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\section*{applications of (higher) category theory}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_geometry}{In geometry}\dotfill \pageref*{in_geometry} \linebreak
\noindent\hyperlink{DiffEqu}{In differential equations}\dotfill \pageref*{DiffEqu} \linebreak
\noindent\hyperlink{in_cohomology}{In cohomology}\dotfill \pageref*{in_cohomology} \linebreak
\noindent\hyperlink{hochschild_cohomology}{Hochschild (co)homology}\dotfill \pageref*{hochschild_cohomology} \linebreak
\noindent\hyperlink{InHomotopyTheory}{In homotopy theory}\dotfill \pageref*{InHomotopyTheory} \linebreak
\noindent\hyperlink{RationalHomotopyTheory}{In rational homotopy theory}\dotfill \pageref*{RationalHomotopyTheory} \linebreak
\noindent\hyperlink{in_ktheory}{In K-theory}\dotfill \pageref*{in_ktheory} \linebreak
\noindent\hyperlink{in_tannaka_duality}{In Tannaka duality}\dotfill \pageref*{in_tannaka_duality} \linebreak
\noindent\hyperlink{in_differential_geometry}{In differential geometry}\dotfill \pageref*{in_differential_geometry} \linebreak
\noindent\hyperlink{InDifferentialCohomology}{In differential cohomology}\dotfill \pageref*{InDifferentialCohomology} \linebreak
\noindent\hyperlink{DeformationTheory}{In deformation theory}\dotfill \pageref*{DeformationTheory} \linebreak
\noindent\hyperlink{Logic}{In logic and type theory}\dotfill \pageref*{Logic} \linebreak
\noindent\hyperlink{Physics}{In physics}\dotfill \pageref*{Physics} \linebreak
\noindent\hyperlink{ClassMech}{Classical mechanics and its geometric quantization}\dotfill \pageref*{ClassMech} \linebreak
\noindent\hyperlink{QuantumMechanics}{Quantum mechanics and quantum information}\dotfill \pageref*{QuantumMechanics} \linebreak
\noindent\hyperlink{GaugeTheory}{Gauge theory}\dotfill \pageref*{GaugeTheory} \linebreak
\noindent\hyperlink{Supergravity}{Supergravity}\dotfill \pageref*{Supergravity} \linebreak
\noindent\hyperlink{BVBRST}{BV-BRST formalism}\dotfill \pageref*{BVBRST} \linebreak
\noindent\hyperlink{QFT}{Quantum field theory}\dotfill \pageref*{QFT} \linebreak
\noindent\hyperlink{holography}{Holography}\dotfill \pageref*{holography} \linebreak
\noindent\hyperlink{3dTFT2dCFT}{3d TFT and 2d CFT}\dotfill \pageref*{3dTFT2dCFT} \linebreak
\noindent\hyperlink{in_your_favorite_topic_here}{In your favorite topic here}\dotfill \pageref*{in_your_favorite_topic_here} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak


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

\begin{quote}%
I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months -- when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration\ldots{} the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it\ldots{} yet it finally surrounds the resistant substance.

[[Alexander Grothendieck]], \emph{[[Récoltes et semailles]]}, 1985--1987, pp. 552-3-1 (``[[The Rising Sea]]'')

I don't want you to think all this is theory for the sake of it, or rather for the sake of itself. It's theory for the sake of other theory.

([[Jacob Lurie|J. Lurie]], \href{http://gowers.wordpress.com/2010/08/31/icm2010-spielman-csornyei-lurie/}{ICM 2010})


\end{quote}
The tools of [[category theory]] and [[higher category theory]] serve to organize other structures. There is a plethora of applications that have proven to be much more transparent when employing the [[nPOV]]. Higher category theory has helped foster entire new fields of study that would have been difficult to conceive otherwise. This page lists and discusses examples.

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

The following is a (incomplete) list of examples of topics for which higher category their has proven to be useful.

\hypertarget{in_geometry}{}\subsubsection*{{In geometry}}\label{in_geometry}

The field of [[differential geometry]] has long managed to avoid the change to an $n$-point of view that had been found to be unavoidable, natural and fruitful in algebraic geometry long ago. But more recently -- not the least due to the recognition of differential [[higher geometry|higher geometric]] structures in the physics of [[gauge theory]] and [[supergravity]] (such as that of [[orbifold]]s and [[orientifold]]s, of smooth [[gerbe]]s and smooth [[principal ∞-bundle]]s) -- [[sheaf and topos theory|sheaf and topos theoretic]] concepts, such as [[synthetic differential geometry]], [[diffeological space]]s and [[differentiable stack]]s are gaining wider recognition and appreciation.

For instance the ordinary category [[Diff]] of [[smooth manifold]]s fails to have all [[pullback]]s, it only has pullbacks along [[transversal map]]s. This observation is usually the starting point for realizing that differential geometry is in need of a bit of [[category theory]] in the form of [[higher geometry]].

In all notions of [[generalized smooth space]]s all pullbacks do exist. But they may still not be the ``right'' pullbacks. For instance cohomology of pullback objects may not have the expected properties. This is solved by passing to smooth [[derived stack]]s, such as [[derived smooth manifold]]s.

Recent developments in [[higher category theory]], such as the concept of higher [[Structured Spaces]] based on [[Higher Topos Theory]], put all these notions of generalized geometries into a unified picture of [[higher geometry]] that realizes old ideas about how category theory provides a language for [[space and quantity]] in great detail and powerful generality and sheds new light on old [[classical mathematics|classical]] problems such the description of the [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves|derived moduli stack of derived elliptic curves]] and the construction of the [[tmf]] [[spectrum]] from it. This construction has benefited tremendously from the adoption of the [[nPOV]]. Using this point of view, the general strategy becomes naturally evident.

\hypertarget{DiffEqu}{}\paragraph*{{In differential equations}}\label{DiffEqu}

Much of [[topological vector space]] theory, e.g., the theory of [[distribution]]s, [[nuclear space]]s, etc. has its origins in [[partial differential equation]] theory and is intensely conceptual (categorical) in spirit. It is routine these days to accept distributional solutions, but it wasn't always so, and it was the efficacy of the abstract TVS theory which changed people's minds.

Way back Cartan studied differential equations in terms of [[exterior differential system]]s. From the $n$POV, these may be understood naturally as sub [[Lie ∞-algebroid]]s of a [[tangent Lie algebroid]].

[[Bill Lawvere]] noticed in the 1960s that the notion of differential equation makes sense in any [[smooth topos]] (as described \href{http://ncatlab.org/nlab/show/differential+equation#InSynthDiff}{here}). In his highly influential article \emph{Categorical dynamics} he promoted the point of view that all things [[differential geometry|differential geometric]] can be formulated in abstract category theory internal to a suitable [[topos]]. This is the origin of [[synthetic differential geometry]]. It may be understood as providing the fundamental characterization of the notion of the [[infinitesimal space|infinitesimal]].

Closely related to both these perspectives, a modern point of view on differential equations that is proving to be very fruitful regards them as part of the theory of [[D-module]]s.

\hypertarget{in_cohomology}{}\subsubsection*{{In cohomology}}\label{in_cohomology}

A multitude of notions of cohomology and its variants are unified from the $n$POV when viewed as [[derived hom space|∞-categorical hom-spaces]] in [[(∞,1)-topoi]]. See [[cohomology]].

\hypertarget{hochschild_cohomology}{}\paragraph*{{Hochschild (co)homology}}\label{hochschild_cohomology}

Specifically, the subject of [[Hochschild cohomology]], when generalized to \emph{higher order Hochschild cohomology} effectively merges into the canonical concept of [[(∞,1)-powering]] of an [[(∞,1)-topos]] over [[∞Grpd]]. See [[Hochschild cohomology]] for details.

\hypertarget{InHomotopyTheory}{}\subsubsection*{{In homotopy theory}}\label{InHomotopyTheory}

The study of [[homotopy theory]] originated in the study of categories such as those of [[topological space]]s and other objects such as [[chain complex]]es whose [[morphism]]s were known to admit a notion of [[homotopy]]. Historically, in a sequence of steps formalisms were proposed that would organize the rich interesting structure found in such situations. As a first approximation the notion of [[homotopy category]] and [[derived category]] was introduced in order to deal with structures ``up to homotopy''. But it was clear that the [[homotopy category]] captured only a very small part of the interesting information. Quillen introduced the notion of [[model category]] as a formalization of the full structure, and this formalization turned out to yield a powerful theory that today provides a powerful toolset for dealing with homotopy theoretic situations.

But also the notion of model category was seen to not be the full answer. For instance a model category in a sense retains \emph{too much} non-intrinsic information. Equivalence classes of model categories under [[Quillen equivalence]] are a more intrinsic characterization of a given [[homotopy theory]]. But this means that one needs some [[higher category theory|higher categorical]] notion for the collection of all model categories. This problem came to be known as the search for the \textbf{homotopy theory of homotopy theories}.

Recently, this problem was fully solved and homotopy theory fully understood as the special case of [[higher category theory]] that deals with [[(∞,1)-category|(∞,1)-categories]]:

\begin{itemize}%
\item the notion of [[model category]], in particular when refined to that of a [[simplicial model category]] serves as a [[presentable (infinity,1)-category|presentation]] of the notion of [[(∞,1)-category]];


\item the ``homotopy theory of homotopy theories'' is accordingly the [[(∞,1)-category of (∞,1)-categories]] $(\infty,1)Cat$;

better yet: there is an [[(∞,n)-category|(∞,2)-category]] of all $(\infty,1)$-categories;


\item in $(\infty,1)Cat$ two $(\infty,1)$-categories presented by model categories are equivalent precisely if the presenting model categories may be connected by a zig-zag sequence of [[Quillen equivalence]]s;


\item all ``homotopy''-constructions in model category theory, such as [[homotopy limit]]s, [[mapping cone]]s etc. are \emph{tools for constructing} the corresponding higher categorical intrinsic notions, such as [[limit in a quasi-category|limit in an (∞,1)-category]].


\item all variant notions find their intrinsic higher categorical interpretation this way: for instance [[stable homotopy theory]] is the study of [[stable (∞,1)-category|stable (∞,1)-categories]];


\item the [[homotopy category]] of a [[model category]] is simply the [[decategorification]] of the corresponding $(\infty,1)$-category to just a [[1-category]];


\item and for instance the notion of homotopy category of a stable $(\infty,1)$-category reproduces the notion of [[triangulated category]], thus incorporating also a large toolset from [[homological algebra]] into the picture.



\end{itemize}
\hypertarget{RationalHomotopyTheory}{}\paragraph*{{In rational homotopy theory}}\label{RationalHomotopyTheory}

\ldots{} The study of [[rational homotopy theory]] is naturally understood as the study of the [[localization of an (∞,1)-category|localizations]] of [[(∞,1)-topos]]es at morphism that induce equivalences in [[cohomology]] with certain line-object coefficients. See [[rational homotopy theory in an (∞,1)-topos]]. \ldots{}

\hypertarget{in_ktheory}{}\subsubsection*{{In K-theory}}\label{in_ktheory}

In full generality, ([[algebraic K-theory|algebraic]]) [[K-theory]] is a universal assignment of [[spectra]] to [[stable (∞,1)-categories]].

\ldots{}

\hypertarget{in_tannaka_duality}{}\subsubsection*{{In Tannaka duality}}\label{in_tannaka_duality}

\ldots{} see [[Tannaka duality]] \ldots{}

\hypertarget{in_differential_geometry}{}\subsubsection*{{In differential geometry}}\label{in_differential_geometry}

See at

\begin{itemize}%
\item [[higher differential geometry applied to plain differential geometry]]

\end{itemize}
\hypertarget{InDifferentialCohomology}{}\subsubsection*{{In differential cohomology}}\label{InDifferentialCohomology}

[[cohesion]] on [[(∞,1)-toposes]] solves the \href{differential%20cohomology%20diagram#SimonsSullivan07}{Simons-Sullivan question} characterization on the characterization of [[generalized (Eilenberg-Steenrod) cohomology|generalized (Eilenberg-Steenrod-type)]] [[differential cohomology]]. See at \emph{[[differential cohomology hexagon]]} for details.

\hypertarget{DeformationTheory}{}\subsubsection*{{In deformation theory}}\label{DeformationTheory}

In deformation theory it was early on recognized that for a good theory the notion of [[Kähler differential]]s has to be generalized to the notion of [[cotangent complex]]. With the advent of the study of derived [[moduli space]]s, such as the [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves|derived moduli space of derived elliptic curves]], this needed to be further generalized to notions of cotangent complexes not just of [[ring]]s, but of [[E-∞-ring]]s.

It turns out that all these concepts are special cases of a construction obtained from a simple higher categorical notion, that of [[left adjoint]] [[section]]s of a [[tangent (∞,1)-category]].

\hypertarget{Logic}{}\subsubsection*{{In logic and type theory}}\label{Logic}

While it is common to view logic as the study of absolute truth, in fact logic can have many different interpretations, or [[semantics]]. A particular semantics for logic can be useful both to inform the study of logic, and to prove facts logically about the semantics. One very fruitful semantics of this sort is \emph{categorical semantics} for logic and [[type theory]], according to which every category (and especially every [[topos]]) has an \href{/nlab/show/type+theory#CategoricalSemantics}{internal language} and [[internal logic]]. Interpreting ``ordinary'' mathematical statements in the internal language of exotic categories can make it much easier to study those categories, while on the other hand it can provide new insight into otherwise mysterious logical notions.

In particular, the internal logic of a category (such as a topos) is, in general, [[constructive mathematics|constructive]], i.e. the principle of [[excluded middle]] (and also stronger statements, such as the [[axiom of choice]]) are generally false. Thus, in order for a theorem to be interpretable internally in such categories, its proof must be constructive. So while the original ``constructivists'' believed that classical mathematics was ``wrong,'' nowadays there are good reasons to care about constructive mathematics even if one believes that excluded middle and the axiom of choice are ``true,'' since regardless of their ``global'' truth they will \emph{not} be true in the internal logic of many interesting categories. Conversely, category-theoretic models have provided new insight into the independence of various axioms in constructive mathematics, such as differing forms of the axiom of choice.

As another example, the [[identity types]] in Martin-L\"o{}f's original constructive dependent [[type theory]] construct, from any [[type]] $A$ and terms $a, b \in A$, a new type $Id_A(a, b)$. According to the [[propositions as types]] interpretation, the elements of $Id_A(a,b)$ are proofs that $a$ and $b$ are propositionally equal; thus $Id_A(a,b)$ is a replacement for the [[truth value]] of the [[proposition]] $(a=b)$. There are type-theoretic [[functions]] $1 \to Id(a, a)$, $Id(b, c) \times Id(a, b) \to Id(a, c)$ and $Id(a, b) \to Id(b, a)$ expressing the [[equivalence relation|reflexivity, transitivity and symmetry]] of this propositional [[equality]], but in general an identity type (even the ``reflexive'' identity type $Id(a,a)$) can have many distinct elements. This has long been a source of discomfort to type theorists. However, from a higher-categorical point of view, it is natural to view the terms of identity types as \emph{isomorphisms} in a [[groupoid]]---or, more precisely, an [[∞-groupoid]], since identity types have their own identity types, and all the laws of associativity, exchange, etc. only hold up to terms of these higher identity types. This suggests that the nonuniqueness of identity proofs should be embraced rather than denigrated, producing a theory at least related to the ``internal logic'' of [[(∞,1)-category]] theory and [[homotopy theory]]; see [[identity type]] for more details.

This is now known as \emph{[[homotopy type theory]]}, see there for more.

\hypertarget{Physics}{}\subsubsection*{{In physics}}\label{Physics}

See also \emph{[[higher category theory and physics]]}.

\hypertarget{ClassMech}{}\paragraph*{{Classical mechanics and its geometric quantization}}\label{ClassMech}

By the end of the 19th century a fairly complete, powerful and elegant mathematical formulation of [[classical mechanics]]: in terms of [[symplectic geometry]]. By the middle of the 20th century, the passage to the corresponding quantum theory was pretty well modeled by the [[geometric quantization]] of symplectic geometries.

But there were some lose ends. Notably the fully general theory involved [[Poisson manifold]]s, not just symplectic manifolds. And the mechanics of relativistic [[classical field theory]] was realized to be more naturally described by [[multisymplectic geometry]].

Both these generalizations have a natural common higher categorical formulation: that of [[Lie ∞-algebroid]]s: a Poisson geometry is naturally encoded in its corresponding [[Poisson Lie algebroid]]. Its higher categorical versions -- the [[n-symplectic manifold]]s -- encode the corresponding multisymplectic geometry.

Moreover, the quantization step of geometric quantization was understood to be effectively the [[Lie integration]] of these [[Lie ∞-algebroid]]s to the corresponding [[Lie ∞-groupoid]]s (currently this is well understood for low $n$).

\hypertarget{QuantumMechanics}{}\paragraph*{{Quantum mechanics and quantum information}}\label{QuantumMechanics}

The basic structure of [[quantum mechanics]] and [[quantum information theory]] is encoded in the theory of [[dagger-compact categories]].

\ldots{}

\hypertarget{GaugeTheory}{}\paragraph*{{Gauge theory}}\label{GaugeTheory}

Maxwell realized that the [[electromagnetic field]] is controlled by a degree 2-cocycle in [[de Rham cohomology]]: the electromagnetic [[field strength]]. Later Dirac noticed that this is one part of a degree 2-cocycle in [[differential cohomology]] that characterize a [[connection on a bundle|connection]] on a [[line bundle]].

Later the [[Yang-Mills field]] was understood to similarly be a [[connection on a bundle]], this time on a $G$-[[principal bundle]] for $G$ some possibly nonabelian [[group]].

While thinking about the mathematical structures possibly underlying [[standard model of particle physics]] and [[gravity]], theoretical physicists considered more general hypothetical [[gauge field]]s, such as the [[Kalb-Ramond field]], the [[RR-field]] or the [[supergravity C-field]]. Today all these gauge fields are understood to be modeled, mathematically, by generalized [[differential cohomology]].

\hypertarget{Supergravity}{}\paragraph*{{Supergravity}}\label{Supergravity}

Theories of [[supergravity]] have been known to require higher [[gauge field]]s in the above sense -- hence the term [[supergravity C-field]]. A powerful formalism for handling these theories is the [[D'Auria-Fre formulation of supergravity]]. As described there, this is secretly (but evidently) nothing but a description of supergravity as a theory of connections on nonabelian $G$-[[principal ∞-bundle]]s for $G$ some super [[Lie ∞-groupoid|Lie ∞-group]]. For instance Cremmer-Scherk 11-dimensional supergravity theory is governed by the super Lie 3-group $G$ whose [[L-∞-algebra]] is the [[supergravity Lie 3-algebra]].

\hypertarget{BVBRST}{}\paragraph*{{BV-BRST formalism}}\label{BVBRST}

The [[BV-BRST formalism]] is secretly a way to talk about the fact that configuraton spaces of [[gauge theory|gauge theories]] are not naive spaces such as [[manifold]]s, but are general [[space]]s in the sense of [[higher geometry]]:

the configuration space is really an object $Conf \in Sh_{(\infty,1)}((dgAlg^-)^{op})$ in the [[∞-stack]] [[(∞,1)-topos]] on the [[(∞,1)-site]] $(dgAlg^-)^{op}$ of certain [[algebra in an (∞,1)-category|∞-algebras]] modeled as [[dg-algebra]]s. The BV-BRST-complex of a physical system is the global [[derived geometry|derived]] function algebra

\begin{displaymath}
\mathcal{O}(Conf) \in dgAlg
  \,.
\end{displaymath}
\begin{quote}%
(many more aspects go here, eventually)\ldots{}


\end{quote}
\hypertarget{QFT}{}\paragraph*{{Quantum field theory}}\label{QFT}

There are essentially two axiomatizations of what [[quantum field theory]] is, both of which are inherently $\infty$-categorical:

\begin{itemize}%
\item in the [[FQFT]] picture -- the \emph{Schr\"o{}dinger picture} -- a quantum field theory is described as an [[(∞,n)-functor]] on an [[(∞,n)-category of cobordisms]]. The [[cobordism hypothesis]] -- now a theorem that characterizes central properties of these [[(∞,n)-categories]], has been a major driving force in the development of [[higher category theory]].


\item in the [[AQFT]]/[[factorization algebra]] picture -- the \emph{Heisenberg picture} -- a quantum field theory is described as an $\infty$-copresheaf of observables on its parameter space.



\end{itemize}
\hypertarget{holography}{}\paragraph*{{Holography}}\label{holography}

\begin{itemize}%
\item [[holographic principle of higher category theory]]

\end{itemize}
\hypertarget{3dTFT2dCFT}{}\paragraph*{{3d TFT and 2d CFT}}\label{3dTFT2dCFT}

3-dimensional [[TFT]] such as [[Chern-Simons theory]] and [[Dijkgraaf-Witten theory]] and the global aspects of 2-dimensional [[conformal field theory]] are inherently governed by the theory of [[modular tensor categories]].

The local aspects of 2-dimensional conformal field theory are governed by [[vertex operator algebra]]s. A [[vertex operator algebra]] is really the [[algebra over an operad]], for the operad of holomorphic pointed spheres (as described there).

\ldots{}

\hypertarget{in_your_favorite_topic_here}{}\subsubsection*{{In your favorite topic here}}\label{in_your_favorite_topic_here}

\ldots{}

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[geometry of physics]]


\item [[fiber bundles in physics]]


\item [[higher category theory and physics]]


\item [[string theory FAQ]]


\item [[twisted smooth cohomology in string theory]]


\item [[motives in physics]]


\item [[Hilbert's sixth problem]]


\item [[model theory and physics]]


\item [[L-infinity algebras in physics]]


\item [[motivation for sheaves, cohomology and higher stacks]]


\item [[applications of (higher) category theory]]


\item [[motivation for higher differential geometry]]


\item [[motivation for cohesion]]



\end{itemize}


\end{document}