org/abs/math.CT/0107188}{arXiv})


\item Tom Leinster, \emph{Higher Operads, Higher Categories} (\href{http://arxiv.org/abs/math/0305049}{arXiv})



\end{itemize}
A grand picture of the theory of higher categories is drawn in

\begin{itemize}%
\item [[Carlos Simpson]], \emph{[[Homotopy Theory of Higher Categories]]} (\href{http://hal.archives-ouvertes.fr/docs/00/44/98/26/PDF/main.pdf}{pdf})

\end{itemize}
Another collection of discussions of definitions of higher categories is given at

\begin{itemize}%
\item [[John Baez]], [[Peter May]] [[Approaching Higher Category Theory]]

\end{itemize}
A brief useful survey of approaches to the definition of higher categories is provided by the set of slides

\begin{itemize}%
\item [[Andre Joyal]], [[Tim Porter]], [[Peter May]], \emph{Weak categories} (\href{https://web.archive.org/web/20150326110254/http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf}{pdf})

\end{itemize}
The theory of [[quasi-categories]] as [[(∞,1)-categories]] has reached a point where it is well developed and being applied to a wealth of problems with

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]} (\href{http://arxiv.org/abs/math.CT/0608040}{arXiv})

\end{itemize}
There's a lot more to add here, even if we restrict ourselves to very general texts. (More specialized stuff should go under more specialized subcategories!)

[[!redirects higher category theory]] [[!redirects higher category]] [[!redirects higher categories]]

[[!redirects higher-order category theory]] [[!redirects higher-order category]] [[!redirects higher-order categories]]

[[!redirects higher-dimensional category theory]] [[!redirects higher-dimensional category]] [[!redirects higher-dimensional categories]]



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