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\begin{document}

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\section*{geometry of physics -- coordinate systems}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

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

previous chapter: \emph{[[geometry of physics -- categories and toposes|categories and toposes]]},

next chapter: \emph{[[geometry of physics -- smooth sets|smooth sets]]}


\end{quote}
As discussed in the chapter \emph{[[geometry of physics -- categories and toposes|categories and toposes]]}, every kind of \emph{[[geometry]]} is modeled on a collection of [[generator|archetypical]] basic [[spaces]] and geometric [[homomorphisms]] between them. In [[differential geometry]] the archetypical spaces are the abstract standard [[Cartesian space|Cartesian coordinate systems]], denoted $\mathbb{R}^n$ (in every [[dimension]] $n \in \mathbb{N}$). The geometric homomorphism between them are [[smooth functions]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$, hence smooth (and possibly degenerate) [[coordinate transformations]].

Here we introduce the basic concept, organizing them in the [[category]] [[CartSp]] of [[Cartesian spaces]] (Prop. \ref{CartSpCategory} below.) We highlight three classical theorems about [[smooth functions]] in Prop. \ref{AlgebraicFactsOfDifferentialGeometry} below, which look innocent but play a decisive role in setting up [[synthetic differential geometry|synthetic differential]] [[supergeometry]] based on the concept of abstract smooth coordinate systems.

At this point these are not yet coordinate systems \emph{on} some other space. But by applying the general machine of [[geometry of physics -- categories and toposes|categories and toposes]] to these, a concept of [[generalized spaces]] modeled on these abstract coordinate systems is induced. These are the \emph{[[smooth sets]]} discussed in the next chapter \emph{[[geometry of physics -- smooth sets]]}.

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

\noindent\hyperlink{CoordinateSystemsLayerMod}{Abstract coordinate systems}\dotfill \pageref*{CoordinateSystemsLayerMod} \linebreak
\noindent\hyperlink{TheContinuumRealWorldLine}{The continuum real line}\dotfill \pageref*{TheContinuumRealWorldLine} \linebreak
\noindent\hyperlink{CartesianSpaces}{Cartesian spaces and smooth functions}\dotfill \pageref*{CartesianSpaces} \linebreak
\noindent\hyperlink{PropertiesOfSmoothFunctions}{The magic properties of smooth functions}\dotfill \pageref*{PropertiesOfSmoothFunctions} \linebreak
\noindent\hyperlink{CoordinateSystemsLayerSem}{The category of abstract coordinate systems}\dotfill \pageref*{CoordinateSystemsLayerSem} \linebreak
\noindent\hyperlink{TheAlgebraicTheoryOfSmoothAlgebras}{The algebraic theory of smooth algebras}\dotfill \pageref*{TheAlgebraicTheoryOfSmoothAlgebras} \linebreak
\noindent\hyperlink{CoverageOfDifferentiallyGoodOpenCovers}{The coverage of differentially good open covers}\dotfill \pageref*{CoverageOfDifferentiallyGoodOpenCovers} \linebreak
\noindent\hyperlink{SliceCategories}{The slice category}\dotfill \pageref*{SliceCategories} \linebreak
\noindent\hyperlink{CoordinateSystemsLayerSyn}{Syntactic Layer}\dotfill \pageref*{CoordinateSystemsLayerSyn} \linebreak
\noindent\hyperlink{Judgments}{Judgments about types and terms}\dotfill \pageref*{Judgments} \linebreak
\noindent\hyperlink{NaturalDeductionRulesForProductTypes}{Natural deduction rules for product types}\dotfill \pageref*{NaturalDeductionRulesForProductTypes} \linebreak
\noindent\hyperlink{DependentSumTypes}{Natural deduction rules for dependent sum types}\dotfill \pageref*{DependentSumTypes} \linebreak
\noindent\hyperlink{DictionaryTypeTheoryCategoryTheory}{Dictionary: type theory / category theory}\dotfill \pageref*{DictionaryTypeTheoryCategoryTheory} \linebreak


\hypertarget{CoordinateSystemsLayerMod}{}\subsection*{{Abstract coordinate systems}}\label{CoordinateSystemsLayerMod}

In this \hyperlink{LayerMod}{Mod Layer} we discuss the [[concrete particulars]] of \emph{[[coordinate systems]]}: the [[continuum]] [[real line]] $\mathbb{R}$, the [[Cartesian spaces]] $\mathbb{R}^n$ formed from it and the [[smooth functions]] between these.

\hypertarget{TheContinuumRealWorldLine}{}\subsubsection*{{The continuum real line}}\label{TheContinuumRealWorldLine}

The fundamental premise of [[differential geometry]] as a model of [[geometry]] in [[physics]] is the following.

\textbf{Premise.} \emph{The abstract [[worldline]] of any [[particle]] is modeled by the [[continuum]] [[real line]] $\mathbb{R}$.}

This comes down to the following sequence of premises.

\begin{enumerate}%
\item There is a [[linear ordering]] of the [[points]] on a [[worldline]]: in particular if we pick points at some intervals on the worldline we may label these in an order-preserving way by [[integers]]

$\mathbb{Z}$.


\item These intervals may each be subdivided into $n$ smaller intervals, for each natural number $n$. Hence we may label points on the [[worldline]] in an order-preserving way by the [[rational numbers]]

$\mathbb{Q}$.


\item This labeling is dense: every point on the worldline is the [[supremum]] of an [[inhabited set|inhabited]] [[bounded subset]] of such labels. This means that a [[worldline]] [[equivalence|is]] the \emph{[[real line]]}, the [[continuum]] of [[real numbers]]

$\mathbb{R}$.



\end{enumerate}
The adjective ``real'' in ``[[real number]]'' is a historical shadow of the old idea that real numbers are related to \emph{observed reality}, hence to [[physics]] in this way. The experimental success of this assumption shows that it is valid at least to very good approximation.

Speculations are common that in a fully exact theory of [[quantum gravity]], currently unavailable, this assumption needs to be refined. For instance in \emph{[[p-adic physics]]} one explores the hypothesis that the relevant [[complete field|completion]] of the rational numbers as above is not the reals, but [[p-adic numbers]] $\mathbb{Q}_p$ for some [[prime number]] $p \in \mathbb{N}$. Or for example in the study of [[QFT on non-commutative spacetime]] one explore the idea that at small scales the smooth [[continuum]] is to be replaced by an object in [[noncommutative geometry]]. Combining these two ideas leads to the notion of [[non-commutative analytic space]] as a potential model for \emph{[[space]]} in [[physics]]. And so forth.

For the time being all this remains speculation and differential geometry based on the [[continuum]] [[real line]] remains the context of all fundamental [[model (in theoretical physics)|model building]] in physics related to observed [[phenomenology]]. Often it is argued that these speculations are necessitated by the very nature of [[quantum theory]] applied to [[gravity]]. But, at least so far, such statements are not actually supported by the standard theory of [[quantization]]: we discuss below in \emph{\href{GeometricQuantization}{Geometric quantization}} how not just [[classical physics]] but also [[quantum theory]], in the best modern version available, is entirely rooted in differential geometry based on the [[continuum]] [[real line]].

This is the motivation for studying models of physics in geometry modeled on the [[continuum]] [[real line]]. On the other hand, in all of what follows our discussion is set up such as to be \emph{maximally independent} of this specific choice (this is what \emph{[[topos theory]]} accomplishes for us, discussed below \emph{\hyperlink{SmoothSpacesLayerSem}{Smooth spaces -- Semantic Layer}}). If we do desire to consider another choice of archetypical spaces for the geometry of physics we can simply ``change the [[site]]'', as discussed \hyperlink{SmoothSpacesLayerSem}{below} and many of the constructions, propositions and theorems in the following will continue to hold. This is notably what we do below in \emph{\hyperlink{SupergeometricCoordinateSystems}{Supergeometric coordinate systems}} when we generalize the present discussion to a flavor of differential geometry that also formalizes the notion of [[fermion]] [[particles]]: ``differential [[supergeometry]]''.

\hypertarget{CartesianSpaces}{}\subsubsection*{{Cartesian spaces and smooth functions}}\label{CartesianSpaces}

\begin{defn}
\label{SmoothFunctions}\hypertarget{SmoothFunctions}{}
A [[function]] of [[sets]] $f : \mathbb{R} \to \mathbb{R}$ is called a \textbf{[[smooth function]]} if, [[coinduction|coinductively]]:

\begin{enumerate}%
\item the [[derivative]] $\frac{d f}{d x} : \mathbb{R} \to \mathbb{R}$ exists;


\item and is itself a smooth function.



\end{enumerate}
We write $C^\infty(\mathbb{R}) \in Set$ for the [[set]] of all smooth functions on $\mathbb{R}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The superscript ``${}^\infty$'' in ``$C^\infty(\mathbb{R})$'' refers to the order of the [[derivatives]] that exist for smooth functions. More generally for $k \in \mathbb{N}$ one writes $C^k(\mathbb{R})$ for the set of $k$-fold [[differentiable functions]] on $\mathbb{R}$. These will however not play much of a role for our discussion here.

\end{remark}
\begin{defn}
\label{CartesianSpaceAndHomomorphism}\hypertarget{CartesianSpaceAndHomomorphism}{}
For $n \in \mathbb{N}$, the \textbf{[[Cartesian space]]} $\mathbb{R}^n$ is the set

\begin{displaymath}
\mathbb{R}^n = \{ (x^1 , \cdots, x^{n}) | x^i \in \mathbb{R} \}
\end{displaymath}
of $n$-[[tuples]] of [[real numbers]]. For $1 \leq k \leq n$ write

\begin{displaymath}
i^k : \mathbb{R} \to \mathbb{R}^n
\end{displaymath}
for the [[function]] such that $i^k(x) = (0, \cdots, 0,x,0,\cdots,0)$ is the [[tuple]] whose $k$th entry is $x$ and all whose other entries are $0 \in \mathbb{R}$; and write

\begin{displaymath}
\mathbb{p}^k : \mathbb{R}^n \to \mathbb{R}
\end{displaymath}
for the function such that $p^k(x^1, \cdots, x^n) = x^k$.

A \textbf{[[homomorphism]]} of Cartesian spaces is a [[smooth function]]

\begin{displaymath}
f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}
  \,,
\end{displaymath}
hence a [[function]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ such that all [[partial derivatives]] exist and are [[continuous map|continuous]] (\ldots{}).

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Regarding $\mathbb{R}^n$ as an $\mathbb{R}$-[[vector space]], every [[linear function]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is in particular a [[smooth function]].

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
But a homomorphism of Cartesian spaces in def. \ref{CartesianSpaceAndHomomorphism} is \emph{not} required to be a [[linear map]]. We do \emph{not} regard the Cartesian spaces here as [[vector spaces]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A [[smooth function]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is called a \textbf{[[diffeomorphism]]} if there exists another smooth function $\mathbb{R}^{n_2} \to \mathbb{R}^{n_1}$ such that the underlying functions of sets are [[inverse]] to each other

\begin{displaymath}
f \circ g = id
\end{displaymath}
and

\begin{displaymath}
g \circ f = id
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
There exists a [[diffeomorphism]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ precisely if $n_1 = n_2$.

\end{prop}
\begin{defn}
\label{CartesianSpacesAndSmoothFunctions}\hypertarget{CartesianSpacesAndSmoothFunctions}{}
We will also say equivalently that

\begin{enumerate}%
\item a [[Cartesian space]] $\mathbb{R}^n$ is an \textbf{[[coordinate system|abstract coordinate system]]};


\item a [[smooth function]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is an \textbf{[[coordinate transformation|abstract coordinate transformation]]};


\item the [[function]] $p^k : \mathbb{R}^{n} \to \mathbb{R}$ is the $k$th \textbf{[[coordinate]]} of the coordinate system $\mathbb{R}^n$. We will also write this function as $x^k : \mathbb{R}^{n} \to \mathbb{R}$.


\item for $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[smooth function]], and $1 \leq k \leq n_2$ we write

\begin{enumerate}%
\item $f^k \coloneqq p^k\circ f$


\item $(f^1, \cdots, f^n) \coloneqq f$.



\end{enumerate}


\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
It follows with this notation that

\begin{displaymath}
id_{\mathbb{R}^n} = (x^1, \cdots, x^n) : \mathbb{R}^n \to \mathbb{R}^n
  \,.
\end{displaymath}
Hence an abstract coordinate transformation

\begin{displaymath}
f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}
\end{displaymath}
may equivalently be written as the tuple

\begin{displaymath}
\left(
    f^1 \left( x^1, \cdots, x^{n_1} \right)
    ,
    \cdots,
    f^{n_2}\left( x^1, \cdots, x^{n_1} \right)
  \right)
  \,.
\end{displaymath}
\end{remark}
\hypertarget{PropertiesOfSmoothFunctions}{}\subsubsection*{{The magic properties of smooth functions}}\label{PropertiesOfSmoothFunctions}

Below we encounter generalizations of ordinary [[differential geometry]] that include explicit ``[[infinitesimals]]'' in the guise of \emph{[[infinitesimally thickened points]]}, as well as ``super-graded infinitesimals'', in the guise of \emph{[[superpoints]]} (necessary for the description of [[fermion fields]] such as the [[Dirac field]]). As we discuss \hyperlink{FieldBundles}{below}, these structures are naturally incorporated into [[differential geometry]] in just the same way as [[Grothendieck]] introduced them into [[algebraic geometry]] (in the guise of ``[[formal schemes]]''), namely in terms of [[formal dual|formally dual]] [[rings of functions]] with [[nilpotent ideals]]. That this also works well for [[differential geometry]] rests on the following three basic but important properties, which say that [[smooth functions]] behave ``more algebraically'' than their definition might superficially suggest:

\begin{prop}
\label{AlgebraicFactsOfDifferentialGeometry}\hypertarget{AlgebraicFactsOfDifferentialGeometry}{}
\textbf{(the three magic algebraic properties of [[differential geometry]])}

\begin{enumerate}%
\item \textbf{[[embedding of smooth manifolds into formal duals of R-algebras|embedding of Cartesian spaces into formal duals of R-algebras]]}

For $X$ and $Y$ two [[Cartesian spaces]], the [[smooth functions]] $f \colon X \longrightarrow Y$ between them (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) are in [[natural bijection]] with their induced algebra [[homomorphisms]] $C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), so that one may equivalently handle [[Cartesian spaces]] entirely via their $\mathbb{R}$-algebras of smooth functions.

Stated more [[category theory|abstractly]], this means equivalently that the [[functor]] $C^\infty(-)$ that sends a [[smooth manifold]] $X$ to its $\mathbb{R}$-[[associative algebra|algebra]] $C^\infty(X)$ of [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) is a \emph{[[fully faithful functor]]}:

\begin{displaymath}
C^\infty(-)
  \;\colon\;
  SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op}
  \,.
\end{displaymath}
(\href{embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93}{Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10})


\item \textbf{[[smooth Serre-Swan theorem|embedding of smooth vector bundles into formal duals of R-algebra modules]]}

For $E_1 \overset{vb_1}{\to} X$ and $E_2 \overset{vb_2}{\to} X$ two [[vector bundle]] (def. \ref{VectorBundle}) there is then a [[natural bijection]] between vector bundle [[homomorphisms]] $f \colon E_1 \to E_2$ and the [[homomorphisms]] of [[modules]] $f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2)$ that these induces between the [[spaces of sections]] (example \ref{ModuleOfSectionsOfAVectorBundle}).

More [[category theory|abstractly]] this means that the [[functor]] $\Gamma_X(-)$ is a [[fully faithful functor]]

\begin{displaymath}
\Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod
\end{displaymath}
(Nestruev 03, theorem 11.29heorem\#Nestruev03))

Moreover, the [[modules]] over the $\mathbb{R}$-algebra $C^\infty(X)$ of [[smooth functions]] on $X$ which arise this way as [[sections]] of [[smooth vector bundles]] over a [[Cartesian space]] $X$ are precisely the [[finitely generated module|finitely generated]] [[free modules]] over $C^\infty(X)$.

(Nestruev 03, theorem 11.32heorem\#Nestruev03))


\item \textbf{[[derivations of smooth functions are vector fields|vector fields are derivations of smooth functions]]}.

For $X$ a [[Cartesian space]] (example \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), then any [[derivation]] $D \colon  C^\infty(X) \to C^\infty(X)$ on the $\mathbb{R}$-[[associative algebra|algebra]] $C^\infty(X)$ of [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) is given by [[differentiation]] with respect to a uniquely defined smooth [[tangent vector field]]: The function that regards [[tangent vector fields]] with [[derivations]] from example \ref{TangentVectorFields}

\begin{displaymath}
\itexarray{
    \Gamma_X(T X)
      &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}&
    Der(C^\infty(X))
    \\
    v &\mapsto& D_v
  }
\end{displaymath}
is in fact an [[isomorphism]].

(This follows directly from the \emph{[[Hadamard lemma]]}.)



\end{enumerate}
\end{prop}
Actually all three statements in prop. \ref{AlgebraicFactsOfDifferentialGeometry} hold not just for [[Cartesian spaces]], but generally for [[smooth manifolds]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below; if only we generalize in the second statement from [[free modules]] to [[projective modules]]. However for our development here it is useful to first focus on just [[Cartesian spaces]] and then bootstrap the theory of [[smooth manifolds]] and much more from that, which we do \hyperlink{FieldBundles}{below}.

\hypertarget{CoordinateSystemsLayerSem}{}\subsection*{{The category of abstract coordinate systems}}\label{CoordinateSystemsLayerSem}

Here we make explicit the \emph{[[category]]} formed by abstract coordinate systems (Prop. \ref{CartSpCategory} below) and mention some of its basic properties. This will serve the discussion of [[smooth sets]] as the \emph{[[sheaves]]} on the category of abstract coordinate systems, in the next chapter \emph{[[geometry of physics -- smooth sets]]}.

$\,$

\begin{prop}
\label{CartSpCategory}\hypertarget{CartSpCategory}{}
\textbf{(the [[category]] [[CartSp]] of abstract [[coordinate systems]]/[[Cartesian spaces]])}

Abstract coordinate systems according to prop. \ref{CartesianSpacesAndSmoothFunctions} form a \emph{[[category]]} (\href{geometry+of+physics+--+Categories+and+Toposes#Categories}{this def.}) -- to be denoted \emph{[[CartSp]]} -- whose

\begin{itemize}%
\item [[objects]] are the abstract coordinate systems $\mathbb{R}^{n}$ (the [[class]] of objects is the [[set]] $\mathbb{N}$ of [[natural numbers]] $n$);


\item [[morphisms]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ are the abstract [[coordinate transformations]] = [[smooth functions]].



\end{itemize}
Composition of morphisms is given by [[composition]] of [[functions]].

Under this identification

\begin{enumerate}%
\item The [[identity morphisms]] are precisely the [[identity functions]].


\item The [[isomorphisms]] are precisely the [[diffeomorphisms]].



\end{enumerate}
\end{prop}
\begin{defn}
\label{OppositeCategoryOfCartSp}\hypertarget{OppositeCategoryOfCartSp}{}
\textbf{([[opposite category]] of [[CartSp]])}

Write [[CartSp]]${}^{op}$ for the [[opposite category]] (\href{geometry+of+physics+--+Categories+and+Toposes#OppositeCategory}{this def.}) of [[CartSp]] (Prop. \ref{CartSpCategory}).

This is the category with the same objects as $CartSp$, but where a morphism $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ in $CartSp^{op}$ is given by a morphism $\mathbb{R}^{n_1} \leftarrow \mathbb{R}^{n_2}$ in $CartSp$.

\end{defn}
We will be discussing below the idea of exploring smooth spaces by laying out abstract coordinate systems in them in all possible ways. The reader should begin to think of the sets that appear in the following definition as the \emph{set of ways} of laying out a given abstract coordinate systems in a given space. This is discussed in more detail below in \emph{\hyperlink{SmoothSpaces}{Smooth spaces}}.

\begin{defn}
\label{}\hypertarget{}{}
A [[functor]] $X : CartSp^{op} \to Set$ (a ``[[presheaf]]'') is

\begin{enumerate}%
\item for each abstract coordinate system $U$ a [[set]] $X(U)$


\item for each [[coordinate transformation]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[function]] $X(f) : X(\mathbb{R}^{n_1}) \to X(\mathbb{R}^{n_2})$



\end{enumerate}
such that

\begin{enumerate}%
\item [[identity]] is respected $X(id_{\mathbb{R}^n}) = id_{X(\mathbb{R}^n)}$;


\item [[composition]] is respected $X(f_2)\circ X(f_1) = X(f_2 \circ f_1)$



\end{enumerate}
\end{defn}
\begin{example}
\label{SmoothSetOfDifferentialPForms}\hypertarget{SmoothSetOfDifferentialPForms}{}
Let $\mathcal{C}$ be a [[category]].

\begin{enumerate}%
\item The following are equivalent:

\begin{enumerate}%
\item $\mathcal{C}$ has a [[terminal object]];


\item the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[right adjoint]]

\begin{displaymath}
\ast 
 \underoverset
 {\underset{}{\longrightarrow}}
 {\overset{}{\longleftarrow}}
 {\bot}
  \mathcal{C}
\end{displaymath}


\end{enumerate}
Under this equivalence, the [[terminal object]] is identified with the image under the right adjoint of the unique object of the [[terminal category]].


\item Dually, the following are equivalent:

\begin{enumerate}%
\item $\mathcal{C}$ has an [[initial object]];


\item the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[left adjoint]]

\begin{displaymath}
\mathcal{C}
 \underoverset
 {\underset{}{\longrightarrow}}
 {\overset{}{\longleftarrow}}
 {\bot}
  \ast
\end{displaymath}


\end{enumerate}
Under this equivalence, the [[initial object]] is identified with the image under the left adjoint of the unique object of the [[terminal category]].



\end{enumerate}
\end{example}
\begin{proof}
Since the unique [[hom-set]] in the [[terminal category]] is [[generalized the|the]] [[singleton]], the hom-isomorphism characterizing the [[adjoint functors]] is directly the [[universal property]] of an [[initial object]] in $\mathcal{C}$

\begin{displaymath}
Hom_{\mathcal{C}}( L(\ast) , X )
  \;\simeq\;
  Hom_{\ast}( \ast, R(X) ) 
  =
  \ast
\end{displaymath}
or of a [[terminal object]]

\begin{displaymath}
Hom_{\mathcal{C}}( X , R(\ast) )
  \;\simeq\;
  Hom_{\ast}( L(X), \ast ) = \ast
  \,,
\end{displaymath}
respectively.

\end{proof}
\hypertarget{TheAlgebraicTheoryOfSmoothAlgebras}{}\subsubsection*{{The algebraic theory of smooth algebras}}\label{TheAlgebraicTheoryOfSmoothAlgebras}

\begin{prop}
\label{}\hypertarget{}{}
\begin{itemize}%
\item The [[category]] [[CartSp]] has all [[finite limit|finite]] [[products]].


\item Every object is a finite [[product]] of the object $\mathbb{R}$ (the [[real line]] itself).


\item The [[terminal object]] is $\mathbb{R}^0$, the [[point]].



\end{itemize}
\end{prop}
Hence [[CartSp]] is (the [[syntactic category]]) of an [[algebraic theory]] (a [[Lawvere theory]]).

This is called the [[theory]] of \emph{[[smooth algebras]]}.

\begin{defn}
\label{}\hypertarget{}{}
A [[product]]-preserving [[functor]]

\begin{displaymath}
A : CartSp \to Set
\end{displaymath}
is a \emph{[[smooth algebra]]}. A [[homomorphism]] of smooth algebras is a [[natural transformation]] between the corresponding functors.

\end{defn}
The basic example is:

\begin{example}
\label{TheSmoothgAlgebraOfFunctionsOnACartesianSpace}\hypertarget{TheSmoothgAlgebraOfFunctionsOnACartesianSpace}{}
For $n \in \mathbb{N}$, the [[smooth algebra]] $C^\infty(\mathbb{R}^n)$ is the functor $CartSp \to Set$ which is [[representable functor|functor corepresented]] by $\mathbb{R}^n \in$ [[CartSp]]. This means that to $\mathbb{R}^k \in CartSp$ it assigns the set

\begin{displaymath}
Hom_{CartSp}(\mathbb{R}^n , \mathbb{R}^k)
  = 
  C^\infty(\mathbb{R}^n, \mathbb{R}^k)
\end{displaymath}
of [[smooth functions]] from $\mathbb{R}^n$ to $\mathbb{R}^k$, and to a [[smooth function]] $f \colon \mathbb{R}^{k_1} \to \mathbb{R}^{k_2}$ it assigns the function

\begin{displaymath}
f\circ (-)
  \colon
  C^\infty(\mathbb{R}^n, \mathbb{R}^{k_1})
  \to
  C^\infty(\mathbb{R}^n, \mathbb{R}^{k_2})
\end{displaymath}
given by postcomposition with $f$.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Example \ref{TheSmoothgAlgebraOfFunctionsOnACartesianSpace} shows how we are to think of a functor $A \colon CartSp \to Set$ as encoding an algebra: such a functor assigns to $\mathbb{R}^n$ a set to be interpreted as a set of ``smooth functions on something with values in $\mathbb{R}^n$'', only that the ``something'' here is not pre-defined, but is instead indirectly characterized by this assignment.

Due to this we will often denote smooth algebras as ``$C^\infty(X)$'', even if ``$X$'' is not a pre-defined object, and write their value on $\mathbb{R}^n$ as $C^\infty(X,\mathbb{R}^n)$.

\end{remark}
This is illustrated by the next example.

\begin{example}
\label{DualNumbers}\hypertarget{DualNumbers}{}
The \textbf{[[ring of dual numbers|smooth algebra of dual numbers]]} $C^\infty(\mathbf{D})$ is the smooth algebra which assigns to $\mathbb{R}^n$ the [[Cartesian product]]

\begin{displaymath}
C^\infty(D,\mathbb{R}^n)

  \coloneqq
  \mathbb{R}^n \times \mathbb{R}^n
\end{displaymath}
of two copies of $\mathbb{R}^n$, which we will write as

\begin{displaymath}
\left\{
    (\epsilon \mapsto (\vec x + \epsilon \vec v))
    |
    \vec x , \vec v \in \mathbb{R}^n
  \right\}
  \,.
\end{displaymath}
Moreover, a [[smooth function]] $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is sent to the function

\begin{displaymath}
C^\infty(D, f)
  \colon
  C^\infty(D, \mathbb{R}^{n_1})
  \to 
  C^\infty(D, \mathbb{R}^{n_2})
\end{displaymath}
given by

\begin{displaymath}
\begin{aligned}
  \left(\epsilon \mapsto \left(\vec x + \epsilon \vec v\right)\right)
  \\
  \left(
    \epsilon \mapsto f(\vec x) + (\mathbf{d}f)(\vec v)
  \right)
  &\mapsto 
  \left(
    \epsilon
    \mapsto 
    \left(
       f\left(\vec x\right) 
        +  
       \sum_{j = 1}^{n_2}
       \left(\sum_{i = 1}^{n_1}\frac{\partial f^j}{\partial x^i} v^i\right)
       \vec e_j  
    \right)
    \right)
  \end{aligned}
  \,.
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
As the notation suggests, we may think of $C^\infty(D)$ as the functions on a first order [[infinitesimal object|infinitesimal neighbourhood]] of the origin in $\mathbb{R}^n$.

\end{remark}
\hypertarget{CoverageOfDifferentiallyGoodOpenCovers}{}\subsubsection*{{The coverage of differentially good open covers}}\label{CoverageOfDifferentiallyGoodOpenCovers}

We discuss a standard structure of a \emph{[[site]]} on the category [[CartSp]]. Following \emph{[[Sketches of an Elephant|Johnstone -- Sketches of an Elephant]]}, it will be useful and convenient to regard a site as a ([[small site|small]]) category equipped with a \emph{[[coverage]]}. This generates a genuine [[Grothendieck topology]], but need not itself already be one.

\begin{defn}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$ the standard [[open ball|open n-ball]] is the subset

\begin{displaymath}
D^n = \{ (x_i)_{i =1}^n \in \mathbb{R}^n | \sum_{i = 1}^n (x_i)^2 \lt 1 \}  \hookrightarrow \mathbb{R}^n
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
There is a [[diffeomorphism]]

\begin{displaymath}
\mathbb{R}^n \stackrel{\simeq}{\to} D^n
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{DifferentiallyGoodOpenCover}\hypertarget{DifferentiallyGoodOpenCover}{}
A \textbf{differentially [[good open cover]]} of a [[Cartesian space]] $\mathbb{R}^n$ is a set $\{U_i \hookrightarrow \mathbb{R}^n\}$ of [[open subset]] inclusions of Cartesian spaces such that these [[open cover|cover]] $\mathbb{R}^n$ and such for each non-empty finite [[intersection]] there exists a [[diffeomorphism]]

\begin{displaymath}
\mathbb{R}^n \stackrel{\simeq}{\to} U_{i_1} \cap \cdots \cap U_{i_k}
\end{displaymath}
that identifies the $k$-fold intersection with a Cartesian space itself.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Differentiably good covers are useful for computations. Their full impact is however on the [[homotopy theory]] of [[simplicial presheaves]] over [[CartSp]]. This we discuss in the chapter on [[geometry of physics -- smooth homotopy types|smooth homotopy types]], around this prop.omotopy types\#DifferentiablyGoodCoverGivesSPlitHyperCoverOverCartSp).

\end{remark}
\begin{prop}
\label{DiffGoodOpenCoversRefineOpenCovers}\hypertarget{DiffGoodOpenCoversRefineOpenCovers}{}
Every [[open cover]] refines to a differentially good open cover, def. \ref{DifferentiallyGoodOpenCover}.

\end{prop}
A proof is at \emph{[[good open cover]]}.

\begin{remark}
\label{}\hypertarget{}{}
Despite its appearance, this is not quite a classical statement. The classical statement is only that every open cover is refined by a \emph{topologically} [[good open cover]]. See the comments \emph{\href{http://ncatlab.org/nlab/show/ball#References}{here in the references-section}} at \emph{[[open ball]]} for the situation concerning this statement in the literature.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The \emph{good} open covers do not yet form a [[Grothendieck topology]] on [[CartSp]]. One of the axioms of a Grothendieck topology is that for every [[covering]] family also its [[pullback]] along any morphism in the category is a covering family. But while the pullback of every [[open cover]] is again an open cover, and hence open covers form a Grothendieck topology on [[CartSp]], not every pullback of a \emph{[[good open cover|good]]} open cover is again \emph{good}.

\end{remark}
\begin{example}
\label{ExampleThatGoodOpenCoversAreNotPullbackStable}\hypertarget{ExampleThatGoodOpenCoversAreNotPullbackStable}{}
Let $\{\mathbb{R}^2\stackrel{\phi_{i}}{\hookrightarrow}\mathbb{R}^2\}_{i \in \{1,2\}}$ be the [[open cover]] of the [[plane]] by an open left half space

\begin{displaymath}
\mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \lt 1 \} \stackrel{\phi_1}{\hookrightarrow} \mathbb{R}^2
\end{displaymath}
and a right open half space

\begin{displaymath}
\mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \gt -1 \} \stackrel{\phi_2}{\hookrightarrow} \mathbb{R}^2
  \,.
\end{displaymath}
The intersection of the two is the open strip

\begin{displaymath}
\mathbb{R}^2 \simeq \{ (x_1, x_2) \in \mathbb{R}^2 | -1 \lt x_1 \lt 1 \}
  \hookrightarrow 
  \mathbb{R}^2
  \,.
\end{displaymath}
So this is a [[good open cover]] of $\mathbb{R}^2$.

But consider then the smooth function

\begin{displaymath}
2(\cos(2 \pi (-)), \sin(2 \pi (-))) \colon \mathbb{R}^1 \to \mathbb{R}^2
\end{displaymath}
which sends the line to a curve in the plane that periodically goes around the [[circle]] of radius 2 in the plane.

Then the pullback of the above good open cover on $\mathbb{R}^2$ to $\mathbb{R}^1$ along this function is an [[open cover]] of $\mathbb{R}$ by two open subsets, each being a disjoint union of countably many open [[intervals]] in $\mathbb{R}$. Each of these open intervals is an open 1-ball hence diffeomorphic to $\mathbb{R}^1$, but their \emph{[[disjoint union]]} is not [[contractible topological space|contractible]] (it does not contract to the point, but to many points!).

So the pullback of the good open cover that we started with is an open cover which is not \emph{good} anymore. But it has an evident \emph{refinement} by a good open cover.

\end{example}
This is a special case of what the following statement says in generality.

\begin{prop}
\label{TheDifferentiallyGoodOpenCoverCoverage}\hypertarget{TheDifferentiallyGoodOpenCoverCoverage}{}
The differentially good open covers, def. \ref{DifferentiallyGoodOpenCover}, constitute a [[coverage]] on [[CartSp]].

Hence [[CartSp]] equipped with that coverage is a [[site]].

\end{prop}
\begin{proof}
By definition of [[coverage]] we need to check that for $\{U_i \hookrightarrow \mathbb{R}^n\}_{i \in I}$ any [[good open cover]] and $f \colon \mathbb{R}^k \to \mathbb{R}^n$ any smooth function, we can find a good open cover $\{K_j \to \mathbb{R}^k\}_{j \in J}$ and a [[function]] $J \to I$ such that for each $j \in J$ there is a smooth function $\phi \colon K_j \to U_{\rho(j)}$ that makes this [[diagram]] [[commuting diagram|commute]]:

\begin{displaymath}
\itexarray{

    K_j &\stackrel{\phi}{\to}& U_{i(j)}
    \\
    \downarrow && \downarrow
    \\
    \mathbb{R}^k &\stackrel{f}{\to}& \mathbb{R}^n
  }
  \,.
\end{displaymath}
To obtain this, let $\{f^* U_i \to \mathbb{R}^k\}$ be the pullback of the original covering family, in that

\begin{displaymath}
f^* U_i \coloneqq \{ x \in \mathbb{R}^k | f(x) \in U_i \}
  \hookrightarrow \mathbb{R}^k
  \,.
\end{displaymath}
This is evidently an [[open cover]], albeit not necessarily a \emph{[[good open cover]]}. But by prop. \ref{DiffGoodOpenCoversRefineOpenCovers} there does exist a good open cover $\{\tilde K_{\tilde j} \hookrightarrow \mathbb{R}^k\}_{\tilde j \in \tilde J}$ \emph{refining} it, which in turn means that for all $\tilde j$ there is

\begin{displaymath}
\itexarray{
    \tilde K_{\tilde j} &\to& K_{j(\tilde j)}
    \\
    \downarrow && \downarrow 
    \\
    \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k
  }
  \,.
\end{displaymath}
Therefore then the [[pasting]] composite of these commuting squares

\begin{displaymath}
\itexarray{
    \tilde K_{\tilde j} &\to& K_{j(\tilde j)} &\to& U_{i(j(\tilde j))}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k
   &\stackrel{f}{\to}& \mathbb{R}^n
  }
\end{displaymath}
solves the condition required in the definition of [[coverage]].

\end{proof}
By example \ref{ExampleThatGoodOpenCoversAreNotPullbackStable} this [[good open cover]] [[coverage]] is not a [[Grothendieck topology]]. But as any coverage, it uniquely completes to one which has the same [[sheaves]].

\begin{prop}
\label{}\hypertarget{}{}
The [[Grothendieck topology]] induced on [[CartSp]] by the differentially good open cover coverage of def. \ref{TheDifferentiallyGoodOpenCoverCoverage} has as covering families the ordinary [[open covers]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
This means that for every sheaf-theoretic construction to follow we can just as well consider the Grothendieck topology of open covers on $CartSp$. The sheaves of the open cover topology are the same as those of the good open cover coverage. But the latter is (more) useful for several computational purposes in the following. It is the \emph{good} open cover coverage that makes manifest, below, that sheaves on $CartSp$ form a [[locally connected topos]] and in consequence then a [[cohesive topos]]. This kind of argument becomes all the more pronounced as we pass \hyperlink{SmoothnGroupoids}{further below} to [[(∞,1)-sheaves]] on [[CartSp]]. This will be discussed in \emph{\hyperlink{InfinityConnectednessOfSmoothInfinityGrpd}{Smooth n-groupoids -- Semantic Layer -- Local Infinity-Connectedness} below.}

\end{remark}
\hypertarget{SliceCategories}{}\subsubsection*{{The slice category}}\label{SliceCategories}

(\ldots{})

\begin{itemize}%
\item [[slice category]] $CartSp_{\mathbb{R}^n}$


\item [[subterminal object]], [[open cover]]



\end{itemize}
(\ldots{})

$\,$

\hypertarget{CoordinateSystemsLayerSyn}{}\subsection*{{Syntactic Layer}}\label{CoordinateSystemsLayerSyn}

In this \hyperlink{LayerSyn}{Syn Layer} we discuss the [[abstract generals]] of \emph{abstract [[coordinate systems]]}, def. \ref{CartesianSpacesAndSmoothFunctions}: the [[internal language]] of a category with [[products]], which is \emph{[[type theory]]} with \emph{[[product types]]}.

\begin{quote}%
This is rough, needs further development.


\end{quote}
\hypertarget{Judgments}{}\subsubsection*{{Judgments about types and terms}}\label{Judgments}

We now introduce a different \emph{notation} for [[objects]] and [[morphisms]] in a [[category]] (such as the category [[CartSp]] of def. \ref{CartSpCategory}). This notation is designed to, eventually, make more transparent what exactly it is that happens when we [[deductive reasoning|reason deductively]] about [[objects]] and [[morphisms]] of a [[category]].

But before we begin to make any actual [[deductions]] about [[objects]] and [[morphisms]] in a category \hyperlink{NaturalDeductionRulesForProductTypes}{below}, in this section here we express the given objects and morphisms at hand in the first place. Such basic statements of the form ``There is an object called $A$'' are to be called \emph{[[judgments]]}, in order not to confuse these with genuine \emph{[[propositions]]} that we eventually formalize \emph{within} this [[metalanguage]].

To express that there is an [[object]]

\begin{displaymath}
X \in \mathcal{C}
\end{displaymath}
in a [[category]] $\mathcal{C}$, we write now equivalently the string of symbols (called a \emph{[[sequent]]})

\begin{displaymath}
\vdash \; X \colon Type
  \,.
\end{displaymath}
We say that these symbols express the \emph{[[judgment]]} that $X$ is a \emph{[[type]]}. We also say that $\vdash \; X \colon Type$ is the \emph{[[syntax]]} of which $X \in \mathcal{C}$ is the \emph{[[categorical semantics]]}.

For instance the [[terminal object]] $* \in \mathcal{C}$ we call the [[categorical semantics]] of the \emph{[[unit type]]} and write [[syntax|syntactically]] as

\begin{displaymath}
\vdash \; Unit \colon Type
  \,.
\end{displaymath}
If we want to express that we do assume that a terminal object indeed exists, hence that we want to be able to \emph{[[deduction|deduce]]} the existence of a terminal object from no hypothesis, then we write this [[judgment]] below a horizontal line

\begin{displaymath}
\frac{}{\vdash \; Unit \colon Type}
  \,.
\end{displaymath}
We will see more interesting such horizontal-line statements \hyperlink{NaturalDeductionRulesForProductTypes}{below}.

Next, to express an [[element]] of the object $X$ in $\mathcal{C}$, hence a [[morphism]]

\begin{displaymath}
* \stackrel{x}{\to} X
\end{displaymath}
in $\mathcal{C}$ we write equivalently the [[sequent]]

\begin{displaymath}
\vdash \; x \colon X
\end{displaymath}
and call this the [[judgment]] that $x$ is a \emph{[[term]]} of [[type]] $X$.

Notice that every object $X \in \mathcal{C}$ becomes the [[terminal object]] in the \emph{[[slice category]]} $\mathcal{C}_{/X}$. Let $A \to X$ be any morphism in $\mathcal{C}$, regarded as an object in the [[slice category]]

\begin{displaymath}
A \in \mathcal{C}_{/X}
  \,.
\end{displaymath}
We declare that the [[syntax]] of which this is the [[categorical semantics]] is given by the [[sequent]]

\begin{displaymath}
x \colon X \;\vdash \;  A(x) \colon Type
  \,.
\end{displaymath}
We say that this expresses the [[judgement]] that $A$ is an \emph{$X$-[[dependent type]]}; or a type in the \emph{[[context]] of a [[free variable]]} $x$ of [[type]] $X$.

Notice that an [[element]] of $A \in \mathcal{C}_{/X}$ is a [[generalized element]] of $A$ in $\mathcal{C}$, namely a morphism $X \to A$ which fits into a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    X &&\stackrel{a}{\to}&& A
    \\
    & {}_{\mathllap{id_X}}\searrow && \swarrow_{}
    \\
    && X
  }
\end{displaymath}
in $\mathcal{C}$.

We declare that the [[syntax]] of which such

\begin{displaymath}
a \in A  \;\;\;\; (in \mathcal{C}_{/X})
\end{displaymath}
is this the [[categorical semantics]] is the [[sequent]]

\begin{displaymath}
x\colon X \;\vdash\; a(x) : A(x)
  \,.
\end{displaymath}
We say that this expresses the [[judgment]] that $a(x)$ is a [[term in context|term depending on]] the [[free variable]] $x$ of [[type]] $X$.

This completes the list of \emph{[[judgment]]} [[syntax]] to be considered. Notice that if the category $\mathcal{C}$ has [[products]] then, even though it does not explicitly appear above, this is sufficient to express any morphism $X \stackrel{f}{\to} Y$ in $\mathcal{C}$ as the [[semantics]] of a [[term]]: we regard this morphism naturally as being the corresponding morphism in the [[slice category]] $\mathcal{C}_{/X}$ which as a [[commuting diagram]] in $\mathcal{C}$ itself is

\begin{displaymath}
\itexarray{
     X && \stackrel{(f,id_X)}{\to} &&  Y\times X
     \\
     & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_2}}
     \\
     && X
  }
  \,.
\end{displaymath}
This is the [[categorical semantics]] for which the [[syntax]] is simply

\begin{displaymath}
x \colon X \;\vdash\; y(x) \colon Y
  \,,
\end{displaymath}
being the [[judgment]] which expresses that $y(x)$ is a [[term in context]] of an $X$-[[dependent type]] $Y$ in the special degenerate case that $Y$ does not actually vary with $x \colon X$.

\hypertarget{NaturalDeductionRulesForProductTypes}{}\subsubsection*{{Natural deduction rules for product types}}\label{NaturalDeductionRulesForProductTypes}

With the \hyperlink{Judgments}{above symbolic notation} for making [[judgments]] about the presence of [[objects]] and [[morphisms]] in a [[category]] $\mathcal{C}$, we now consider a system of rule of [[deduction]] that tells us how we may \emph{process} these symbols (how to do \emph{[[computations]]}) such that the new symbols we obtain in turn express new objects and new morphisms in $\mathcal{C}$ that we can build out of the given ones by \emph{[[universal constructions]]} in the sense of [[category theory]].

This way of \emph{deducing} new expressions from given ones is very basic as well as very natural and hence goes by the technical term \emph{[[natural deduction]]}. For every kind of [[type]] (every [[universal construction]] in [[category theory]]) there is, in [[natural deduction]], one set of rules for how to [[deductive reasoning|deductively reason]] about it. This set of rules, in turn, always consists of four kinds of rules, called the

\begin{enumerate}%
\item [[type formation rule]]


\item [[term introduction rule]]


\item [[term elimination rule]]


\item [[computation rule]].



\end{enumerate}
These are going to be the [[syntax]] in \emph{[[type theory]]} of which [[universal constructions]] in [[category theory]] is the [[categorical semantics]].

In our running example where $\mathcal{C} =$ [[CartSp]], the only [[universal construction]] available is that of forming [[products]]. We therefore introduce now the [[natural deduction]] rules by way of example of the special case of [[product types]].

\textbf{1. [[type formation rule]]} Let

\begin{displaymath}
A , B \in \mathcal{C}
\end{displaymath}
be two [[objects]] in a [[category]] with [[products]]. Then there exists [[generalized the|the]] [[product]] [[object]]

\begin{displaymath}
A \times B \in \mathcal{C}
  \,.
\end{displaymath}
We now declare that the [[syntax]] of which this state of affairs is the [[categorical semantics]] is the collection of symbols of the form

\begin{displaymath}
\frac{A \colon Type \;\;\;\;\; B \colon Type}{ A \times B \colon Type}
  \,.
\end{displaymath}
Here on top of the horizontal line we have the two [[judgments]] which express that, [[syntax|syntactically]], $A$ is a [[type]] and $B$ is a [[type]], and [[semantics|semantically]] that $A \in \mathcal{C}$ and $B \in \mathcal{C}$. Below the horizontal line is, in turn, the [[judgment]] which expresses that there is, syntactically, a [[product type]], which semantically is the [[product]] $A \times B \in \mathcal{C}$. The horizontal line itself is to indicate that if we are given the (symbols of) the collection of judgments on top, then we are entitled to also obtain the judgment on the bottom.

\textbf{Remark (Computation)} All this may seem, on first sight, like being a lot of fuss about something of grandiose banality. To see what is gradually being accomplished here despite of this appearance, as we proceed in this discussion, the reader can and should think of this as the first steps in the definition of a [[programming language]]: the notion of judgment is a syntactic rule for strings of symbols that a computer is to read in, and a [[natural deduction]]-step as the [[type formation rule]] above is an operation that this computer validates as being an allowed step of transforming a memory state with a given collection of such strings into a new memory state to which the string below the horizontal line is added. As we add the remaining rules below, what looks like a grandiose banality so far will remain grandiose, but no longer be a banality. The reader feeling in need of more motivational remarks along these lines might want to take a break here and have a look at the entry \emph{[[computational trinitarianism]]} first, that provides more pointers to the grandiose picture which we are approaching here.

Next, the second [[natural deduction]] rule for [[product types]] is the

\textbf{2. [[term elimination rule]]}. The fact that $A \times B \in \mathcal{C}$ is equipped with two [[projection]] [[morphisms]]

\begin{displaymath}
\itexarray{
    A \stackrel{p_1}{\leftarrow} A \times B \stackrel{p_2}{\to} B
  }
\end{displaymath}
means that from every [[element]] $t$ of $A \times B$ we may deduce the existence of [[elements]] $p_1(t)$ and $p_2(t)$ of $A$ and $B$, respectively. We declare now that this is the [[categorical semantics]] of which the [[natural deduction]] [[syntax]] is:

\begin{displaymath}
\frac{\vdash \; t \colon A \times B}{\vdash \; p_1(t) \colon A}
  \;\;\;\;\;\;\;\;\;
  \frac{\vdash \; t \colon A \times B}{\vdash \; p_2(t) \colon B}
  \,.
\end{displaymath}
As before, this is to say that if syntactically we are given strings of symbols expressing [[judgments]] as on the top of these horizontal lines, then we may ``[[natural deduction|naturally deduce]]'' also the judgment of the string of symbols on the bottom of this line.

\textbf{3. [[term introduction rule]]}. The first part of the [[universal property]] of the [[product]] in [[category theory]] is that for $Q \in \mathcal{C}$ any other [[object]] equipped with morphisms

\begin{displaymath}
\itexarray{
    && Q
    \\
    & {}^{\mathllap{a}}\swarrow && \searrow^{\mathrlap{b}}
    \\
    A  && &&  B
  }
\end{displaymath}
in $\mathcal{C}$, we obtain a canonical morphism

\begin{displaymath}
Q \to A \times B
\end{displaymath}
in $\mathcal{C}$. This is now declared to be the [[categorical semantics]] of which the [[natural deduction]] [[syntax]] is

\begin{displaymath}
\frac{
    \vdash\; a \colon A \;\;\;\;\;\; \vdash\; b \colon B
  }{
   \vdash (a,b) \colon A \times B
  }
  \,.
\end{displaymath}
With the [[elements]] that are the [[semantics]] of the terms appearing here made explicit, this is the syntax for a [[diagram]]

\begin{displaymath}
\itexarray{
     && Q
     \\
     & {}^{\mathllap{a}}\swarrow &\downarrow^{\mathrlap{(a,b)}}& \searrow^{\mathrlap{b}}
     \\
     A && A \times B && B
  }
  \,.
\end{displaymath}
\textbf{4. [[computation rule]]}. The next part of the [[universal property]] of the [[product]] in [[category theory]] is that the resulting [[diagram]]

\begin{displaymath}
\itexarray{
    && Q
    \\ 
    & {}^{\mathllap{a}}\swarrow &\downarrow& \searrow^{\mathrlap{b}}
    \\
    A &\stackrel{p_1}{\leftarrow}& A \times B & \stackrel{p_2}{\to} & B
  }
\end{displaymath}
is in fact a \emph{[[commuting diagram]]}. Syntactically this is, clearly, the rule that the following identifications of strings of symbols are to be enforced

\begin{displaymath}
p_1(a,b) = a \;\;\;\;\;\; p_2(a,b) = b
  \,.
\end{displaymath}
This concluces the description of the [[natural deduction]] about [[objects]], [[morphisms]] and [[products]] in a [[category]] using its [[type theory]] [[syntax]]. We summarize the dictionary between [[category theory]] and [[type theory]] discussed so far \hyperlink{DictionaryTypeTheoryCategoryTheory}{below}.

In the \hyperlink{SmoothSpaces}{next section} we promote our running example category $\mathcal{C}$, which admits only very few [[universal constructions]] (just [[products]]), to a richer category, the [[sheaf topos]] over it. That richer category then accordingly comes with a richer [[syntax]] of [[natural deduction]] inside it, namely with full [[dependent type theory]]. This we discuss in the \hyperlink{SmoothSpacesLayerSyn}{Syn Layer below}.

\hypertarget{DependentSumTypes}{}\subsubsection*{{Natural deduction rules for dependent sum types}}\label{DependentSumTypes}

(\ldots{}) [[dependent sum type]] (\ldots{})

\hypertarget{DictionaryTypeTheoryCategoryTheory}{}\subsubsection*{{Dictionary: type theory / category theory}}\label{DictionaryTypeTheoryCategoryTheory}

The dictionary between [[dependent type theory|dependent]] [[type theory]] with [[product types]] and [[category theory]] of categories with [[products]].

[[!include judgements for types and terms - table]]

$\,$

[[!include substitution natural deduction - table]]

$\,$

[[!include product natural deduction - table]]

$\,$

[[!include dependent sum natural deduction - table]]

Below in \hyperlink{SmoothSpacesLayerSyn}{Smooth spaces - Syntactic Layer} we complete this dictionary to one between [[dependent type theory]] with [[dependent products]] and [[toposes]].



\end{document}