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\section*{geometry of physics -- categories and toposes}

\begin{quote}%
This entry is one chapter of \emph{[[geometry of physics]]}.

next chapters: \emph{[[geometry of physics -- smooth sets|smooth sets]]}, \emph{[[geometry of physics -- supergeometry|supergeometry]]}


\end{quote}
$\,$


\vspace{.5em} \hrule \vspace{.5em}
$\,$

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

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

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

[[category theory|Category theory]] and [[topos theory]] concern the general abstract structure underlying [[algebra]], [[geometry]] and [[logic]]. They are ubiquituous in and indispensible for organizing conceptual mathematical frameworks.

We give here an introduction to the basic concepts and results, aimed at providing background for the [[synthetic differential geometry|synthetic]] [[higher differential geometry|higher]] [[geometry of physics -- supergeometry|supergeometry]] of relevance in formulations of fundamental [[physics]], such as used in the chapters \emph{[[geometry of physics -- perturbative quantum field theory|on perturbative quantum field theory]]} and \emph{[[geometry of physics -- fundamental super p-branes|on fundamental super p-branes]]}. For quick informal survey see \emph{[[schreiber:Introduction to Higher Supergeometry]]}.

This makes use of the following curious dictionary between [[category theory]]/[[topos theory]] and the [[geometry]] of [[generalized spaces]], which we will explain in detail (following \href{functorial+geometry#Grothendieck65}{Grothendieck 65}, \href{space+and+quantity#Lawvere86}{Lawvere 86, p. 17}, [[Some Thoughts on the Future of Category Theory|Lawvere 91]]):

$\,$

\begin{tabular}{l|l|l}
$\phantom{A}$[[category theory]]&Rmk. \ref{PresaheavesAsGeneralizedSpaces}&$\phantom{A}$[[geometry]] of [[generalized spaces]]\\
\hline 
$\phantom{A}$[[presheaf]]&Expl. \ref{CategoryOfPresheaves}&$\phantom{A}$[[generalized space]]\\
$\phantom{A}$[[representable presheaf]]$\phantom{A}$&Expl. \ref{RepresentablePresheaves}$\phantom{A}$&$\phantom{A}$model [[space]]  $\phantom{A}$regarded as [[generalized space]]\\
$\phantom{A}$[[Yoneda lemma]]&Prop. \ref{YonedaLemma}$\phantom{A}$&$\phantom{A}$sets of probes of [[generalized spaces]]  $\phantom{A}$are indeed  $\phantom{A}$sets of maps from model [[spaces]] $\phantom{A}$\\
$\phantom{A}$[[Yoneda embedding]] $\phantom{A}$&Prop. \ref{YonedaEmbedding}$\phantom{A}$&$\phantom{A}$nature of model [[spaces]] is preserved when  $\phantom{A}$regarding them as [[generalized spaces]] $\phantom{A}$\\
$\phantom{A}$[[Yoneda embedding]] is$\phantom{A}$  $\phantom{A}$[[free co-completion]]$\phantom{A}$&Prop. \ref{FreeCocompletion}&$\phantom{A}$[[generalized spaces]] really are$\phantom{A}$  $\phantom{A}$glued from ordinary [[spaces]]$\phantom{A}$\\
&&\\
$\phantom{A}$\textbf{[[topos theory]]}&\textbf{Rmk. \ref{SheafConditionAsLocality}}&$\phantom{A}$\textbf{[[local-global principle]] for [[generalized spaces]]}$\phantom{A}$\\
$\phantom{A}$[[coverage]]&Defn. \ref{Coverage}&$\phantom{A}$notion of locality\\
$\phantom{A}$[[sheaf&sheaf condition]]&Defn. \ref{Sheaf}$\phantom{A}$  Prop. \ref{CechGroupoidCoRepresents}\\
$\phantom{A}$[[comparison lemma]]&Prop. \ref{ComparisonLemma}&$\phantom{A}$notion of [[generalized spaces]]  $\phantom{A}$independent under change of model [[space]]\\
$\phantom{A}$\textbf{[[gros topos}&gros topos theory]]**&\textbf{Rmk. \ref{CohesiveGeneralizedSpacesAsFoundations}}\\
$\phantom{A}$[[cohesion]]&Defn. \ref{CohesiveTopos}&$\phantom{A}$[[generalized spaces]] obey  $\phantom{A}$principles of [[differential topology]]\\
$\phantom{A}$[[differential cohesion]]&Defn. \ref{DifferentialCohesion}&$\phantom{A}$[[generalized spaces]] obey  $\phantom{A}$principles of [[differential geometry]]\\
$\phantom{A}$super cohesion$\phantom{A}$&Defn. \ref{SuperDifferentialCohesion}&$\phantom{A}$[[generalized spaces]] obey  $\phantom{A}$principles of [[supergeometry]]\\
\end{tabular}

The perspective is that of \emph{[[functorial geometry]]} (\hyperlink{functorial+geometry#Grothendieck65}{Grothendieck 65}). (For more exposition of this point see also at \emph{[[motivation for sheaves, cohomology and higher stacks]]}.) This dictionary implies a wealth of useful tools for handling and reasoning about [[geometry]]:

We discuss \hyperlink{BasicNotionsOfToposTheory}{below} that [[sheaf toposes]], regarded as [[categories]] of [[generalized spaces]] via the above disctionary, are ``convenient contexts'' for geometry (Prop. \ref{PropertiesOfSheafToposes} below), in the technical sense that they provide just the right kind of generalization that makes all desireable constructions on spaces actually exist:

\begin{tabular}{l|l}
$\phantom{A}$[[sheaf topos]]$\phantom{A}$&$\phantom{A}$as [[category]] of [[generalized spaces]] $\phantom{A}$\\
\hline 
$\phantom{A}$[[Yoneda embedding]]: $\phantom{A}$&$\phantom{A}$contains and generalizes ordinary [[spaces]] $\phantom{A}$\\
$\phantom{A}$has all [[limits]]: $\phantom{A}$&$\phantom{A}$contains all [[Cartesian products]] and [[intersections]] $\phantom{A}$\\
$\phantom{A}$has all [[colimits]]: $\phantom{A}$&$\phantom{A}$contains all [[disjoint unions]] and [[quotients]]\\
$\phantom{A}$[[cartesian closed category&cartesian closure]]: $\phantom{A}$\\
$\phantom{A}$[[locally cartesian closed category&local cartesian closure]]: $\phantom{A}$\\
\end{tabular}

Notably [[mapping spaces]] play a pivotal role in [[physics]], in the guise of [[spaces of field histories]], but fall outside the applicability of traditional formulations of [[geometry]] based on just [[manifolds]]. [[topos theory|Topos theory]] provides their existence (Prop. \ref{PropertiesOfSheafToposes} below) and the relevant infrastructure, for example for the construction of [[transgression of differential forms]] to mapping spaces of [[smooth sets]], that is the basis for [[sigma-model]]-[[field theories]]. This is discussed in the following chapters \emph{[[geometry of physics -- smooth sets|on smooth sets]]} and \emph{[[geometry of physics -- supergeometry|on supergeometry]]}.

In conclusion, one motivation for [[category theory]] and [[topos theory]] is \emph{a posteriori}: As a matter of experience, there is just no other toolbox that allows to deeply understand and handle the [[geometry of physics]]. Similar comments apply to a wealth of other topics of mathematics.

We offer also an \emph{a priori} motivation:

 \emph{Category theory is the theory of duality.}

\emph{[[duality|Duality]]} is of course an ancient notion in [[philosophy]]. At least as a term, it makes a curious re-appearance in the conjectural [[theory (physics)|theory]] of fundamental [[physics]] formerly known as \emph{[[string theory]]}, as \emph{[[duality in string theory]]}. In both cases, the literature left some room in delineating what precisely is meant. But the philosophically inclined mathematician could notice (see \href{adjoint+functor#Lambek82}{Lambek 82}) that an excellent candidate to make precise the idea of \emph{[[duality]]} is the mathematical concept of \emph{[[adjoint functor|adjunction]]}, from [[category theory]]. This is particularly pronounced for \emph{[[adjoint triples]]} (Remark \ref{AdjointTriples} below) and their induced [[adjoint modalities]] ([[Some Thoughts on the Future of Category Theory|Lawvere 91]], see Def. \ref{AdjointModality} below), which exhibit a given ``[[modality|mode of being]]'' of any object $X$ as intermediate between two dual opposite extremes (Prop. \ref{ComparisonMorphismBetweenOppositeExtremes} below):

\begin{displaymath}
\Box X 
    \overset{\phantom{AAAA}}{\longrightarrow}
  X 
    \overset{\phantom{AAAA}}{\longrightarrow}
  \bigcirc X
\end{displaymath}
For example, \emph{[[cohesion|cohesive]]} [[geometry|geometric]] [[structure]] on [[generalized spaces]] is captured, this way, as [[modality]] in between the [[discrete object|discrete]] and the [[codiscrete object|codiscrete]] (Example \ref{DiscreteTopologicalSpaces}, and Def. \ref{CohesiveTopos} below).

Historically, [[category theory]] was introduced in order to make precise the concept of \emph{[[natural transformation]]}: The concept of \emph{[[functors]]} was introduced just so as to support that of natural transformations, and the concept of \emph{[[categories]]} only served that of functors (see e.g. \href{category+theory#Freyd65}{Freyd 65, Part II}).

But natural transformations are, in turn, exactly the basis for the concept of \emph{[[adjoint functors]]} (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} below), equivalently \emph{[[adjunctions]] between categories} (Prop. \ref{AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat} below), shown on the left. All \emph{[[universal constructions]]}, the heart of category theory, are special cases of [[adjoint functors]] -- hence of dualities, if we follow \href{adjoint+functor#Lambek82}{Lambek 82}: This includes the concepts of \emph{[[limits]]} and \emph{[[colimits]]} (Def. \ref{Limits} below), [[ends]] and [[coends]] (Def. \ref{EndAndCoendInTopcgSmash} below) [[Kan extensions]] (Prop. \ref{TopologicalLeftKanExtensionBCoend} below), and the behaviour of these constructions, such as for instance the [[free co-completion]] nature of the [[Yoneda embedding]] (Prop. \ref{FreeCocompletion} below).

$\,$

$\,$

$\,$

Therefore it makes sense to regard category theory as the \textbf{theory of adjunctions},  hence the \textbf{theory of duality}:



 $\,$

\newline | $\phantom{A}$[[adjoint triple|adjunction of adjunctions]]$\phantom{A}$  $\phantom{AA}$[[adjoint triple|duality of dualities]]$\phantom{A}$ | $\phantom{A}$Def. \ref{AdjunctionofAdjunction}$\phantom{A}$ | | $\phantom{A}$Def. \ref{QuillenAdjointTriple}$\phantom{A}$ |\newline | $\phantom{A}$ [[adjoint equivalence]]$\phantom{A}$  $\phantom{AA}$[[adjoint equivalence|dual equivalence]] $\phantom{AA}$ | $\phantom{A}$ Def. \ref{AdjointEquivalenceOfCategories} $\phantom{A}$ | $\phantom{A}$ Def. \ref{EnrichedEquivalenceOfCategories} $\phantom{A}$ | $\phantom{A}$Def. \ref{QuillenEquivalence} | | $\phantom{A}$ [[adjoint functor|adjunction]]$\phantom{A}$  $\phantom{AA}$[[duality]]$\phantom{A}$ | $\phantom{A}$ Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} $\phantom{A}$ | $\phantom{A}$ Def. \ref{EnrichedAdjunction} $\phantom{A}$ | $\phantom{A}$Def. \ref{QuillenAdjunction} | | $\phantom{A}$ [[natural transformation]] $\phantom{A}$ | $\phantom{A}$ Def. \ref{NaturalTransformations} $\phantom{A}$ | $\phantom{A}$ Def. \ref{EnrichedNaturalTransformation} $\phantom{A}$ | | | $\phantom{A}$ [[functor]] $\phantom{A}$ | $\phantom{A}$ Def. \ref{Functors} $\phantom{A}$ | $\phantom{A}$ Def. \ref{TopologicallyEnrichedFunctor} $\phantom{A}$ | | | $\phantom{A}$ [[category]] $\phantom{A}$ | $\phantom{A}$ Def. \ref{Categories} $\phantom{A}$ | $\phantom{A}$ Def. \ref{TopEnrichedCategory} $\phantom{A}$ | $\phantom{A}$ Def. \ref{ModelCategory} |

The pivotal role of [[adjunctions]] in [[category theory]] (\href{adjoint+functor#fn:1}{Lawvere 08}) and in the [[foundations of mathematics]] ([[Adjointness in Foundations|Lawvere 69]], [[Cohesive Toposes and Cantor's lauter Einsen|Lawvere 94]] ) was particularly amplified by [[F. W. Lawvere]]\footnote{``the universality of the concept of adjointness, which was first isolated and named in the conceptual sphere of category theory'' (\href{adjoint+functor#Lawvere69#Lawvere69}{Lawvere 69}) ``In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role.'' (first line from \emph{\href{https://ncatlab.org/nlab/show/William+Lawvere#Interview07}{An interview with William Lawvere}}, paraphrasing the first paragraph of \emph{\href{William+Lawvere#TakingCategoriesSeriously}{Taking categories seriously}})} . Moreover, [[Lawvere]] saw the future of category theory ([[Some Thoughts on the Future of Category Theory|Lawvere 91]]) as concerned with [[adjunctions]] expressing systems of archetypical dualities that reveal foundations for [[geometry]] (\href{cohesive+topos#LawvereAxiomatic}{Lawvere 07}) and [[physics]] ([[Toposes of laws of motion|Lawvere 97]], see Def. \ref{CohesiveTopos} and Def. \ref{DifferentialCohesion} below). He suggested (\href{objective+and+subjective+logic#Lawvere94}{Lawvere 94}) this as a precise formulation of core aspects of the \emph{theory of everything} of early 19th century [[philosophy]]: [[Hegel]]`s \emph{[[Science of Logic]]}.

These days, of course, \emph{[[theories of everything]]}, such as [[string theory]], are understood less ambitiously than Hegel's ontological process, as mathematical formulations of fundamental theories of physics, that could conceptually unify the hodge-podge of currently available ``standard models'' [[standard model of particle physics|of particle physics]] and [[standard model of cosmology|of cosmology]] to a more coherent whole.

The idea of \emph{[[duality in string theory]]} refers to different perspectives on physics that appear dual to each other while being \emph{equivalent}. But one of the basic results of category theory (Prop. \ref{EveryEquivalenceOfCategoriesComesFromAnAdjointEquivalence}, below) is that equivalence is indeed a special case of adjunction. This allows to explore the possibility that there is more than a coincidence of terms.

Of course the usage of the term \emph{[[duality in string theory]]} is too loose for one to expect to be able to refine each occurrence of the term in the literature to a mathematical adjunction. However, we will see mathematical formalizations of core aspects of key string-theoretic dualities, such as \emph{[[topological T-duality]]} and the \emph{[[duality between M-theory and type IIA string theory]]}, in terms of [[adjunctions]]. Indeed, at the heart of these \emph{[[dualities in string theory]]} is the phenomenon of \emph{[[double dimensional reduction]]}, which turns out to be formalized by one of the most fundamental adjunctions in ([[higher category theory|higher]]) [[category theory]]: \emph{[[base change]]} along the point inclusion into a [[classifying space]]. All this is discussed in the chapter on \emph{[[geometry of physics -- fundamental super p-branes|fundamental super p-branes]]}.

This suggests that there may be a deeper relation here between the superficially alien uses of the word ``duality'', that is worth exploring.

In this respect it is worth noticing that core structure of string/M-theory arises via [[universal constructions]] from the [[superpoint]] (as explained in the chapter \emph{[[geometry of physics -- fundamental super p-branes|on fundamental super p-branes]]}), while the superpoint itself arises, in a sense made precise by [[category theory]], ``from nothing'', by a system of twelve [[adjunctions]] (explained in the chapter [[geometry of physics -- supergeometry|on supergeometry]]).

$\,$

Here we introduce the requisites for understanding these statements.

$\,$

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



[[!include geometry of physics – basic notions of category theory]]

$\,$

[[!include geometry of physics - basic notions of categorical algebra]]

$\,$

[[!include geometry of physics - universal constructions]]

$\,$

[[!include geometry of physics - basic notions of topos theory]]

$\,$

[[!include geometry of physics - cohesive toposes]]

$\,$

[[!include geometry of physics -- homotopy types]]

$\,$

[[!include geometry of physics -- basic notions of higher topos theory]]

[[!redirects geometry of physics -- categories and toposes]] [[!redirects geometry of physics -- Categories and Toposes]]



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