\section{Conclusion}
The aim of this paper was to introduce a Galois theory for general differential equations in characteristic zero. To accomplish this we had to introduce a large body of algebraic concepts.
\subsection{Summary}
We are giving a biref summary of the concepts introduced.
\subsubsection{Algebras and coalgebras}
The primer was clearly the introduction of algebras and a certain type of algebra extensions, the so called Ore-extensions. In summary, for a given algebra $A$ over a ring $R$, an algebra automorphism $\alpha \in \trm{Aut}_{R-\trm{alg}}(A)$ and an $\alpha$-derivation $\delta \in \trm{End}_R(A)$ which is
$$\delta(a b) = \alpha(a) \delta(b) + \delta(a) b,\ \forall a, b \in A$$
the Ore extension $A[X,\alpha,\delta]$ is an $A$-algebra, with
$$X a = \alpha(a) X - \delta(a)$$
for all $a \in A$. In addition, we provided the basic concepts of Lie algebras and their enveloping algebras. A Lie algebra is an $R$-module over a ring $R$ with a antisymmetric multiplication map $\mu$ fulfulling the Jacobi-identity. Their enveloping algebras are unital associative algebras in the above sense. Lie algebras are intimitely connected to derivations and differential rings.\\
Coalgebras a categorically dual to algebras, that is the dual module $C^*$ for any coalgebra $C$ is an algebra over the same ring. Furthermore, the module $\trm{Hom}_R(C,A)$ is an algebra as well, with convolution
$$\mu : \trm{Hom}_R(C,A) \otimes \trm{Hom}_R(C,A) \longrightarrow \trm{Hom}_R(C,A),\ f \otimes g \longmapsto \mu_A \circ (f \otimes g) \circ \Delta_C$$
as multiplication and $\eta_A \circ \eps_C$ as unit.
\subsubsection{Bialgebras, module algebras and Hopf algebras}
A bialgebra $(D,\mu_D,\eta_D,\Delta_D,\eps_D)$ has both, the structure of an algebra $(D,\mu_D,\eta_D)$ and that of a coalgebra $(D,\Delta_D,\eps_D)$, such that multiplication and unit map are homomorphisms of coalgebras and comultiplication and counit are homomorphisms of algebras.\\
For a bialgebra $D$, a $D$-module algebra $(A,\Psi_A)$ is an algebra $(A,\mu_A,\eta_A)$ such that
$$\Psi (a b) = \sum_{(d)} \mu_A(d_{(1)}(a) \otimes d_{(2)}(b)),\ \Psi_A(1_D \otimes a) = \eps_D(1_D) a$$
holds.\\
A Hopf algebra $D$ is a bialgebra, with a bialgebra antihomomorphism $S$:
$$S : D \longrightarrow D^{\trm{copop}}$$
such that $\eta \eps(d) = \mu(id_D \otimes S(\Delta(d))) = \mu(S \otimes id(\Delta(d)))$ holds for all $d \in D$.
\subsubsection{Differential modules and their constructs}
We introduced the concept of derivations over arbitrary modules over differential rings. Next, we introduced the ring of differential operators and showed that this ring is an Ore extension of the differential ring. This is followed by the defintion of the ring of differential polynmials. We showed that this ring is a module algebra over the ring of differential operators.\\
Furthermore, we discussed the basic Picard-Vessiot theory for linear differential equations in characteristic zero. We showed that any simple differential ring containing all solutions for suchs an equations is isomorphic to the PV ring. Moreover, algebraic elements over the field of constants are constant over the PV ring and any constant algebraic element over the PV ring is already algebraic over the field of constants. We defined the Galois group for this type of equations to be all $k^\partial$-linear bijections commuting with the derivation $\partial$. Concluding this section, we showed to simple examples over a non-trivial and a trivial differential ring:
$$L_1 = \partial - a, \ k = \currfield(z),\ L_2 = \partial^2 - a, \ k = \currfield,\ a \in \currfield^\times$$
in both cases. We computed the PV rings $R_1 = \currfield(z)[y,y^{-1}], \partial(y) = a y$ and $R_2 = \currfield[y,y^{-1}], \partial(y) = \sqrt{a} y$ and with Galois group $\currfield^\times$.
\subsubsection{General theory by Heiderich}
Next, we introduced the concept of iterative derivations and the universal Taylor homomorphism for differential ring extensions in characteristic zero. This was followed by the definition of the Umemura functor which assigns to each commutative algebra over a differential ring extension the group of algebra automorphisms leaving the image of the differential ring under the Taylor homomorphism fixed and making the following diagram commutative:
%diagram!
This led us the the definition of the Lie-Ritt functor assigning to each non-reduced algebra over $K$ the group of infinitesimal coordinate transformation. Heiderich proved that the Umemura functor is such a Lie-Ritt functor. Lastly, we compared the general theory with the PV theory and revisited our previous example $L_2$. There, we computed the Hopf algebra $H = R \otimes R^{\Psi_R \otimes \Psi_R}$ and saw that its set of prime ideals is indeed isomorphic to the spectrum of the ring of Laurent polynomials over $\currfield$.
\subsection{Outlook}
As we only computed a linear example over a trivial differential ring, it would be most interesting to extend this to non-trivial differential rings. Furthermore, the theory is general enough to deal with non-linear differential equations. Rather simple examples as $\partial(x) - x^2$ could be a starting point to further our understanding of this intriguing theory.\\
Secondly, the theory developed by Umemura and Heiderich does not involve non-commutative cases as the definition of the Lie-Ritt functors heavily depends on commutativity of the underlying algebras. An inverstigation  into this would seem promissing.
%We broadly introduced the concepts of module algebras and their Galois theory according to \cite{Heid10,Heid13}. This concept got applied to a rather simple example. In addition, we compared the classical Galois theory in the sense of Picard-Vessiot with the expanded theory.\\
%\indent To conclude, this theory provides alternative means not only to linear differential equations but rather to a wide range of similar problems, as iterative derivatives, difference equations and, of course, non-linear (ordinary or partial) differential equations.
\newpage
\section*{Acknowledgements}
First of all, I would like to thank professor Gro\ss{}e-Kl\"onne for offering me this challenging topic and providing invaluable hints, as well as doctor R\"uhling. Furthermore, I am grateful to Greta, Uwe, Anne and Anne W. for supporting me almost endlessly. In addition, Verena and Alex are not to be forgotten, last but not least Christian and Otis.
\newpage
\section{Appendix}\label{appendix}
%\subsection{Basic category theory}
%To generalize or abstract certain statements it is convenient to introduce the notion of categories. Here, we give a brief introduction for the sake of clarity following loosely \cite{Awo} and \cite{Borc}. % In short, a category is a class of sets $\trm{Obj}$ sharing a common mathematical structure. Maps for two objects in $\trm{Obj}$ are called morphism on $\trm{Obj}$ if they preserve the structure. More formally,
% A category $\mathcal{C}$ conists of the class of objects $\trm{Obj}(\mathcal{C})$ and morphisms for two objects $A, B$ in $\trm{Obj}(\mathcal{C})$ which is denoted by $\mathcal{C}(A,B)$, $\mathcal{M_C}$ or $\trm{Morph}_{\mathcal{C}}(A,B)$. Moreover, if $A, B, C$ are three (not necessarily distinct) objects in $\mathcal{C}$, the composition of two morphisms $f: A \longrightarrow B$ and $g : B \longrightarrow C$ is denoted by $g \circ f : A \longrightarrow C$ and is a morphism in $\mathcal{C}(A,C)$. Lastly, the identity map $id : A \longrightarrow A$ is a morphism for all objects $A$ in $\trm{Obj}(\mathcal{C})$.%if $X$ and $Y$ are two objects in $\trm{Obj}$ %(with structure maps $\phi$, $\psi$ or tuples of structure maps), then a map $f : X \longrightarrow Y$ is called a morphism in the category of $\trm{Obj}$ if and only there is a map $\wt{f} : \im \phi \longrightarrow \im \psi$ such that the diagram commutes\index{Index}{category}\index{Index}{morphism}
%%$$\xymatrix{
%%X \ar[r]^f \ar[d]_\phi & Y \ar[d]^\psi\\
%%\trm{im} \phi \ar[r]_{\wt{f}} & \im \psi.\\
%%}$$
%%The class of
%\bsp Some prominent examples
%\bn
%\item the category of sets with morphisms simply all maps between two sets.
%\index{Index}{category!of sets}
%\item the category of pointed spaces $\trm{PSpc}$, which can be considered as all non-empty sets with a designated element - the basepoint - and its morphisms are maps preserving the basepoints (the mandatory element in each object).
%\item the category of abelian groups $\trm{Abel}$, where morphisms are group homomorphisms,
%\index{Index}{category!of groups!abelian}
%\item the category of non-abelian groups $\trm{NAbel}$, where morphisms are also group homomorphisms, in case we do not know if a given group belongs to either of the two categories we simply assign it to $\trm{Grp}$,
%\index{Index}{category!of groups}
%\index{Index}{category!of groups!non-abelian}
%\item the category of rings $\trm{Rng}$, with ring homomorphisms as morphisms (note this is a proper subcategory of $\trm{Abel}$), with its prominent subcategory $\trm{CRng}$, the category of commutative rings and unital rings $\trm{URng}$.
%\index{Index}{category!of rings}
%\index{Index}{category!of rings!commutative}
%\index{Index}{category!of rings!unital}
%\item the category of differential manifolds, with smooth maps as morphisms,
%\index{Index}{category!of differential manifolds}
%\item the category of topological spaces $\trm{Top}$, with continuous maps as morphisms,
%\index{Index}{category!of topological spaces}
%\en
%We note, that in general there is no concept of union, products etc. - that is the union or product of to objects in the same category does not necessarily constitute an other object in same category, respectively. Therefore, we avoid talking about sets of certain mathematical objects (rather classes).
%\begin{defi}[Covariant and contravariant functors]
%Let $\mathcal{C}$ and $\mathcal{D}$ be two categories with morphisms $\mathcal{M_C}$ and $\mathcal{M_D}$, respectively. A (co/contravariant) functor $F$ is a pairing of $\mathcal{C}$ and $\mathcal{D}$ such that
%\bd
%\item[covariant]
%\bn
%\item for all $X \in \mathcal{C}$ the image $F(X)$ is an object in $\mathcal{D}$,
%\item for all morphisms $f : X \longrightarrow Y$, $X, Y \in \mathcal{C}$, the map $F(f) : F(X) \longrightarrow F(Y)$ is a morphism of $\mathcal{D}$
%\item $F(id_X) = id_{F(X)}$.
%\en
%\item[contravariant] as in covariant except:
%\bn
%\item for all morphisms $f : X \longrightarrow Y$, $X, Y \in \mathcal{C}$, the map $F(f) : F(Y) \longrightarrow F(X)$ is a morphism of $\mathcal{D}$, as well as
%\item $F(id_{F(X)}) = id_X$.
%\en
%\ed
%\index{Index}{functor!covariant}
%\index{Index}{functor!contravariant}
%\end{defi}
%\bsp Classical examples are
%\bn
%\item from main theorem of (classical Galois theory): let $L/K$ be a field extension
%$$\trm{Fix} : \trm{Grp} \longrightarrow \trm{CRng},\ H \longmapsto L^H := \{x \in L : g x = x\ \forall g \in H\}$$
%$$\trm{Gal} : \trm{CRng} \longrightarrow \trm{Grp},\ M \longmapsto \{\varphi \in \trm{Aut}_K(M) : \varphi\mid_K = id_K\},$$
%both define functors. Although, these functors are only defined on subcategories. In particular, the second functor is only well defined for normal separable fields $K \subset M \subset L$.
%\item in algebraic geometry, the (pre-) sheave defines a functor from the category of topolocial subspaces (of a variety - affine, projective, ...) to the category of associative algebras over a given ground field,
%\item furthermore, the so called forgetful functor: let $\mathcal{C}$ be a subcategory of $\mathcal{D}$, then $F : \mathcal{C} \longrightarrow \mathcal{D}$ is called the forgetful functor (informally, we simply omit some of the structure maps from $\mathcal{C}$), e.g
%$$F : \trm{Grp} \longrightarrow \trm{Set},$$
%\item the set of all automorphisms $\trm{Aut}$ on a given algebraic category (e.g. $\trm{Vec}$, $\trm{Grp}$, $\trm{Rng}$,...) is also a functor - assigning to an object $X$ the set of all structure preserving bijections $(X,\phi) \longrightarrow (X,\phi)$, i.e.
%$$\trm{Aut} : \trm{AlgCat} \longrightarrow \trm{Grp}.$$
%More generally, any structure preserving bijection (i.e. each map having a two-sided inverse which still preserves $\phi$) defines the $\trm{Aut}$ functor on their respective category.
%\en
%\begin{defi}
%For a given category $\mathcal{C}$ we construct a new category $\mathcal{C}^{\trm{op}}$ by simply reversing arrows and composition order of morphisms:
%$$f, g \in \mathcal{M_C},\ g \circ f \in \mathcal{M_C} \RA f^{\trm{op}}, g^{\trm{op}} \in \mathcal{M}_{\mathcal{C}^{\trm{op}}},\  (g \circ f)^{\trm{op}} := f^{\trm{op}} \circ g^{\trm{op}} \in \mathcal{M}_{\mathcal{C}^{\trm{op}}}.$$
%We call $\mathcal{C}^{\trm{op}}$ the opposite or dual category of $\mathcal{C}$.
%\index{Index}{category!dual}
%\end{defi}
%We already encountered the opposite category in case of $C$-coalgebras and $C$-algebras (i.e. coalgebras are dual to the algebras).
%\begin{defi}
%We call a category small if it defines an actual set. Otherwise it is called a large category. In addition, we call a category $\mathcal{C}$ a category with initial object, if there is an object $I$ in $\mathcal{C}$ such that for all $X$ in $\mathcal{C}$ there is exactly one morphism $\iota : I \longrightarrow X$. The dual notion is the category with terminal object: if there is an object $T$ in $\mathcal{C}$ such that for every object $X$ in $\mathcal{C}$ there is one morphism $\tau : X \longrightarrow T$.
%\index{Index}{category!small}
%\index{Index}{category!large}
%\index{Index}{category!with initial object}
%\index{Index}{category!with terminal object}
%\end{defi}
%\bsp To illustrate the last definitions:
%\bn
%\item $\trm{Set}$ is a large category.
%\item Usually the class of morphisms for a given category $\mathcal{C}$ is also large. However, some counter examples are for instance module homomorphisms.
%\item The category of unital rings is a category with initial object $(\zz,+,\cdot,1)$.
%\item Dually, the category of schemes over unital rings is a category with terminal object (by duality: $\trm{Spec}(\zz)$).
%\en
%\begin{defi}
%A natural transformation $\zeta$ for two given categories $\mathcal{C}$, $\mathcal{D}$ and two functors $F, G$ between the two categories is a family of morphisms such that
%\bn
%\item it assigns to each object $X \in \mathcal{C}$ a morphism $\zeta_X : F(X) \longrightarrow G(X)$ and
%\item for each morphism $f : X \longrightarrow Y$ in $\mathcal{C}$ we have
%$$\bao{cc}
%\xymatrix{
%F(X) \ar[r]^{F(f)} \ar[d]_{\zeta_X} & F(Y)\ar[d]^{\zeta_Y}\\
%G(X) \ar[r]_{G(f)} & G(Y)\\
%} &
%\xymatrix{
%F(Y) \ar[r]^{F(f)} \ar[d]_{\zeta_Y} & F(X)\ar[d]^{\zeta_X}\\
%G(Y) \ar[r]_{G(f)} & G(X)\\
%}\\
%\trm{covariant} & \trm{contravariant},\\
%\ea$$
%where covariant stands for covariant functors $F, G$ and contravariant stands for the other case.
%\index{Index}{natural transformation}
%\en
%\end{defi}
%Let $\mathcal{C}$ be a category. For two objects $A, B$ in $\mathcal{C}$ we denote the class of morphisms $\mathcal{M}_{\mathcal{C}}(A,B)$ simply by $\trm{Hom}_{\mathcal{C}}(A,B)$ or $\trm{Hom}(A,B)$ if there is no ambiguity. Every object $A \in \mathcal{C}$ defines a functor $F_A := \trm{Hom}(A,\_) : \mathcal{C} \longrightarrow \trm{Set}, B \longmapsto \trm{Hom}(A,B)$ such that for all maps $\varphi : B \longrightarrow B'$:
%$$F_A(\varphi)(f) := \varphi \circ f,\ \forall f \in F_A(B).$$
%Each morphism $\varphi : A' \longrightarrow A$ in $\mathcal{C}$ defines a map $f \longmapsto f \circ \varphi : F_A(B) \longrightarrow F_{A'}(B)$ being natural wrt $B$, i.e. this map is a natural transformation. In particular, the pairing $A \longmapsto F_A$ is a contravariant functor.
%\begin{defi}
%A functor $F : \mathcal{C} \longrightarrow \trm{Set}$ is called representable if it is isomorphic to $F_A$ for some $A \in \mathcal{C}$.
%\index{Index}{functor!representable}
%\end{defi}
%\subsubsection{Direct products and coproducts}
% %From now on, let $\mathcal{C}$ is a category with finite products.
%Given a category $\mathcal{C}$ and an index set $I$ we define
%\begin{defi}[direct product]
%for a family of objects $\{X_i : i \in I\}$ in $\mathcal{C}$ the direct product $X$ to be the object in $\mathcal{C}$ such that for each canonical projection $\pi_i : X \longrightarrow X_i$, $i \in I$ and an indexed family of morphisms $f_i : Y \longrightarrow X_i$ for all $i \in I$ and $Y \in \mathcal{C}$, there is a unique morphism $f : Y \longrightarrow X$ making
%$$\xymatrix{
%Y \ar[rd]_f\ar[r]^{f_i}&X_i\\
%&X\ar[u]_{\pi_i}\\
%}$$
%commutative. Sometimes the direct product is denoted by $\prod_{i \in I} X_i$.
%\index{Index}{product!direct}
%\end{defi}
%The coproduct is simply:
%\begin{defi}[coproduct]
%For a family of objects $\{X_i : i \in I\}$ in $\mathcal{C}$ the coproduct $X$ to be the object in $\mathcal{C}$ such that for each (not necessarily injective) inclusion $\iota_i : X_i \longrightarrow X$ and a family of morphisms $f_i : X_i \longrightarrow Y$ for all $i \in I$ and $Y \in \mathcal{C}$ there exists a unique $f : X \longrightarrow Y$ making
%$$\xymatrix{
%Y &X_i\ar[l]_{f_i}\ar[d]^{\iota_i}\\
%&X\ar[lu]^f\\
%}$$
%commutative. The coproduct $X$ is sometimes denoted by $\coprod_{i \in I} X_i$ or $\bigoplus_{i \in I} X_i$.
%\index{Index}{coproduct}
%\end{defi}
%\bmk For a finite index set $I$ both notions are equivalent. Consider for instance for any ring $R$ the left module $\prod_{i \leq n} R$ and $\bigoplus_{i \leq n} R$. However, if $I$ is not finite, then the coproduct is a strict subset of the direct product. Furthermore, the products and coproducts are only unique up to isomorphism (within their respective category).
%\begin{defi}
%$\mathcal{C}$ is called a category with finite products, if for any finite subcategory of $\mathcal{C}$ its coproduct is in $\mathcal{C}$ and there exists a finite object in $\mathcal{C}$, denoted by $*$ - called the empty product - and an isomorphism:
%$$S \times * \simeq S \simeq * \times S$$
%for all $S \in \mathcal{C}$.
%\index{Index}{category!with finite product}
%\index{Index}{category!empty object}
%\end{defi}
%$\trm{PSpc}$, the category of pointed spaces is an example of a category with finite products with one element sets as empty products.
%\subsubsection{Co-/limits, formal schemes and group schemes}
%
%To complete our defintions we need:
%\begin{defi} Let $\mathcal{C}$ be a category.
%\bn
%\item A diagram of type $\mathcal{J}$ is a functor $F : \mathcal{J} \longrightarrow \mathcal{C}$, where $\mathcal{J}$ is an index category and $F$ indexes objects and morphisms in $\mathcal{C}$. A diagram $F$ of type $\mathcal{J}$ is called small or finite if $\mathcal{J}$ is a small or finite category.
%\item Let $F : \mathcal{J} \longrightarrow \mathcal{C}$ be a diagram of type $\mathcal{J}$. A cone to $F$ is an object $N$ in $\mathcal{C}$ and a family of morphisms $\psi_X : N \longrightarrow F(X)$ indexed by $X$ in $\mathcal{J}$ such that for all morphisms $f : X \longrightarrow Y$ in $\mathcal{J}$ we get
%$$F(f) \circ \psi_X = \psi_Y.$$
%A cone is denoted by $(N,\psi)$.
%\item A co-cone is dual to cone: co-cone of a diagram $F$ is an object $N$ in $\mathcal{C}$ and family of morphisms $\psi_X : F(X) \longrightarrow N$ for every $X$ in $\mathcal{J}$ such that for any morphism $f : X \longrightarrow Y$ in $\mathcal{J}$ we have: $\psi_X \circ F(f) = \psi_Y$.
%The pair $(N,\phi)$ denotes the co-cone.
%\item A limit of a diagram of type $\mathcal{J}$ is a cone $(L,\phi)$ of $F$ such that for any other cone $(N,\psi)$ of $F$ there exists a unique morphism $u : N \longrightarrow L$ such that following diagram commutes:
%$$\xymatrix{
%& N \ar[ldd]_{\psi_X} \ar[d]^u \ar[rdd]^{\psi_Y}&\\
%&L\ar[ld]^{\phi_X} \ar[rd]_{\phi_Y}&\\
%F(X) \ar[rr]_{F(f)} & & F(Y)\\
%}$$
%\item A colimit of a diagram $F$ is a co-cone $(L,\phi)$ of $F$ such that for any other co-cone $(N,\psi)$ of $F$ there is a unique morphism $u : N \longrightarrow L$ such that the following diagram commutes:
%$$\xymatrix{
%F(X) \ar[rd]^{\phi_X}\ar[rdd]_{\psi_X}\ar[rr]^{F(f)} && F(Y)\ar[ld]_{\phi_Y}\ar[ldd]^{\psi_Y}\\
%&L\ar[d]_u&\\
%&N&\\
%}$$
%\en
%\index{Index}{diagram}
%\index{Index}{cone}
%\index{Index}{cocone}
%\index{Index}{limit}
%\index{Index}{colimit}
%\end{defi}
%\bmk A cone $(N,\psi)$ of a diagram $F : \mathcal{J} \longmapsto \mathcal{C}$ is characterized by the following commutative diagram:
%$$\xymatrix{
% & F(X) \ar[dd]^{F(f)} & X \ar[l]_F \ar[dd]^f\\
%N \ar[ru]^{\psi_X} \ar[rd]_{\psi_Y}& &\\
%& F(Y) & Y\ar[l]^F.\\
%}$$
%A co-cone $(N,\phi)$ of a diagram $F : \mathcal{J} \longrightarrow \mathcal{C}$ is characterized by the following commutative diagram:
%$$\xymatrix{
% & F(X) \ar[ld]_{\phi_X}\ar[dd]^{F(f)} & X \ar[l]_F \ar[dd]^f\\
%N & &\\
%& F(Y) \ar[lu]^{\phi_Y}& Y\ar[l]^F.\\
%}$$
%\subsubsection{Monoidal and group category}
%Let $\mathcal{C}$ is a category with finite products.
%\begin{defi}
%A monoid $G$ in $\mathcal{C}$ is a triple $(G, m, e)$, with operation morphism $m : G \times G \longrightarrow G$ and unit morphism $e : * \longrightarrow G$ satisfying the following commutative diagrams:
%\bd
%\item[associativity] $$\xymatrix{
%G \times G \times G \ar[rr]^{id_G \times m}\ar[d]_{m \times id_G} && G\times G\ar[d]^m\\
%G \times G \ar[rr]_m && G,\\
%}$$
%\item[unit] $$\xymatrix{
% & G \ar[ld]_{\simeq} \ar[rd]^{\simeq} & \\
% \ast \times G\ar[d]_{e \times id_G} & & G \times \ast \ar[d]^{id_G \times e}\\
% G \times G \ar[rd]_{m} & & G \times G \ar[ld]^m\\
% & G &\\
%}$$
%\ed
%We denote by $\trm{Mon}$ the monoidal category.
%\index{Index}{category!monoidal}
%\end{defi}
%\bmk Compare the two diagrams with the definition of associative unital algebras over some ring $R$ (replacing direct products with tensor products, then here the empty product is indeed $* = R$ and unit $\eta = e$).\\
%For each monoid $G$ we define the opposite monoid $G^{\trm{op}}$ being the same set with operation $(g, h) \longmapsto h g$.
%\begin{defi}
%A group $G$ in $\mathcal{C}$ is a quadruple $(G, m, e, S)$ being a monoid in $\mathcal{C}$ and $S : G \longrightarrow G$ defines a commuting diagram:
%$$\xymatrix{
%&G\ar[ld]_\Delta\ar[rd]^\Delta&\\
%G \times G \ar[rd]^{S \times id}\ar[dd]_{\pi \times \pi} && G \times G\ar[ld]_{id \times S}\ar[dd]^{\pi \times \pi}\\
%& G \times G \ar[d]_m &\\
%\ast \ar[r]_e & G & \ast\ar[l]^e,\\
%}$$
%where $\Delta : G \longrightarrow G \times G$ is the diagonal map, $f_1 \times f_2 = \left[(g,h) \longmapsto \left(f_1(g),f_2(h)\right)\right]$ and $\pi : G \longrightarrow \ast \simeq G/G$ is the trivial projection, such that
%$$S : G \longrightarrow G^{\trm{op}}$$
%is a group homomorphism.
%\index{Index}{category!of groups}
%\end{defi}
%\subsubsection{Schemes}
%Now, we introduce some important notations in the field of algebraic geometry. Let $\trm{UCRng}$ denote the category of unital commutative rings, $\trm{Mod}_R$ the category of $R$-modules over $R$ in $\trm{UCRng}$ and $\trm{Mod}_R \cap R$ the subcategory of $R$-submodules in $R$ (i.e. ideals). We have
%\begin{defi}[Spectrum of ring]
%The functor
%$$\trm{Spec} : \trm{UCRng} \longrightarrow \trm{Set},\ R \longmapsto \{\mathfrak{p} \in \trm{Mod}_R \cap R :  \trm{Ann}(R/\mathfrak{p}) = 0\}$$
%assigns to each commutative unital ring $R$ its spectrum, i.e. the set of its prime ideals.
%\index{Index}{spectrum}
%\end{defi}
%\bsp We have:
%\bn
%\item $\spec \zz = \left\{(p) : p \trm{~prime~number}\right\} \cup \{0\}$,
%\item for $n \geq 2$, $\spec \zz_n = \left\{(\ov{p}) : p \mid n \wedge p \trm{~prime}\right\} \cup \{\ov{0}\}$, in particular the first subset may be empty if $n$ is prime,
%\item $\spec \zz_2[X]$:
%$$\left\{\left<f = \sum_{i \leq n} f_i X^i\right> :\exists m \in \nz,\ X^m \mid (f + \ov{1}) \wedge |\{f_i : f_i \neq \ov{0}\}| \in 2 \nz + 1\right\} \cup \left\{\left<\ov{0}\right>, \left<X\right>, \left<X + \ov{1}\right>\right\},$$ in words:
%all polynomials $f$ with odd non-zero coefficients and $f \equiv 1 \mod X^m$ for some $m \geq 1$, zero and the two polynomials of degree one.
%\item in general, for any field $k$ we have
%$$\spec k[X] = \{(f) : f \trm{~irreducible}\} \cup \{0\},$$
%in case of algebraic closeness, $\spec k[X] = \{\left<X - a\right> : a \in k\} \cup \{\left<0\right>\}$.
%\item $R$ integral domain, iff $\{0\} \in \spec R$,
%\item $R$ a field, iff $\{0\} = \spec R$.
%\en
%Note, that the non-unitary ring $R := \left(\bao{cc} 0 & k\\0 & 0\\\ea\right)$ for some field $k$ has no prime ideal (but a maximal ideal (!)). Thus, unitality ensures the existence of prime ideals. % However, let us recall some further definitions.
%%\begin{defi}
%%Let $R = k$ be a algebraically closed field and identify the affine space $A_k^n$ with the coordinate space.
%%\bn
%%\item For some $S \subset k[X]$, the set $Z(S) = \{x \in k^n : f(x) = 0 \forall f \in S\}$ is called the zero set or algebraic set of $S$.
%%\item For each $Z \subset k^n$ an algebraic set, we have the ideal $I(Z) := \{f \in k[X] : f(x) = 0 \forall x \in Z\}$ and call $A(Z) := k[X]/I(Z)$ the coordinate ring of $Z$.
%%\item If $A(Z)$ is an integral domain, we call $Z$ an affine variety.
%%\item For $k^{n+1}$ we define projective space $\prjn_k$ the over $k$ as the quotient
%%$$\prjn_k := \left(k^{n+1} \bsl \{0\}\right)^2 /\sim,\ \sim := \left\{(x,y) \in \left(k^{n+1} \bsl \{0\}\right)^2 : \exists \lambda \in k^\times, y = \lambda x\right\}.$$
%%\item A polynomial is called homogeneous of degree $n$, if we have
%%$$f = \sum_{|\alpha| = n} f_\alpha X_\alpha, f_\alpha \in k, X_\alpha = \prod_{i \leq k} X^{s_i}_{\alpha_i}\ \trm{and}\ \sum s_i = n.$$
%%An ideal is called homogeneous, if it is generated by homogeneous polynomials and a projective algebraic set is simply the zero set of a homogeneous polynomial $f$, i.e. $Z(f) := \{x \in \prjn_k : f(x) = 0\}$. A projective variety is simply the zero set of homogeneous prime ideal.
%%\item Let $Z \subset k^n$ be an affine variety and $f \in k[X]$ some polynomial such that $Z(f) \nsubset Z$. We call $U_f := Z\bsl Z(f)$ the principal open sets of $Z$.
%%\item For some affine variety $Z$, the structure presheaf is defined as the functor $\trm{Top} \longrightarrow \trm{Alg}$ with
%%$$\bao{rrcl}
%%\mathcal{O}_Z :& \{U : U \subset Z\trm{~open}\} &\longrightarrow &\mathcal{O}_Z(U) := S^{-1}A(Z),\\
%%&&&\\
%%&S &:=& \{f \in A(Z) : f(x) \neq 0 \forall x \in U\}\\
%%\ea$$
%%assigning to every open set $U$ the ring of regular functions on $U$ with the following condition:
%%\bn
%%\item $\mathcal{O}_Z(\emptyset) = \{0\}$,
%%\item $\mathcal{O}_Z\mid_{V}(U) = \mathcal{O}_Z(V)$ for all $V \subset U$ open,
%%\item $\mathcal{O}_Z\mid_W \circ \mathcal{O}_Z\mid_V = \mathcal{O}_W$ for all $W \subset V \subset U$ open.
%%\en
%%\item We call a (structure) presheaf of an affine variety $X$ a sheaf, if there is the following glueing property. Let $\mathcal{U}$ be an open cover of some open $U \subset X$. For all $V, W \in \mathcal{U}$ and $f_V \in \mathcal{O}_X(V), f_W \in \mathcal{O}_X(W)$ such that $f_V\mid_{V \cap W} = f_W\mid_{V \cap W}$ then there exists a unique $f \in \mathcal{O}_X(U)$ such that $f\mid_V = f_V$ and $f\mid_W = f_W$.
%%\item A ringed space for some topological space $X$ consists of the pair $(X, \mathcal{O}_X)$, where $\mathcal{O}_X$ is the structure sheaf of $X$.
%%\en
%%\end{defi}
%%First, we note that our definition of algebraic sets induces a topology on $k^n$, the so called Zariski-topology by simply putting our algebraic sets as closed subsets of $k^n$. In particular, any affine variety defines an irreducible topological space. The definition of projective sheafs is omitted but the interested reader may consult \cite{Hart}\\
%%Furthermore, we may reformulate affine varieties as follows:
%We are omitting the definitions of presheafs, sheafs and ringed spaces and refer the interested reader to \cite{Hart}.
%%\begin{defi}
%%Let $(X, \mathcal{O}_X)$ be a ringed space. We call $X$ a scheme, if for any open cover $\mathcal{U}$ of $X$ and $U \in \mathcal{U}$ the ringed space $(U, \mathcal{O}_X\mid_{U})$ is isomorphic to some affine scheme.
%%\end{defi}
%\begin{defi}
%For some ideal $I \subset R$, the affine scheme $X_I$ is the set of all prime ideals $\mathfrak{p} \supset I$.
%%$$X_I :=  \{\mathfrak{p} \in \tmr{Spec}(R) : a \notin \mthfrak{p}\}$
%A principal open set $U(a)$, for some $a \in R\bsl R^\times$, is $\{\mathfrak{p} \in \trm{Spec} R : a \notin \mathfrak{p}\}$. A scheme $X$ is simply some ringed space $(X,\mathcal{O}_X)$ such that for every open cover $\mathcal{U}$ of $X$ the restrictions $\mathcal{O}_X\mid_U$ as ringed spaces $(U, \mathcal{O}_X\mid_U)$ define affine schemes for all $U \in \mathcal{U}$.
%\index{Index}{scheme}
%\index{Index}{scheme!affine}
%\index{Index}{Set!principal open}
%\end{defi}
%In short, a scheme is a (topological) space with structure sheaf $\mathcal{O}_X$ that is locally isomorphic to some affine variety.
%\bsp For any ring $R$ in $\trm{UCRng}$:
%$$S^{-1}_{\mathfrak{p}}(R),\ S_{\mathfrak{p}} = R \bsl \mathfrak{p},$$
%the pair $(\trm{Spec}(R), S^{-1})$ is a scheme.% Here in obuse of notation, we use the multiplative system $S_{\mathfrak{p}}$ as a functor $\trm{Set} \longmapsto \trm{Mon}$.
%\begin{lemm}
%The following statements are equivalent:
%\bn
%\item the ringed space $(X, \mathcal{O}_X)$ is an affine variety,
%\item and:
%\bn
%\item $X$ is an irreducible topological space,
%\item $\mathcal{O}_X$ is a structure sheaf,
%\item $X$ is isomorphic to an affine variety.
%\en
%\en
%\end{lemm}
%A proof can be found in \cite{Hart}.
%\begin{defi}
%A formal scheme is a functor $X : \trm{CRng} \longrightarrow \trm{Set}$, that is a small filtered colimit of affine schemes. Its category is denoted by $\trm{FSch}$ - its morphisms are natural transformations. Given a scheme $S$, we define the formal schemes over $S$ as follows, all objects are morphisms $X \longrightarrow S$ of formal schemes and as morphisms between $X \longrightarrow S$ and $Y \longrightarrow S$ all morphisms $X \longrightarrow Y$, such that
%$$\xymatrix{
%X \ar[r] \ar[rd]& Y\ar[d]\\
%&S\\
%}$$
%commutes. We denote the formal schemes over $S$ with $\trm{FSch}_S$.
%\index{Index}{scheme!formal scheme}
%\end{defi}
%\bmk According to \cite{Strickl} a formal scheme to $\mathcal{X}$ is as follows: given a small filtered category $\mathcal{J}$ and a functor $i \longmapsto X_{i}$ from $\mathcal{J}$ to $\mathcal{X} = \{\trm{CRng}, \trm{Set}\}$ such that
%$$X = \lim_{\substack{\longrightarrow\\i}} X_{i} \in \mathcal{X}$$
%or equivalently $X(R) = \lim_{\substack{\longrightarrow\\i}} X_{i}(R)$ for all $R$.
%\bsp Two examples to illustrate the definition of formal schemes (following \cite{Strickl}):
%\bn
%\item Let $R$ be an object in $\trm{CRng}$ with unit and $N(R)$ denote its nilradical. The functors
%$$\hat{\mathbb{A}}_n = \left[R \longmapsto N(R)^n\right]$$
%are a prominent example.
%\item Let $X$ be some scheme and $Y = V(I)$ a closed subscheme then
%$$X_{\hat{Y}} := \lim_{\substack{\longrightarrow\\N}} V(I^N)$$
%defines a formal scheme.
%\en
%%An scheme is simply a ringed space $(Z,\mathcal{O}_Z)$ such that for ... open cover $\mathcal{U}$ the restriction $\mathcal{O}_Z\mid_U$ is an affine scheme for all $U \in \mathcal{U}$.
%\begin{defi}
%A group scheme is a scheme $X = X(G)$ which has a group structure $G$ as well, i.e. a functor $m : X \times X \longrightarrow X$. A principal homogeneous space - or torsor - for a given group (group scheme) $G$ is a pair $(G,X)$, with $X$ some set, such that the map
%$$\alpha : (X, G) \longrightarrow (X, X),\ (x, g) \longmapsto (x, g x)$$
%is a bijection.
%\index{Index}{group scheme}
%\index{Index}{space!principal homogeneous}
%\index{Index}{torsor}
%\end{defi}
%\begin{koro}
%The following statements are equivalent.
%\bn
%\item $(G, X)$ is a principle homogeneous space.
%\item The action $\alpha : G \times X \longrightarrow X$ is transitive on $X$ and its stabilizer $\trm{Stab}_G(X)$ is trivial.
%\en
%\end{koro} \bws The proof is simply the application of the above definitions.
%
%%Again, our notation, not incidentally, resembles the notation of Hopf-algebras. Now an important classification of affine groups:
%\begin{defi}
%An affine group is a representable functor $G : \trm{Alg} \longrightarrow \trm{Set}$ with natural transformation $\mu : G \times G \longrightarrow G$ such that $(G(A),\mu(A),e(A))$ is a group for all $A \in \trm{Alg}$. $G$ is called affine algebraic group if $G$ is represented by a finite presented algebra $A$.
%\index{Index}{affine group}
%\index{Index}{affine algebraic group}
%\end{defi}
%Formal group laws will be discussed in a later section.
%\newpage
\subsection{Topological basics}
We take some notions from topology as given.
\subsubsection{Basis and neighborhood basis}
\begin{defi}
Let $(X, \tau)$ be a topological space.
\bd
%\item[Filter] A subset $F \subset \tau$ is called a filter if:
%\bn
%\item for all $A, B \in F$, we have $A \cap B \in F$,
%\item the empty set $\emptyset$ is not in $F$,
%\item if $A \in F$ and $A \subset B$, then $B \in F$ for all $B \subset X$.
%\en
\item[Basis] A subset $\beta \subset \tau$ is called an open neighborhood basis for $x \in X$ if:% ein topologischer Raum. Eine Teilmenge $\beta \subset \tau$ hei\ss{}t Basis von $x \in X$, oder auch Umgebungsbasis von $x$, falls
\bn
\item for every open $V \subset X$ with $x \in V$ there is an $U \in \beta$ open in $X$ such that $x \in U \subset V$.%f\"ur alle offene $V \subset X$ offen, mit $x \in V$ gibt es ein $U \in \beta$ mit $x \in U \subset V$ offen,
\item for all $U, V \in \beta$ we have $x \in U \cap V$.%f\"ur $U_1, U_2 \in \beta$ ist $x \in U_1 \cap U_2$.
\en
A subset $\beta \subset \tau$ is called (open) basis of $X$ if
%Eine Teilmenge $\beta \subset \tau$ hei\ss{}t Basis von $X$, falls
\bn
\item all $U \in \beta$ are open in $X$,
\item every open subset of $X$ is a union of elements in $\beta$.%jede offene Teilmenge $V \in \tau$ eine Vereinigung von $U \in \beta$ ist.
\en
\ed
A neighborhood basis $\beta$ of $x \in X$ is called a fundamental basis of $x \in X$ if every neighborhood $x \in U$ is a finite intersection of elements in $\beta$.%Eine Umgebungsbasis $\beta$ von $x \in X$ eines topologischen Raums $(X, \tau)$ hei\ss{}t  Fundamentalbasis, falls jede offene Umgebung $V$ von $x$ einen endlichen Durchschnitt von $U_i \in \beta$ enth\"alt.
\index{Index}{filter}
\index{Index}{basis}
\index{Index}{basis!neighborhood}
\index{Index}{basis!fundamental}
\end{defi}
\subsubsection{Linear topological rings}
Topological rings $R$ are topological spaces $(R,\tau_R)$ such that addition and multiplication are continuous wrt. the product topology:
$$+, \cdot \in C(R\times R,R).$$%Topologische Ringe sind Ringe $R$, mit einer Topologie $\tau$, sodass
%$$+ : R \times R \longrightarrow R,\ \cdot : R \times R \longrightarrow R$$
%stetige Abbildungen in der Produkttopologie $\tau_{R \times R}$ sind. Die Menge der Ideale in $R$ definiert eine Umgebungsbasis der $0 \in R$. %Im Allgemeinen ist diese Basis abgeschlossen in $R$,
\index{Index}{topological ring}
The set of ideals in $R$ is neighborhood basis of $0 \in R$. In general, the elements generated by union are not ideals but are contained in larger ideals (in general $I \cup J$ is not an ideal but is contained in its sum $I + J$).
\begin{defi}[Linear topological rings]
A topological ring $R$ is called linear if there is a fundamental neighborhood basis $\beta$ of $0 \in R$.
%Ein topologischer Ring $R$ hei\ss{}t linear, falls es eine fundamentale Umgebungsbasis von $0 \in R$ gibt.
\index{Index}{topological ring!linear}
\end{defi}
Is $R$ a linear topological ring with fundamental neigborhood basis $\beta(0)$ then any open neighborhood of zero contains at least one ideal, trivially $(0)$, since the intersection of ideals is again an ideal. Let $\{I_i \in \eta(0) : i \in \mathcal{I}\}$ be a system of ideals then the union:
$$\bigcup_{i \in \mathcal{I}'} I_i,\ \forall \mathcal{I}' \subset \mathcal{I},\ |\mathcal{I}'| < \infty$$
is a subset of $R$ containing each $I_i$. Consequently, $\beta(0)$ is an open neighborhood basis for zero. In addition, elements in $\beta(0)$ are also intersection stable. Hence, we get a clopen basis.
%Ist nun $R$ ein linear topologischer Ring, mit fundamentaler Umgebungsbasis $\beta(0)$, dann enth\"alt jede offene Umgebung der Null mindestens ein Ideal $I \in \beta(0)$, trivialerweise mindestens $(0)$, da der Schnitt beliebiger Ideale wieder ein Ideal ergibt. Sei $\{I_i \in \beta(0) : i \in \mathcal{I}\}$ ein System von Idealen, dann ist deren Vereinigung eine Teilmenge von $R$, die alle Ideale der Form
%$$\bigcap_{i \in \mathcal{I}'} I_i, \forall \mathcal{I}' \subset \mathcal{I}\ \trm{und}\ |\mathcal{I}'| < \infty$$
%enth\"alt und definiert damit eine offene Umgebung der Null. Andererseits sind die Schnitte der Ideale auch Umgebungen der Null, d.h. damit sind alle Ideale \textit{clopen} in $R$.
\begin{defi}
Let $R$ be a linear topological ring with fundamental neighborhood basis of zero: $\beta(0)$. $\hat{R}$ is called the completion of $R$, if
%Sei $R$ ein linear topologischer Ring mit Fundamental-UB $\beta(0)$. Ein Ring $\hat{R}$ hei\ss{}t Vervollst\"andigung von $R$, falls
$$\hat{R} \simeq \lim_{\substack{\longleftarrow\\I \in \beta(0)}} R/I,$$
i.e. the profinite limit of $(R/I)_{I \in \beta(0)}$, with ring homomorphisms $R/I \longrightarrow R/J$ for all $I \subset J$ and $\beta(0)$ ordered by wrt. inclusion.
%ist, d.h. der pro-endliche (oder inverse) Limes von  $(R/I)_{I \in \beta(0)}$, mit Ringmorphismen $R/I \longrightarrow R/J$ f\"ur alle $I \subset J$ und $\beta(0)$ angeordnet bzgl. Inklusion.
\index{Index}{topological ring!completion of}
\end{defi}
\bmk In terms of co/limits, the completion of a linear topological ring $R$ with neighborhood basis of zero is simply the limit.
\bsp Consider $(0) \neq \idealp = (p) \subset \zz$ and $S := \prod_{n \geq 1} \zz/\idealp^n$ with component-wise ring operations and inclusion map:
$$\zz \longrightarrow S, z \longmapsto (z \mod p^n)_{n\geq1}.$$
Furthermore, there are projections $\pi_{ij} : \zz/\idealp^i \longrightarrow \zz/\idealp^j, x \mod p^i \longmapsto x \mod p^j$ for all $i \geq j$. Clearly, $\beta(0) := \{\idealp^n : n \geq 1\}$ defines a neighborhood system of zero (all intersections contain the zero ideal) and via inclusion a partial order on $\beta(0)$ (in this cases total).
%\subsection{Commutative algebra}
%
%\subsubsection{Invariant and equivariant rings}
%Given a finite family of polynomials $\mathcal{F} \subset k[x]$, its invariant group is defined to be
%the set of all $k$-linear maps $\varphi : k[x] \longrightarrow k[x]$ such that $f \circ \varphi = f$ for all $f \in \mathcal{F}$. Conversely:
%\begin{defi}
%the invariant ring of a given group $G$ is the set of all polynomials in $k[x]$ such that $f g = f$ for all $g \in G$. This ring is denoted by
%$$k[x]^G.$$
%\end{defi}
%Since the invariant group only acts on the monomials of degree $\geq 1$ we have $g\mid_{k.1_{k[x]}} = id_{k.1_{k[x]}}$. For $n \in \nz$ the most prominent example is the ring of symmetric polynomials:
%$$k[x]^{S_n} = k[s_1,\ldots,s_n],$$
%with $s_i = \sum_{\substack{\alpha \in \nz_0^n\\|\alpha| = i}} x^\alpha$ the symmetric polynomials. They are of utmost importance in classical Galois theory.
%\begin{defi}
%For a given finite family of polynomials $\mathcal{F}$ we call the set of all $k$-linear maps $\varphi$ commuting with all $f \in \mathcal{F}$ the equivariant group of $\mathcal{F}$. Conversely, for a given group $G$ the set of all polynomials in $k[x]$ commuting with all $g \in G$ is called the equivariant ring of $G$ and is denoted by
%$$k[x]^G_G.$$
%\end{defi}
%\bsp Consider $\varphi = [x^i \longmapsto (-x)^i] \in \trm{End}(k[x])$, with $n = 1$ for all $i \geq 0$. The invariant ring is obviously $k[x^2]$.

%Although rather complicated, we simply recall our definition of linear topological rings $(R,+,\cdot,1,\tau)$, where the completion, in the above sense, is a colimit, indexed via some neighborhood basis $\beta(0)$ of zero (also filtered, as all ideals contain zero ideal):
%\bd
%\item[Index category] all neighborhood basis of zero:
%$$\mathcal{J} \subset \trm{Mod}_R(R) \subset \trm{Mod}_R$$
%clearly form a sub-category of all $R$-submodules.
%\item[Diagram to $F$]
%\item[Co-cone to $F$]
%\ed
%\newpage
%\subsection{Formal group laws}
%Here we follow \cite{Strickl}.
%\begin{defi}
%Let $C$ be a commutative ring and $n \in \nz$. An $n$-dimensional formal group law $F$ is a formal power series in $C[[x_1,\ldots,x_n,y_1,\ldots,y_n]]^n := C[[x,y]]^n$ such that
%\bn
%\item $F(0,x) = x \in C[[x]]^n$,
%\item $F(x,y) = F(y,x) \in C[[x,y]]^n$,
%\item $F(F(x,y),z) = F(x,F(y,z)) \in C[[x,y,z]]^n$ and
%\item there is an map $m$ on $C[[x]]$ such that $F(m(x),x) = 0$.
%\en
%\index{Index}{formal group law}
%\end{defi}
%\bsp Some examples taken from Strickland 2011:
%\bn
%\item the map $F(x,y) = x + y \in C[[x,y]]^n$ is called the $n$-dimensional formal additive group law, with $m = [x \longmapsto -x]$,
%\item let $c \in C$, the map $F(x,y) = x + y + c x y \in C[[x,y]]$ is a 1-dimensional formal group law, with $m = [x \longmapsto -x/(1 + c x)]$ (recall if the constant term of a power series is a unit, then the formal power series itself is a unit - hence $1 + c x$ is a unit),
%\item if $c \in C^\times$, then $F(x,y) = \frac{x + y}{1 + \frac{xy}{c^2}}$ is a formal group law, with
%$m$ as in the first example. It is well-known in relativistic geometry - the so called Lorenz-FGL.
%\en
%\begin{lemm}
%Let $F$ be an $n$-dimensional formal group law over some commutative ring $R$. There exists a $\Psi \in R[[x_1,\ldots,x_n]]^n$ such that $\Psi(0) = 0$ and
%$$F(u, \Psi(u)) = F(\Psi(u),u) = 0.$$
%\end{lemm}
%A proof is given in \cite{Serr}.
%\bmk For any $n$-dimensional formal group law $F \in R[[x,y]]^n$, we define a formal group scheme over $\trm{Spec}(R)$ via
%$$\mathbf{F} : \trm{CAlg}_R \longrightarrow \trm{Grp},\ A \longmapsto N(A)^n$$
%where the operation is given via $(u,v) \longmapsto F(u,v)$.


\chapter{Products in dagger categories with complete ordered Hom-sets}

\fxwarning{This is a rough draft. It is not yet checked for errors.}

\begin{note}
  What I previously denoted $\prod F$ is now denoted $\bigodot^{\text{proj}}_{\sqcap} F$ (and
  likewise for $\mathord{\coprod}$). The other draft chapters referring to
  this chapter may be not yet updated.
\end{note}

\begin{prop}
  ~
  \fxwarning{Should we move this to volume 1?}
  \begin{enumerate}
    \item Every entirely defined monovalued morphism is metamonovalued and metacomplete.
    \item Every surjective injective morphism is metainjective and co-metacomplete.
  \end{enumerate}

\end{prop}

\begin{proof}
Let's prove the first (the second follows from duality):

Let $f$ be an entirely defined monovalued morphism.

$\left( \bigsqcap G \right) \circ f \sqsubseteq \bigsqcap_{g \in G} (g \circ
f)$ by monotonicity of composition.

Using the fact that $f$ is monovalued and entirely defined:

$\left( \bigsqcap_{g \in G} (g \circ f) \right) \circ f^{\dagger} \sqsubseteq
\bigsqcap_{g \in G} (g \circ f \circ f^{\dagger}) \sqsubseteq \bigsqcap G$;

$\bigsqcap_{g \in G} (g \circ f) \sqsubseteq \left( \bigsqcap_{g \in G} (g
\circ f) \right) \circ f^{\dagger} \circ f \sqsubseteq \left( \bigsqcap G
\right) \circ f$.

So $\left( \bigsqcap G \right) \circ f = \bigsqcap_{g \in G} (g \circ f)$.

Let $f$ be a entirely defined monovalued morphism.

$f \circ \left( \bigsqcup G \right) \sqsupseteq \bigsqcup_{g \in G} (f \circ
g)$ by monotonicity of composition.

Using the fact that $f$ is entirely defined and monovalued:

$f^{\dagger} \circ \left( \bigsqcup_{g \in G} (f \circ g) \right) \sqsupseteq
\bigsqcup_{g \in G} (f^{\dagger} \circ f \circ g) \sqsupseteq \bigsqcap G$;

$\bigsqcup_{g \in G} (f \circ g) \sqsupseteq f \circ f^{\dagger} \circ
\bigsqcup_{g \in G} (f \circ g) \sqsupseteq f \circ \left( \bigsqcup G
\right)$.

So $f \circ \left( \bigsqcup G \right) = \bigsqcup_{g \in G} (f \circ g)$.
\end{proof}

\section{General product in partially ordered dagger category}

To understand the below better, you can restrict your imagination to the case
when $\mathcal{C}$ is the category $\mathbf{Rel}$.

\subsection{Products}

Let $\mathcal{C}$ be a dagger category, each Hom-set of which is a complete
lattice (having order agreed with the dagger).

We will designate some morphisms as \emph{principal} and require that
principal morphisms are both metacomplete and co-metacomplete. (For a
particular example of the category $\mathbf{Rel}$, all morphisms are
considered principal.)

Let $\prod^{(Q)} X$ be an object for each indexed family $X$ of objects.

Let $\pi$ be a partial function mapping elements $X \in \dom \pi$ (which
consists of small indexed families of objects of $\mathcal{C}$) to indexed
families $\prod^{(Q)} X \rightarrow X_i$ of principal morphisms (called
\emph{projections}) for every $i \in \dom X$.

We will denote particular projections as $\pi^X_i$.

\begin{defn}
  If $\pi$ is defined at $\lambda
  j \in n : \Src F_j$ and $\lambda j \in n : \Dst F_j$, then
  \[ \bigodot^{\text{proj}}_{\sqcap} F = \bigsqcap_{i \in \dom F} ((\pi^{\Dst \circ
   F_{}}_i)^{\dagger} \circ F_i \circ \pi^{\Src \circ F}_i) . \]
\end{defn}

If $F_i:Y\to X_i$ for all~$i$ for some object~$Y$:
\[
  \prod^{\text{proj}}_{\sqcap} F = \bigsqcap_{i \in \dom F} ((\pi^{\Dst\circ F}_i)^{\dagger} \circ F_i) .
\]

If $F_i:X_i\to Y$ for all~$i$ for some object~$Y$:
\[
  \coprod^{\text{proj}}_{\sqcup} F = \bigsqcup_{i \in \dom F} (F_i \circ \pi^{\Src\circ F}_i) .
\]

\begin{rem}
  \begin{align*}
  (\pi^{\Dst \circ
   F_{}}_i)^{\dagger} \circ F_i \circ \pi^{\Src \circ F}_i \in \Hom \left(
  \prod^{(Q)}_{j \in n} \Src F_j , \prod^{(Q)}_{j \in n} \Dst F_j
  \right);\\
  (\pi^{\Dst \circ
   F_{}}_i)^{\dagger} \circ F_i \in \Hom \left(
  Y , \prod^{(Q)}_{j \in n} \Dst F_j
  \right);\\
F_i \circ \pi^{\Src \circ F}_i \in \Hom \left(
  \prod^{(Q)}_{j \in n} \Src F_j , Y \right).
  \end{align*}
  are properly defined and have the same sources and destination
  (whenever $i \in \dom F$ is), thus the meet in the formulas is
  properly defined.
\end{rem}

\begin{rem}
  Thus, for example,
\begin{multline*}
  F_0 \odot^{\text{proj}}_{\sqcap} F_1 = ((\pi^{(\Dst F_0 , \Dst
  F_1)}_0)^{\dagger} \circ F_0 \circ \pi^{(\Src F_0 , \Src
  F_1)}_0) \sqcap\\ ((\pi^{(\Dst F_0 , \Dst F_1)}_1)^{\dagger}
  \circ F_1 \circ \pi^{(\Src F_0 , \Src F_1)}_1)
\end{multline*}
  that is product is defined by a pure algebraic formula.
\end{rem}

\begin{lem}
$F\mapsto\bigsqcup_{i\in\dom F}\phi(F_i)$ for ordinal variadic~$F$ is infinitely
associative for any function~$\phi$ defined on all values~$F_i$.
\end{lem}

\begin{proof}
I will denote $t(F)=\bigsqcup_{i\in\dom F}\phi(F_i)$. We need
to prove:
\begin{widedisorder}
\item[$t(t\circ S)=t(\concat S)$]
$t(\concat S)=
\bigsqcup_{i\in\dom(\concat S)}\phi((\concat S)_i)=
\bigsqcup_{i\in\dom(\uncurry(S))}\phi((\uncurry(S))_i)$.

$t(t\circ S)=
\bigsqcup_{i\in\dom S}\phi(tS_i)=
\bigsqcup_{i\in\dom S}\bigsqcup_{j\in\dom F}\phi((S_i)_j)$.

So, obviously $t(t\circ S)=t(\concat S)$.

\item[$t(\llbracket x\rrbracket)=x$] Obvious.
\end{widedisorder}
\end{proof}

\begin{cor}
All three above defined products are infinitely associative
for ordinal variadic families~$F$.
\end{cor}

\begin{proof}
An obvious consequence taking into account duality.
\end{proof}

\begin{prop}
  $\bigodot^{\text{proj}}_{\sqcap} F = \max \setcond{ \Phi \in \Hom \left( \prod^{(Q)}_{j \in
  n} \Src F_j , \prod^{(Q)}_{j \in n} \Dst F_j \right)
  }{ \forall i \in n : \Phi \sqsubseteq (\pi^{\Dst\circ F}_i)^{\dagger} \circ F_i \circ \pi^{\Src\circ F}_i }$.
\end{prop}

\begin{proof}
  By definition of meet on a complete lattice.
\end{proof}

\begin{thm}
  Let $\pi^X_i$ be metamonovalued morphisms. Let~$I$ be an index set. If $S \in \subsets \prod_{i\in I}\Hom(A_i, B_i)$ for some objects $A_i$, $B_i$ where $i\in I$ then
  \begin{gather*}
    \bigsqcap_{f\in S} \bigodot^{\operatorname{proj}}_{\sqcap}f =
    \bigodot_{i\in I}\bigsqcap_{f\in S}f_i=
    \bigodot_{i\in I}\bigsqcap\Pr_i S;\\
    \bigsqcap_{f\in S} \prod^{\operatorname{proj}}_{\sqcap} f =
    \prod_{i\in I}\bigsqcap_{f\in S}f_i=
    \prod_{i\in I}\bigsqcap\Pr_i S;\\
    \bigsqcap_{f\in S} \coprod^{\operatorname{proj}}_{\sqcap} f =
    \coprod_{i\in I}\bigsqcap_{f\in S}f_i=
    \coprod_{i\in I}\bigsqcap\Pr_i S.
  \end{gather*}
\end{thm}

\begin{proof}
Let us consider for example the first formula (two others
are similar):
  \begin{align*}
  \bigsqcap_{f\in S} \bigodot^{\operatorname{proj}}_{\sqcap} & = \\
  \bigsqcap_{f\in S} \bigsqcap_{i\in I}((\pi^{\Dst\circ f_i}_i)^{\dagger} \circ f_i \circ \pi^{\Src\circ f_i}_i) & = \\
  \bigsqcap_{i\in I}\bigsqcap_{f\in S}((\pi^{\Dst\circ f_i}_i)^{\dagger} \circ f_i \circ \pi^{\Src\circ f_i}_i) & = \\
  \bigsqcap_{i\in I}((\pi^{\Dst\circ f_i}_i)^{\dagger} \circ \bigsqcap_{f\in S}f_i \circ \pi^{\Src\circ f_i}_i) & = \\
  \bigodot_{i\in I}\bigsqcap_{f\in S}f_i & = \\
  \bigodot_{i\in I}\bigsqcap\Pr_i S.
  \end{align*}
\end{proof}

\begin{cor}
~
\begin{enumerate}
\item
  $(a_0 \odot^{\operatorname{proj}}_{\sqcap} b_0) \sqcap (a_1 \odot^{\operatorname{proj}}_{\sqcap} b_1) = (a_0 \sqcap a_1)
  \odot^{\operatorname{proj}}_{\sqcap} (b_0 \sqcap b_1)$;
\item
  $(a_0 \times^{\operatorname{proj}}_{\sqcap} b_0) \sqcap (a_1 \times^{\operatorname{proj}}_{\sqcap} b_1) = (a_0 \sqcap a_1)
\times^{\operatorname{proj}}_{\sqcap} (b_0 \sqcap b_1)$;
\item
  $(a_0 \amalg^{\operatorname{proj}}_{\sqcap} b_0) \sqcap (a_1 \amalg^{\operatorname{proj}}_{\sqcap} b_1) = (a_0 \sqcap a_1)
  \amalg^{\operatorname{proj}}_{\sqcap} (b_0 \sqcap b_1)$;
\end{enumerate}
\end{cor}

\subsection{Product for endomorphisms}

Let $F$ is an indexed family of endomorphisms of $\mathcal{C}$.

I will denote $\Ob f$ the object (source and destination) of an
endomorphism $f$.

Let also $\pi^X_i$ be a monovalued entirely defined morphism (for each $i \in
\dom F$).

Then
\begin{gather*}
\bigodot^{\text{proj}}_{\sqcap} F = \bigsqcap_{i \in \dom F} ((\pi^{\lambda j \in n :
\Ob F_j}_i)^{\dagger} \circ F_i \circ \pi^{\lambda j \in n : \Ob
F_j}_i);\\
\bigodot^{\text{proj}}_{\sqcup} F = \bigsqcup_{i \in \dom F} ((\pi^{\lambda j \in n :
\Ob F_j}_i)^{\dagger} \circ F_i \circ \pi^{\lambda j \in n : \Ob
F_j}_i)
\end{gather*}
(if $\pi$ is defined at $\lambda j \in n : \Ob F_j$).

Abbreviate $\pi_i = \pi^{\lambda j \in n : \Ob F_j}_i$.

So
\begin{gather*}
\bigodot^{\text{proj}}_{\sqcap} F = \bigsqcap_{i \in \dom F} ((\pi_i)^{\dagger} \circ
F_i \circ \pi_i);\\
\bigodot^{\text{proj}}_{\sqcup} F = \bigsqcup_{i \in \dom F} ((\pi_i)^{\dagger} \circ
F_i \circ \pi_i).
\end{gather*}

$\bigodot^{\text{proj}}_{\sqcap} F = \max \setcond{ \Phi \in \End \left( \prod^{(Q)}_{j \in n}
\Ob F_j \right) }{ \forall i \in n : \Phi
\sqsubseteq (\pi_i)^{\dagger} \circ F_i \circ \pi_i }$.

$\bigodot^{\text{proj}}_{\sqcup} F = \min \setcond{ \Phi \in \End \left( \prod^{(Q)}_{j \in n}
\Ob F_j \right) }{ \forall i \in n : \Phi
\sqsupseteq (\pi_i)^{\dagger} \circ F_i \circ \pi_i }$.

Taking into account that $\pi_i$ is a monovalued entirely defined morphism, we
get:

\begin{obvious}
$\bigodot^{\text{proj}}_{\sqcap} F = \max \setcond{ \Phi \in \End \left( \prod^{(Q)}_{j \in
n} \Ob F_j \right) }{ \forall i \in n : \pi_i
\in \mathrm{C} (\Phi , F_i) }$.
\end{obvious}

\begin{obvious}
$\bigodot^{\text{proj}}_{\sqcup} F = \min \setcond{ \Phi \in \End \left( \prod^{(Q)}_{j \in
n} \Ob F_j \right) }{ \forall i \in n : \pi_i
\in \mathrm{C}_{\ast} (\Phi , F_i) }$.
\end{obvious}

\begin{rem}
  The above formulas may allow to define the product for non-dagger categories
  (but only for endomorphisms). In this writing I don't introduce a notation
  for this, however.
\end{rem}

\begin{cor}
  $\pi_i \in \mathrm{C} \left( \bigodot^{\text{proj}}_{\sqcap} F , F_i \right)$ and
  $\pi_i \in \mathrm{C}_{\ast} \left( \bigodot^{\text{proj}}_{\sqcup} F , F_i \right)$
  for every $i \in \dom F$.
\end{cor}

\subsection{Category of continuous morphisms}

\begin{defn}
  The category $\cont (\mathcal{C})$ is defined as follows:
  \begin{itemize}
    \item Objects are endomorphisms of the category $\mathcal{C}$.

    \item Morphisms are triples $(f , a , b)$ where $a$ and $b$ are objects
    and $f : \Ob a \rightarrow \Ob b$ is an entirely defined
    monovalue principal morphism of the category $\mathcal{C}$ such that $f
    \in \mathrm{C} (a , b)$ (in other words, $f \circ a \sqsubseteq b \circ
    f$).

    \item Composition of morphisms is defined by the formula $(g , b , c)
    \circ (f , a , b) = (g \circ f , a , c)$.

    \item Identity morphisms are $(a , a , 1^{\mathcal{C}}_a)$.
  \end{itemize}
\end{defn}

It is really a category:

\begin{proof}
  We need to prove that: composition of morphisms is a morphism, composition
  is associative, and identity morphisms can be canceled on the left and on
  the right.

  That composition of morphisms is a morphism by properties of generalized
  continuity.

  That composition is associative is obvious.

  That identity morphisms can be canceled on the left and on the right is
  obvious.
\end{proof}

\begin{rem}
  The ``physical'' meaning of this category is:
  \begin{itemize}
    \item Objects (endomorphisms of $\mathcal{C}$) are spaces.

    \item Morphisms are continuous functions between spaces.

    \item $f \circ a \sqsubseteq b \circ f$ intuitively means that $f$
    combined with an infinitely small is less than infinitely small combined
    with $f$ (that is $f$ is continuous).
  \end{itemize}
\end{rem}

\begin{defn}
  $\pi^{\cont (\mathcal{C})}_i = \left( \bigodot^{\text{proj}}_{\sqcap} F , F_i ,
  \pi_i \right)$.
\end{defn}

\begin{prop}
  $\pi_i$ are continuous, that is $\pi^{\cont}
  (\mathcal{C})_i$ are morphisms.
\end{prop}

\begin{proof}
  We need to prove $\pi_i \in \mathrm{C} \left( \bigodot^{\text{proj}}_{\sqcap} F , F_i \right)$
  but that was proved above.
\end{proof}

Let further $\mathcal{C}$ have sets as objects and $\prod^{(Q)}X=\prod X$ for an indexed family~$X$ of sets and $\pi_i = \Pr_i$ (for $i \in \dom F$).

\begin{lem}
  $f \in \Hom_{\cont (\mathcal{C})} \left( Y ,
  \bigodot^{\text{proj}}_{\sqcap} F \right)$ is continuous iff all $\pi_i \circ f$ are continuous.
\end{lem}

\begin{proof}
  ~
  \begin{description}
    \item[$\Rightarrow$] Let $f \in \Hom_{\cont
    (\mathcal{C})} \left( Y , \bigodot^{\text{proj}}_{\sqcap} F \right)$. Then $f \circ Y
    \sqsubseteq \left( \bigodot^{\text{proj}}_{\sqcap} F \right) \circ f$; $\pi_i \circ f \circ Y
    \sqsubseteq \pi_i \circ \left( \bigodot^{\text{proj}}_{\sqcap} F \right) \circ f$; $\pi_i
    \circ f \circ Y \sqsubseteq \left( \bigodot^{\text{proj}}_{\sqcap} F \right) \circ \pi_i \circ
    f$. Thus $\pi_i \circ f$ is continuous.

    \item[$\Leftarrow$] Let all $\pi_i \circ f$ be continuous. Then
    $\pi^{\cont (\mathcal{C})}_i \circ f \in
    \Hom_{\cont (\mathcal{C})} (Y , F_i)$;
    $\pi^{\cont (\mathcal{C})}_i \circ f \circ Y \sqsubseteq
    F_i \circ \pi^{\cont (\mathcal{C})}_i \circ f$. We need
    to prove $Y \sqsubseteq f^{\dagger} \circ \left( \bigodot^{\text{proj}}_{\sqcap} F \right)
    \circ f$ that is
    \[ Y \sqsubseteq f^{\dagger} \circ \bigsqcap_{i \in n} ((\pi_i)^{\dagger}
       \circ F_i \circ \pi_i) \circ f \]
    for what is enough (because $f$ is metamonovalued)
    \[ Y \sqsubseteq \bigsqcap_{i \in n} (f^{\dagger} \circ (\pi_i)^{\dagger}
       \circ F_i \circ \pi_i \circ f) \]
    what follows from $Y \sqsubseteq \bigsqcap_{i \in n} (f^{\dagger} \circ
    (\pi_i)^{\dagger} \circ \pi_i \circ f \circ Y)$ what is obvious.
  \end{description}
\end{proof}

\begin{thm}\label{cont-pr-pr}
$\prod^{\text{proj}}_{\sqcap}$ together with $\pi$ is a categorical product in the category $\cont (\mathcal{C})$.
\end{thm}

\begin{proof}
Check
\url{http://math.stackexchange.com/questions/102632/how-to-check-whether-it-is-a-direct-product/102677\#102677}

I will denote $(\prod f)x=\prod_{i\in\dom f}f_i x$ for an
indexed family~$f$ of functions.

We need to prove:
\begin{enumerate}
\item $\pi_k\circ\prod f=f_k$;
\item $\prod_{i\in\dom f}(\pi_i\circ f)=f$.
\end{enumerate}
But it follows from the fact that $\pi_i=\Pr_i$.
\end{proof}

\section{On duality}

We will consider duality where both the category $\mathcal{C}$ and orders on
Hom-sets are replaced with their dual. I will denote $A
\xleftrightarrow{\dual} B$ when two formulas $A$ and $B$ are dual with
this duality.

\begin{prop}
  $f \in \mathrm{C} (\mu, \nu) \xleftrightarrow{\dual} f^{\dagger}
  \in \mathrm{C} (\nu^{\dagger} , \mu^{\dagger})$.
\end{prop}

\begin{proof}
  $f \in \mathrm{C} (\mu, \nu) \Leftrightarrow f \circ \mu
  \sqsubseteq \nu \circ f \xleftrightarrow{\dual} \mu^{\dagger}
  \circ f^{\dagger} \sqsupseteq f^{\dagger} \circ^{\dagger} \nu^{- 1}
  \Leftrightarrow f^{\dagger} \in \mathrm{C} (\nu^{\dagger} ,
  \mu^{\dagger})$.
\end{proof}

$f \text{ is entirely defined} \Leftrightarrow f^{\dagger} \circ f \sqsupseteq
1_{\Src f} \xleftrightarrow{\dual} f^{\dagger} \circ f \sqsubseteq
1_{\Src f} \Leftrightarrow f \text{ is injective} \Leftrightarrow
f^{\dagger} \text{ is monovalued}$.

$f \text{ is monovalued} \Leftrightarrow f \circ f^{\dagger} \sqsubseteq
1_{\Dst f} \xleftrightarrow{\dual} f \circ f^{\dagger} \sqsupseteq
1_{\Dst f} \Leftrightarrow f \text{ is surjective} \Leftrightarrow
f^{\dagger} \text{ is entirely defined}$.

\section{Dual products}

The below is the dual of the above, proofs are omitted as they are dual.

Let $\coprod^{(Q)}X$ be an object for each indexed family~$X$ of objects.

I will denote \emph{coprincipal} morphisms~$f^{\dagger}$ where $f$~is principal. (Usually there is no distinction between
principal and coprincipal.)

Let $\iota$ be a partial function mapping elements $X\in\dom\iota$ (which consist of small indexed families of objects of~$\mathcal{C}$) to indexed families $X_i\to\coprod^{(Q)}X$ of coprincipal morphisms (called \emph{injections}) for every $i\in\dom X$.

We will denote particular morphisms as $\iota_i^X$.

If (but we won't assume this below) $\iota_i =
(\pi_i)^{\dagger}$. We have the above equivalent to $\pi_i$ being principal.

We will define $\bigodot^{\operatorname{inj}}$, $\prod^{\operatorname{inj}}$, $\coprod^{\operatorname{inj}}$
by analogy with $\operatorname{proj}$ counterparts replacing~$\pi$ by~$\iota^{\dagger}$.

We will also define $\bigodot^{\operatorname{proj}}_{\sqcup}$, $\bigodot^{\operatorname{inj}}_{\sqcup}$, etc.\ by replacing~$\bigsqcap$ by~$\bigsqcup$.

\subsection{Dual products for endomorphisms}

\begin{prop}
  $\bigodot^{\operatorname{inj}}_{\sqcup} F = \min \setcond{ \Phi \in \End \left( \coprod^{(Q)}_{j
  \in n} \Ob F_j \right) }{ \forall i \in n :
  \Phi \sqsupseteq \iota^{\lambda j \in n : \Src F_j}_i \circ
  F_i^{\dagger} \circ (\iota^{\lambda j \in n : \Dst F_j}_i)^{\dagger} }$.
\end{prop}

\begin{proof}
  By duality.
\end{proof}

Let $F$ be an indexed family of endomorphisms of $\mathcal{C}$.

\begin{defn}
  $\bigodot^{\operatorname{inj}}_{\sqcup} F = \bigsqcup_{i \in \dom F} (\iota^{\lambda j \in n :
  \Ob F_j}_i \circ F_i^{\dagger} \circ (\iota^{\lambda j \in n :
  \Ob F_j}_i)^{\dagger})$.
\end{defn}

Abbreviate $\iota_i = \iota^{\lambda j \in n : \Ob F_j}_i$.

So $\bigodot^{\operatorname{inj}}_{\sqcup} F = \bigsqcup_{i \in \dom F} (\iota_i \circ F_i^{\dagger}
\circ (\iota_i)^{\dagger})$.

$\bigodot^{\operatorname{inj}}_{\sqcup} F = \min \setcond{ \Phi \in \End \left( \coprod^{(Q)}_{j \in n}
\Ob F_j \right) }{ \forall i \in n : \Phi
\sqsupseteq \iota_i \circ F_i^{\dagger} \circ (\iota_i)^{\dagger} }$.

Taking into account that $\iota_i$ is a monovalued entirely defined morphism,
we get:

\begin{obvious}
$\coprod^{(L)} = \min \setcond{ \Phi \in \End \left( \coprod^{(Q)}_{j \in
n} \Ob F_j \right) }{ \forall i \in n : \iota_i
\in \mathrm{C} (F_i^{\dagger} , \Phi) }$.{\hspace*{\fill}}{\medskip}
\end{obvious}

\begin{cor}
  $\iota_i \in \mathrm{C} \left( F_i , \coprod^{(L)} F \right)$ for every $i
  \in \dom F$.
\end{cor}

\begin{rem}
The last two theorems don't require that our category is dagger. I omit the proof.
\end{rem}

\subsection{Category of continuous morphisms}

Let $\iota_i$ be canonical injections.

\begin{defn}
$\iota^{\cont(\mathcal{C})}_i = \left(F_i,\bigodot^{\operatorname{inj}}_{\sqcup}F,\iota_i\right)$.
\end{defn}

\begin{obvious}
$\iota_i$ are continuous that is $\iota^{\cont(\mathcal{C})}_i$ are morphisms.
\end{obvious}

\begin{thm}
$\coprod^{\operatorname{inj}}_{\sqcup}$ together with $\iota$ is a categorical coproduct in the category~$\cont(\mathcal{C})$.
\end{thm}

\begin{proof}
Dual to theorem~\ref{cont-pr-pr}.
\end{proof}

\section{Applying this to the theory of funcoids and reloids}

\subsection{Funcoids}

\begin{defn}
  $\mathbf{Fcd} \eqdef \cont \mathsf{FCD}$.
\end{defn}

Let $F$ be a family of endofuncoids.

\subsection{Reloids}

\begin{defn}
  $\mathbf{Rld} \eqdef \cont \mathsf{RLD}$.
\end{defn}

Let $F$ be a family of endoreloids.

It is trivial?? that for uniform spaces infimum product of reloids coincides
with product uniformilty.

\section{Initial and terminal objects}

\subsection{Of category~$\mathbf{Fcd}$}

Initial object of $\mathbf{Fcd}$ is the endofuncoid
$\bot^{\mathsf{FCD} (\emptyset , \emptyset)}$. It is
initial because it has precisely one morphism $o$ (the empty set considered as
a function) to any object $Y$. $o$ is a morphism because $o \circ
\bot^{\mathsf{FCD} (\emptyset , \emptyset)} \sqsubseteq Y
\circ o$.

\begin{prop}
  Terminal objects of $\mathbf{Fcd}$ are exactly
  $\uparrow^{\mathscr{F}} \{ \ast \} \times^{\mathsf{FCD}}
  \uparrow^{\mathscr{F}} \{ \ast \} = \uparrow^{\mathsf{FCD}} \{ (\ast
  , \ast) \}$ where $\ast$ is an arbitrary point.
\end{prop}

\begin{proof}
  In order for a function $f : X \rightarrow \uparrow^{\mathsf{FCD}} \{
  (\ast , \ast) \}$ be a morphism, it is required exactly $f \circ X
  \sqsubseteq \uparrow^{\mathsf{FCD}} \{ (\ast , \ast) \} \circ f$

  $f \circ X \sqsubseteq (f^{- 1} \circ \uparrow^{\mathsf{FCD}} \{
  (\ast , \ast) \})^{- 1}$; $f \circ X \sqsubseteq (\{ \ast \}
  \times^{\mathsf{FCD}} \langle f^{- 1} \rangle \{ \ast \})^{- 1}$; $f
  \circ X \sqsubseteq \langle f^{- 1} \rangle \{ \ast \}
  \times^{\mathsf{FCD}} \{ \ast \}$ what true exactly when $f$ is a
  constant function with the value $\ast$.
\end{proof}

If $n = \emptyset$ then; $\bigodot^{\text{proj}}_{\sqcap} \emptyset = \prod^{\text{proj}}_{\sqcap} \emptyset = \coprod^{\text{proj}}_{\sqcap} \emptyset = \max
\mathsf{FCD} (\{\emptyset\}, \{\emptyset\}) = \uparrow^{\mathscr{F}} \{ \emptyset \}
\times^{\mathsf{FCD}} \uparrow^{\mathscr{F}} \{ \emptyset \} =
\uparrow^{\mathsf{FCD}} \{ (\emptyset , \emptyset) \}$.

\subsection{Of category~$\mathbf{Rld}$}

Initial object of $\mathbf{Rld}$ is the endofuncoid
$\bot^{\mathsf{RLD} (\emptyset , \emptyset)}$. It is
initial because it has precisely one morphism $o$ (the empty set considered as
a function) to any object $Y$. $o$ is a morphism because $o \circ
\bot^{\mathsf{RLD} (\emptyset , \emptyset)} \emptyset \sqsubseteq Y
\circ o$.

\begin{prop}
  Terminal objects of $\mathbf{Rld}$ are exactly
  $\uparrow^{\mathscr{F}} \{ \ast \} \times^{\mathsf{RLD}}
  \uparrow^{\mathscr{F}} \{ \ast \} = \uparrow^{\mathsf{RLD}} \{ (\ast
  , \ast) \}$ where $\ast$ is an arbitrary point.
\end{prop}

\begin{proof}
  In order for a function $f : X \rightarrow \uparrow^{\mathsf{RLD}} \{
  (\ast , \ast) \}$ be a morphism, it is required exactly $f \circ X
  \sqsubseteq \uparrow^{\mathsf{RLD}} \{ (\ast , \ast) \} \circ f$

  $f \circ X \sqsubseteq (f^{- 1} \circ \uparrow^{\mathsf{RLD}} \{
  (\ast , \ast) \})^{- 1}$; $f \circ X \sqsubseteq (\{ \ast \}
  \times^{\mathsf{RLD}} \langle f^{- 1} \rangle \{ \ast \})^{- 1}$; $f
  \circ X \sqsubseteq \langle f^{- 1} \rangle \{ \ast \}
  \times^{\mathsf{RLD}} \{ \ast \}$ what true exactly when $f$ is a
  constant function with the value $\ast$.
\end{proof}

If $n = \emptyset$ then; $\bigodot^{\text{proj}}_{\sqcap} \emptyset = \prod^{\text{proj}}_{\sqcap} \emptyset = \coprod^{\text{proj}}_{\sqcap} \emptyset = \max
\mathsf{RLD} (\{\emptyset\}, \{\emptyset\}) = \uparrow^{\mathscr{F}} \{ \emptyset \}
\times^{\mathsf{RLD}} \uparrow^{\mathscr{F}} \{ \emptyset \} =
\uparrow^{\mathsf{RLD}} \{ (\emptyset , \emptyset) \}$.

\section{Canonical product and subatomic product}

\fxwarning{Confusion between filters on products and multireloids.}

\begin{prop}
  $\Pr^{\mathsf{RLD}}_i |_{\mathfrak{F} (Z)} = \langle \pi_i \rangle$
  for every index $i$ of a cartesian product $Z$.
\end{prop}

\begin{proof}
  If $\mathcal{X} \in \mathfrak{F} (Z)$ then $(\Pr^{\mathsf{RLD}}_i
  |_{\mathfrak{F} (Z)}) \mathcal{X} = \Pr^{\mathsf{RLD}}_i  \mathcal{X}
  = \bigsqcap^{\mathscr{F}} \rsupfun{\Pr_i} \mathcal{X} =
  \bigsqcap \langle \pi_i \rangle \up \mathcal{X} = \langle \pi_i
  \rangle \mathcal{X}$.
\end{proof}

\begin{prop}
  $\prod^{(A)} F = \bigsqcap_{i \in n} \left( \left( \pi^{\mathsf{FCD}
  \left( \prod_{i \in n} \Dst F \right)}_i \right)^{- 1} \circ F_i \circ
  \pi^{\mathsf{FCD} \left( \prod_{i \in n} \Src F \right)}_i
  \right)$.
\end{prop}

\begin{proof}
  $a \mathrel{\left[ \prod^{(A)} F \right]} b \Leftrightarrow \forall i \in
  \dom F : \Pr^{\mathsf{RLD}}_i a \mathrel{[F_i]}
  \Pr^{\mathsf{RLD}}_i b \Leftrightarrow \forall i \in \dom F :
  \left\langle \pi^{\mathsf{FCD} \left( \prod_{i \in n}
  \Dst F \right)}_i \right\rangle a \mathrel{[F_i]}
  \left\langle \pi^{\mathsf{FCD} \left( \prod_{i \in n} \Src F
  \right)}_i \right\rangle b \Leftrightarrow \forall i \in \dom F : a
  \mathrel{\left[ \left( \pi^{\mathsf{FCD} \left( \prod_{i \in n}
  \Dst F \right)}_i \right)^{- 1} \circ F_i \circ
  \pi^{\mathsf{FCD} \left( \prod_{i \in n} \Src F \right)}_i
  \right]} b \Leftrightarrow a \mathrel{\left[ \bigsqcap_{i \in n} \left(
  \left( \pi^{\mathsf{FCD} \left( \prod_{i \in n} \Dst F
  \right)}_i \right)^{- 1} \circ F_i \circ \pi^{\mathsf{FCD} \left(
  \prod_{i \in n} \Src F \right)}_i \right) \right]} b$ for ultrafilters
  $a$ and $b$.
\end{proof}

\begin{cor}
  $\bigodot^{\text{proj}}_{\sqcap} F = \prod^{(A)} F$ is $F$ is a small indexed family of
  funcoids.
\end{cor}

\section{Further plans}

Coordinate-wise continuity.

\section{Cartesian closedness}

We are not only to prove (or maybe disprove) that our categories are cartesian closed, but also to find (if any) explicit formulas for exponential transpose and evaluation.

''Definition'' A category is //cartesian closed// iff:
\begin{enumerate}
\item It has finite products.
\item For each objects $A$, $B$ is given an object $\operatorname{HOM} ( A , B)$ (//exponentiation//) and a morphism $\varepsilon_{A, B} : \operatorname{HOM} ( A , B) \times A \rightarrow B$.
\item For each morphism $f : Z \times A \rightarrow B$ there is given a morphism (//exponential transpose//) $\sim f : Z \rightarrow \operatorname{HOM} ( A , B)$.
\item $\varepsilon_{B,C} \circ ( \sim f \times 1_A) = f$ for $f : A \rightarrow B \times C$.
\item $\sim ( \varepsilon_{B,C} \circ ( g \times 1_A)) = g$ for $g : A \rightarrow \operatorname{HOM} ( B , C)$.
\end{enumerate}

We will also denote $f\mapsto (-f)$ the reverse of the bijection $f\mapsto (\sim f)$.

Our purpose is to prove (or disprove) that categories $\mathbf{Dig}$, $\mathbf{Fcd}$, and $\mathbf{Rld}$ are cartesian closed. Note that they have finite (and even infinite) products is already proved.

Alternative way to prove:
you can prove that the functor $-\times B$ is left adjoint to the exponentiation $-^B$ where the counit is given by the evaluation map.

\subsection{Definitions}

Categories $\mathbf{Dig}$, $\mathbf{Fcd}$, and $\mathbf{Rld}$ are respectively categories of:
\begin{enumerate}
\item discretely continuous maps between digraphs;
\item (proximally) continuous maps between endofuncoids;
\item (uniformly) continuous maps between endoreloids.
\end{enumerate}

''Definition'' //Digraph// is an endomorphism of the category $\mathbf{Rel}$.

For a digraph $A$ we denote $\operatorname{Ob} A$ the set of vertexes or $A$ and $\operatorname{GR} A$ the set of edges or $A$.

''Definition'' Category $\mathbf{Dig}$ of digraphs is the category whose objects are digraphs and morphisms are discretely continuous maps between digraphs. That is morphisms from a digraph $\mu$ to a digraph $\nu$ are functions (or more precisely morphisms of $\mathbf{Set}$) $f$ such that $f \circ \mu \sqsubseteq \nu \circ f$ (or equivalently $\mu \sqsubseteq f^{- 1} \circ \nu \circ f$ or equivalently $f \circ \mu \circ f^{- 1} \sqsubseteq \nu$).

''Remark'' Category of digraphs is sometimes defined in an other (non equivalent) way, allowing multiple edges between two given vertices.

\subsection{Conjectures}

\begin{conjecture}
  The categories $\mathbf{Fcd}$ and $\mathbf{Rld}$ are
  cartesian closed (actually two conjectures).
\end{conjecture}

\url{http://mathoverflow.net/questions/141615/how-to-prove-that-there-are-no-exponential-object-in-a-category}
suggests to investigate colimits to prove that there are no exponential
object.

Our purpose is to prove (or disprove) that categories $\mathbf{Dig}$, $\mathbf{Fcd}$, and $\mathbf{Rld}$ are cartesian closed. Note that they have finite (and even infinite) products is already proved.

Alternative way to prove:
you can prove that the functor $-\times B$ is left adjoint to the exponentiation $-^B$ where the counit is given by the evaluation map.

See \url{http://www.springer.com/us/book/9780387977102} for another way to prove Cartesian closedness.

\subsection{Category Dig is cartesian closed}

Category of digraphs is the simplest of our three categories and it is easy to demonstrate that it is cartesian closed. I demonstrate cartesian closedness of $\mathbf{Dig}$ mainly with the purpose to show a pattern similarly to which we may probably demonstrate our two other categories are cartesian closed.

Let $G$ and $H$ be digraphs:
\begin{itemize}
\item $\operatorname{Ob} \operatorname{HOM} ( G , H) = ( \operatorname{Ob} H)^{\operatorname{Ob} G}$;
\item $( f , g) \in \operatorname{GR} \operatorname{HOM} ( G , H) \Leftrightarrow \forall ( v , w) \in \operatorname{GR} G : ( f ( v) , g ( w)) \in \operatorname{GR} H$ for every $f, g \in \operatorname{Ob} \operatorname{HOM} ( G , H) = ( \operatorname{Ob} H)^{\operatorname{Ob} G}$;
\end{itemize}

$\operatorname{GR} 1_{\operatorname{HOM} ( B , C)} = \operatorname{id}_{\operatorname{Ob} \operatorname{HOM} ( B , C)} = \operatorname{id}_{( \operatorname{Ob} H)^{\operatorname{Ob} G}}$

Equivalently

$( f , g) \in \operatorname{GR} \operatorname{HOM} ( G , H) \Leftrightarrow \forall ( v , w) \in \operatorname{GR} G : g \circ \{ ( v , w) \} \circ f^{- 1} \subseteq \operatorname{GR} H$

$( f , g) \in \operatorname{GR} \operatorname{HOM} ( G , H) \Leftrightarrow g \circ ( \operatorname{GR} G) \circ f^{- 1} \subseteq \operatorname{GR} H$

$( f , g) \in \operatorname{GR} \operatorname{HOM} ( G , H) \Leftrightarrow \langle f \times^{( C)} g \rangle \operatorname{GR} G \subseteq \operatorname{GR} H$

The transposition (the isomorphism) is uncurrying.

$\sim f = \lambda a \in Z \lambda y \in A : f ( a , y)$ that is $( \sim f) ( a) ( y) = f ( a , y)$.

$( - f) ( a , y) = f ( a) ( y)$

If $f : A \times B \rightarrow C$ then $\sim f : A \rightarrow \operatorname{HOM} ( B , C)$

''Proposition'' Transposition and its inverse are morphisms of $\mathbf{Dig}$.

''Proof'' It follows from the equivalence $\sim f : A \rightarrow \operatorname{HOM} ( B , C) \Leftrightarrow \forall x, y : ( x A y \Rightarrow ( \sim f) x ( \operatorname{HOM} ( B , C))  ( \sim f) y) \Leftrightarrow \\ \forall x, y : ( x A y \Rightarrow \forall ( v , w) \in B : ( ( \sim f) x v , ( \sim f) y w) \in C) \Leftrightarrow \\ \forall x, y, v, w : ( x A y \wedge v B w \Rightarrow ( ( \sim f) x v , ( \sim f) y w) \in C) \Leftrightarrow \\ \forall x, y, v, w : ( ( x , v)  ( A \times B)  ( y , w) \Rightarrow ( f ( x , v) , f ( y , w)) \in C) \Leftrightarrow f : A \times B \rightarrow C$.

Evaluation $\varepsilon : \operatorname{HOM} ( G , H) \times G \rightarrow H$ is defined by the formula:

Then evaluation is $\varepsilon_{B, C} = \mathop{\sim} 1_{\operatorname{HOM}(B,C)}$.

So $\varepsilon_{B, C} ( p , q) = ( \mathop{\sim} 1_{\operatorname{HOM}(B,C)}) ( p , q) = 1_{\operatorname{HOM}(B,C)} ( p) ( q) = p ( q)$.

''Proposition'' Evaluation is a morphism of $\mathbf{Dig}$.

''Proof'' Because $\varepsilon_{B, C} ( p , q) = \mathop{\sim} 1_{\operatorname{HOM}(B,C)}$.

It remains to prove:
\begin{itemize}
\item $\varepsilon_{B, C} \circ ( \sim f \times 1_{A}) = f$ for $f : A \rightarrow B \times C$;
\item $\sim ( \varepsilon_{B, C} \circ ( g \times 1_{A})) = g$ for $g : A \rightarrow \operatorname{HOM} ( B , C)$.
\end{itemize}

''Proof'' $\varepsilon_{B, C} ( \sim f \times 1_{A}) ( a , p) = \varepsilon_{B, C} ( ( \sim f) a , p) = ( \sim f) a p = f ( a , p)$. So $\varepsilon_{B, C} \circ ( \sim f \times 1_{A}) = f$.

  $(\sim ( \varepsilon_{B, C} \circ ( g \times 1_{A}))) ( p) ( q) = ( \varepsilon_{B, C} \circ ( g \times 1_{A})) ( p , q) = \varepsilon_{B, C} ( g \times 1_{A}) ( p , q) = \varepsilon_{B, C} ( g p , q) = g ( p) ( q)$. So $\sim ( \varepsilon_{B, C} \circ ( g \times 1_{A})) = g$.

\subsection{New attempt}

We will take $\times^{(C)}$ as the product?? in the category $\mathbf{Fcd}$

\begin{prop}
$\rsupfun{\bigcup\rsupfun{\curry f}X}Y=
\rsupfun{f}(X\times Y)$
[Is the left part always defined?]
\end{prop}

\begin{proof}
$\rsupfun{\bigcup\rsupfun{\curry f}X}Y=\setcond{\rsupfun{\bigcup\rsupfun{\curry f}X}\{y\}}{y\in Y}=
\setcond{\rsupfun{\bigcup\left(\bigcup_{x\in X}(\curry f)x\right)}\{y\}}{y\in Y}=
\setcond{\bigcup\rsupfun{\bigcup_{x\in X}(\curry f)x}\{y\}}{y\in Y}=
\setcond{\bigcup\bigcup_{x\in X}\rsupfun{(\curry f)x)}\{y\}}{y\in Y}=
\setcond{\bigcup_{x\in X}(\curry f)x)y}{y\in Y}=
\setcond{((\curry f)x)y}{x\in X,y\in Y}$.

$\rsupfun{f}(X\times Y)=\setcond{f(x,y)}{x\in X,y\in Y}$.

So the thesis.
\end{proof}

\begin{prop}
$\supfun{\bigsqcup\supfun{\curry f}x}y =
\supfun{f}(x\times^{\mathsf{RLD}}y)$.
\end{prop}

\begin{proof}
$\supfun{\bigsqcup\supfun{\curry f}x}y =
\bigsqcap_{Y\in\up y}\bigsqcup\supfun{\bigsqcup\supfun{\curry f}x}Y=
\bigsqcap_{Y\in\up y}\bigsqcup\supfun{\bigsqcap_{X\in\up x}\bigsqcup\supfun{\curry f}X}Y=
\bigsqcap_{Y\in\up y}\bigsqcap_{X\in\up x}\bigsqcup\supfun{\bigsqcup\supfun{\curry f}X}Y=
\bigsqcap_{Y\in\up y}\bigsqcap_{X\in\up x}\bigcup\rsupfun{\bigcup\rsupfun{\curry f}X}Y=
\bigsqcap_{Y\in\up y}\bigsqcap_{X\in\up x}\supfun{f}(X\times Y)$.
By properties of generalized filter bases
$\bigsqcap_{Y\in\up y}\bigsqcap_{X\in\up x}\supfun{f}(X\times Y)=\supfun{f}(x\times^{\mathsf{RLD}} y)$
\end{proof}

Let $G$ and $H$ be endofuncoids. By definition:
\begin{itemize}
\item $\operatorname{Ob} \operatorname{HOM} ( G , H) = ( \operatorname{Ob} H)^{\operatorname{Ob} G}$;
\item $( f , g) \in \operatorname{GR} \operatorname{HOM} ( G , H) \Leftrightarrow \forall v,w\in\atoms^{\mathscr{F}}\Ob G : \supfun{f}v \times^{\mathsf{FCD}} \supfun{g}w \sqsubseteq H$ for every $f, g \in \operatorname{Ob} \operatorname{HOM} ( G , H) = ( \operatorname{Ob} H)^{\operatorname{Ob} G}$;
\end{itemize}

$\operatorname{GR} 1_{\operatorname{HOM} ( B , C)} = \operatorname{id}_{\operatorname{Ob} \operatorname{HOM} ( B , C)} = \operatorname{id}_{( \operatorname{Ob} H)^{\operatorname{Ob} G}}$

Equivalently

$( f , g) \in \operatorname{GR} \operatorname{HOM} ( A , B) \Leftrightarrow \forall v,w\in\atoms^{\mathscr{F}}\Ob A : g \circ (v \times^{\mathsf{FCD}} w) \circ f^{- 1} \sqsubseteq \operatorname{GR} B$

$( f , g) \in \operatorname{GR} \operatorname{HOM} ( A , B) \Leftrightarrow g \circ A \circ f^{- 1} \sqsubseteq B$

$( f , g) \in \operatorname{GR} \operatorname{HOM} ( A , B) \Leftrightarrow \langle f \times^{( C)} g \rangle A \sqsubseteq B$

\begin{lem}
$F\suprel{\operatorname{HOM}(A,B)}G \Leftrightarrow G\circ A\circ F^{-1}\sqsubseteq B$ for sets $F$,~$G$ of functions.
\end{lem}

\begin{proof}
$F\suprel{\operatorname{HOM}(A,B)}G \Leftrightarrow
\exists f\in F,g\in G:(f,g)\in\operatorname{HOM}(A,B)\Leftrightarrow
\exists f\in F,g\in G:g\circ A\circ f^{-1}\sqsubseteq B\Leftrightarrow
G\circ A\circ F^{-1}\sqsubseteq B$.
\end{proof}

\begin{prop}
$\mathcal{F}\suprel{\operatorname{HOM}(A,B)}\mathcal{G} \Leftrightarrow \mathcal{G}\circ A\circ \mathcal{F}^{-1}\sqsubseteq B$.
\end{prop}

\begin{proof}
$\mathcal{F}\suprel{\operatorname{HOM}(A,B)}\mathcal{G} \Leftrightarrow
\forall F\in\up\mathcal{F},G\in\up\mathcal{G}:F\suprel{\operatorname{HOM}(A,B)}G
\Leftrightarrow
\forall F\in\up\mathcal{F},G\in\up\mathcal{G}:G\circ A\circ F^{-1}\sqsubseteq B$ what by properties of generalized filter
bases is equivalent to $\mathcal{G}\circ A\circ \mathcal{F}^{-1}\sqsubseteq B$.
\end{proof}

Let $\sim f=\bigsqcup\circ\supfun{\curry f}$. Here we consider $\bigsqcup\circ\supfun{\curry f}$ as a principal funcoid that is binary relation:

\begin{prop}
$\bigsqcup\circ\supfun{\curry f}$ is a complete and co-complete pointfree funcoid.
\end{prop}

\begin{proof}
It is obviously a pointfree funcoid.

Let~$X$ be a principal filter.
$\bigsqcup\supfun{\curry f}X$ is obviously principal.
So, it's co-complete.

And it is obviously complete.
\end{proof}

\begin{obvious}
$\supfun{\supfun{\sim f}x}y = \supfun{f}(x\times^{\mathsf{RLD}}y)$.
\end{obvious}

Let $f\in\Hom(A\times B,C)$ that is $f\in Z^{A\times B}$. Then
$\curry f\in (C^B)^A$, $\supfun{\curry f}X\in\subsets(C^B)$,
$\bigsqcup\supfun{\curry f}X\in C^B$.
Thus $\sim f\in (\mathsf{FCD}(B,C))^{\mathscr{F}(A)}$.

If $f : A \times B \rightarrow C$ then $\sim f : \mathscr{F}(A) \rightarrow \operatorname{HOM} ( B , C)$

\fxwarning{Rewrite below:}

\begin{prop}
Transposition and its inverse are morphisms of $\mathbf{Fcd}$.
\end{prop}

\begin{proof}
It follows from the equivalence $\sim f : A \rightarrow \operatorname{HOM} ( B , C) \Leftrightarrow \forall x,y\in\atoms^{\mathscr{F}}\Ob A : ( x\suprel{A}y \Rightarrow \supfun{\sim f} x \suprel{\operatorname{HOM}(B,C)} \supfun{\sim f} y) \Leftrightarrow \\
\forall x,y\in\atoms^{\mathscr{F}}\Ob A : (x\suprel{A}y \Rightarrow (\supfun{\sim f}y)\circ B\circ(\supfun{\sim f}x)^{-1} \sqsubseteq C) \Leftrightarrow \\
\forall x,y\in\atoms^{\mathscr{F}}\Ob A : (x\suprel{A}y \Rightarrow \forall p,q\in\atoms^{\mathscr{F}}\Ob B : (
p\times^{\mathsf{FCD}}q\sqsubseteq B \Rightarrow
(\supfun{\sim f}y)\circ(p\times^{\mathsf{FCD}}q)\circ(\supfun{\sim f}x)^{-1} \sqsubseteq C) \Leftrightarrow \\
\forall x,y\in\atoms^{\mathscr{F}}\Ob A, p,q\in\atoms^{\mathscr{F}}\Ob B:
(x\suprel{A}y\land p\suprel{B}q \Rightarrow (\supfun{\sim f}y)\circ(p\times^{\mathsf{FCD}}q)\circ(\supfun{\sim f}x)^{-1} \sqsubseteq C) \Leftrightarrow \\
\forall x,y\in\atoms^{\mathscr{F}}\Ob A, p,q\in\atoms^{\mathscr{F}}\Ob B:
(x\suprel{A}y\land p\suprel{B}q \Rightarrow \supfun{\supfun{\sim f}x}p\times^{\mathsf{FCD}}\supfun{\supfun{\sim f}y}q \sqsubseteq C) \Leftrightarrow \\
\forall x,y\in\atoms^{\mathscr{F}}\Ob A, p,q\in\atoms^{\mathscr{F}}\Ob B:
(x\times^{\mathsf{RLD}}p\suprel{A\times^{(C)}B}y\times^{\mathsf{RLD}}q \Rightarrow \supfun{f}(x\times^{\mathsf{RLD}}p)\times^{\mathsf{FCD}}\supfun{f}(y\times^{\mathsf{RLD}}q) \sqsubseteq C) %\Leftrightarrow \\
\forall x,y\in\atoms^{\mathscr{F}}\Ob A, p,q\in\atoms^{\mathscr{F}}\Ob B:
((x\times^{\mathsf{RLD}}p)\times^{\mathsf{FCD}}(y\times^{\mathsf{RLD}}q)\sqsubseteq A\times^{(C)}B \Rightarrow f\circ((x\times^{\mathsf{RLD}}p)\times^{\mathsf{FCD}}(y\times^{\mathsf{RLD}}q))\circ f^{-1} \sqsubseteq C)
\Leftarrow \\
\forall t,s\in\atoms(\Ob A\times^{\mathsf{RLD}}\Ob B):
(t\times^{\mathsf{FCD}}s\sqsubseteq A\times^{(C)}B \Rightarrow f\circ(t\times^{\mathsf{FCD}}s)\circ f^{-1} \sqsubseteq C)
\Leftrightarrow \\
f\circ(A\times^{(C)}B)\circ f^{-1} \sqsubseteq C) \Leftrightarrow
f : A\times^{(C)}B \rightarrow C
$

But:

$f : A\times^{(C)}B \rightarrow C \Leftrightarrow f\circ(A\times^{(C)}B)\circ f^{-1} \sqsubseteq C \Leftrightarrow??
\supfun{f}A\times^{\mathsf{FCD}}\supfun{f}B\sqsubseteq C
\Rightarrow
\forall x,y\in\atoms^{\mathscr{F}}\Ob A, p,q\in\atoms^{\mathscr{F}}\Ob B:(x\times^{\mathsf{RLD}}p\sqsubseteq A\land y\times^{\mathsf{RLD}}q\sqsubseteq B \Rightarrow
\supfun{f}(x\times^{\mathsf{RLD}}p)\times^{\mathsf{FCD}}
\supfun{f}(y\times^{\mathsf{RLD}}q) \sqsubseteq C) \Leftrightarrow
\forall x,y\in\atoms^{\mathscr{F}}\Ob A, p,q\in\atoms^{\mathscr{F}}\Ob B:((x\times^{\mathsf{RLD}}p)\times^{\mathsf{FCD}}(y\times^{\mathsf{RLD}}q)\sqsubseteq A\times^{(C)}B \Rightarrow
\supfun{f}(x\times^{\mathsf{RLD}}p)\times^{\mathsf{FCD}}
\supfun{f}(y\times^{\mathsf{RLD}}q) \sqsubseteq C)$.

Thus the thesis follows.
\end{proof}

Evaluation $\varepsilon : \operatorname{HOM} ( G , H) \times G \rightarrow H$ is defined by the formula:

Then evaluation is $\varepsilon_{B, C} = \mathop{\sim} 1_{\operatorname{HOM}(B,C)}$.

So $\varepsilon_{B, C} ( p , q) = ( \mathop{\sim} 1_{\operatorname{HOM}(B,C)}) ( p , q) = 1_{\operatorname{HOM}(B,C)} ( p) ( q) = p ( q)$.

''Proposition'' Evaluation is a morphism of $\mathbf{Dig}$.

''Proof'' Because $\varepsilon_{B, C} ( p , q) = \mathop{\sim} 1_{\operatorname{HOM}(B,C)}$.

It remains to prove:
\begin{itemize}
\item $\varepsilon_{B, C} \circ ( \sim f \times 1_{A}) = f$ for $f : A \rightarrow B \times C$;
\item $\sim ( \varepsilon_{B, C} \circ ( g \times 1_{A})) = g$ for $g : A \rightarrow \operatorname{HOM} ( B , C)$.
\end{itemize}

''Proof'' $\varepsilon_{B, C} ( \sim f \times 1_{A}) ( a , p) = \varepsilon_{B, C} ( ( \sim f) a , p) = ( \sim f) a p = f ( a , p)$. So $\varepsilon_{B, C} \circ ( \sim f \times 1_{A}) = f$.

  $(\sim ( \varepsilon_{B, C} \circ ( g \times 1_{A}))) ( p) ( q) = ( \varepsilon_{B, C} \circ ( g \times 1_{A})) ( p , q) = \varepsilon_{B, C} ( g \times 1_{A}) ( p , q) = \varepsilon_{B, C} ( g p , q) = g ( p) ( q)$. So $\sim ( \varepsilon_{B, C} \circ ( g \times 1_{A})) = g$.

\subsection{By analogy with the proof that Dig is cartesian closed}

The most obvious way for proof attempt that $\mathbf{Fcd}$ is cartesian closed is an analogy with the proof that
$\mathbf{Dig}$ is cartesian closed.

Consider the long formula above. The proof would arise if we replace $x$ and $y$ in this formula with filters and operations and relations on set element with operations and relations on filters.

This proof could be simplified in either of two ways:
\begin{itemize}
\item replace $x$ and $y$ with ultrafilters, see [[Proof for Fcd using ultrafilters]];
\item replace $x$ and $y$ with sets (principal filter), see [[Proof for Fcd using sets]].
\end{itemize}

This is not quite easy however, because we need to calculate uncurrying for a entirely defined monovalued principal funcoid (what is essentially the same as a function of a $\mathbf{Set}$-morphisms) taking either ultrafilters or principal filters as arguments. Such (generalized) uncurrying is not quite easy.

To sum what we need to prove:
\begin{itemize}
\item Transposition is a morphism.
\item Evaluation is a morphism.
\item $\varepsilon_{B,C} \circ ( \sim f \times 1_A) = f$ for $f : A \rightarrow B \times C$.
\item $\sim ( \varepsilon_{B,C} \circ ( g \times 1_A)) = g$ for $g : A \rightarrow \operatorname{HOM} ( B , C)$.
\end{itemize}

\subsection{Attempt to describe exponentials in Fcd}

\begin{itemize}
\item Exponential object $\operatorname{HOM}(A,B)$ is the following endofuncoid:
\item\begin{itemize}
\item Object $\operatorname{Ob}\operatorname{HOM}(A,B) = (\operatorname{Ob} B)^{\operatorname{Ob} A}$;
\item Graph is $\operatorname{GR} \operatorname{HOM} ( A , B) = \uparrow^{\mathsf{FCD}} \setcond{ ( f , g) }{ f, g \in (\operatorname{Ob}B)^{\operatorname{Ob} A} \wedge \uparrow^{\mathsf{FCD}}g \circ A \circ \uparrow^{\mathsf{FCD}}f^{- 1} \sqsubseteq B }$.
\end{itemize}
\item Transposition is uncurrying.
\item Evaluation is $\varepsilon_{A, B} x = \langle \dom x \rangle \im x$.
\end{itemize}

We need to prove that the above defined are really an exponential and an evaluation.

Possible ways to prove that $\mathbf{Fcd}$ is cartesian closed follow:

NEW IDEA: Instead take $\GR\operatorname{HOM}(A,B) =
\uparrow^{\mathsf{FCD}}\setcond{(\dom p,\im p)}{
p\in\End(B)^{\End(A)}\land\supfun{p}A\sqsubseteq B}$ (what's about other
kinds of projections?)

\subsection{Proof for Fcd using sets}

Currying for sets is $\langle f \rangle ( X \times Y) = \bigcup \langle \langle \sim f \rangle X
\rangle Y$ (as it's easy to prove). This simple formula gives hope, but...

It does not work with sets because an analogy for sets of the last equality of the above mentioned long formula would be:

$\forall X, Y, V, W \in \mathscr{P} \operatorname{Ob} A : \left( X \times V \mathrel{[
A \times B]^{\ast}} Y \times W \Rightarrow \langle f \rangle ( X \times V)
\mathrel{[ C]^{\ast}} \langle f \rangle ( Y \times W) \right) \Rightarrow \\ f : A
\times B \rightarrow C$

but this implication seems false.

The most obvious way for proof attempt that $\mathbf{Fcd}$ is cartesian closed is an analogy with the proof that Dig is cartesian closed.

Use the exponential object, transposition, and evaluation as defined in [[this page|Is category Fcd cartesian closed?]]

\subsection{Reducing to the fact that Dig is cartesian closed}
It is probably a simpler way to prove that $\mathbf{Fcd}$ is cartesian closed by embedding it into $\mathbf{Dig}$ (which is [[already known to be cartesian closed|Category Dig is cartesian closed]]).

$\mathbf{Fcd}$ can be embedded into $\mathbf{Dig}$ by the formulas:
\begin{itemize}
\item $A \mapsto \langle A \rangle$;
\item $f \mapsto \langle f \rangle$.
\end{itemize}

That this really maps a morphism of $\mathbf{Fcd}$ into a morphism of $\mathbf{Dig}$ follows from the fact that $\langle g\circ f\rangle = \langle g\rangle\circ\langle f\rangle$.

Obviously this embedding (denote it $T$) is an injective (both on objects and morphisms) functor.

We will define:
\begin{itemize}
\item $\varepsilon^{\mathbf{Fcd}}_{A, B} = T^{-1} \varepsilon^{\mathbf{Dig}}_{T A, T B}$;
\item $\sim^{\mathbf{Fcd}} f = T^{-1} \sim^{\mathbf{Dig}} T f$.
\end{itemize}

Due to functoriality and injectivity of $T$ it is enough to prove that above defined $\varepsilon^{\mathbf{Fcd}}_{A, B}$ and $\sim^{\mathbf{Fcd}} f$ exist and are morphisms of $\mathbf{Fcd}$.

$\varepsilon^{\mathbf{Dig}}_{T A, T B} \ne T\varepsilon^{\mathbf{Fcd}}_{A, B}$ because $\varepsilon^{\mathbf{Dig}}_{T A, T B}$ accepts ordered pairs as the argument and $T \varepsilon^{\mathbf{Fcd}}_{A, B}$ accepts sets as the argument. So this is a dead end. Can the proof idea be salvaged?

\section{Is category Rld cartesian closed?}

We may attempt to prove that $\mathbf{Rld}$ is cartesian closed by embedding it into supposedly cartesian closed category $\mathbf{Fcd}$ by the function $\rho$:

$\langle \rho f \rangle x = f \circ x \quad \text{and} \quad \langle \rho f^{- 1} \rangle y = f^{- 1} \circ y$.

TODO: More to write on this topic.


\documentclass{article}

\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{hyperref}
\usepackage{enumerate}
\usepackage[margin=2cm]{geometry}

\title{Augmented Kalman filter specialization}
\author{Darjus Hosszejni}
\date{\today}

\begin{document}

\maketitle

\section{Model specifications}

\subsection{General}

The model, conditional on mixture states (from de Jong), is
\begin{align*}
y_t & = X_t\beta+Z_t\alpha_t+G_tu_t, \qquad t=1,2,\dots,n, \\
\alpha_{t+1} & = W_t\beta+T_t\alpha_t+H_tu_t, \qquad t=0,1,2,\dots,n,
\end{align*}
where $\alpha_0=0$, and $u_t\sim \mathcal{N}(0,\sigma_g^2I)$.

\subsection{Our case}

Denote $\gamma_{ts}=d_t\rho\sigma\exp(m_{ts}/2)$. Depending on the mixture state $s$, we have
\begin{align*}
y_t & = m_{ts}+\alpha_t+v_{ts}u_{t1}, \\
\alpha_{t+1} & = \mu(1-\phi)+a_{ts}\gamma_{ts}+\phi\alpha_t+b_{ts}v_{ts}\gamma_{ts}u_{t1}+\sigma\sqrt{1-\rho^2}u_{t2}.
\end{align*}

\subsection{Pattern matching}

The parameters from the general setup are substituted here according to below.
\begin{align*}
\beta & = \begin{pmatrix} 1 \\ 1 \\ \mu(1-\phi) \end{pmatrix}, \\
Z_t & = 1, \\
T_t & = \phi, \\
\sigma_g & = 1, \\
G_t & = \begin{pmatrix} v_{ts} & 0 \end{pmatrix}, \\
H_0 & = \begin{pmatrix} 0 & \frac{\sigma}{\sqrt{1-\phi^2}} \end{pmatrix}, \\
H_t & = \begin{pmatrix} b_{ts}v_{ts}\gamma_{ts} & \sigma\sqrt{1-\rho^2} \end{pmatrix}, \\
W_0 & = \begin{pmatrix} 0 & 0 & \frac1{1-\phi} \end{pmatrix}, \\
W_t & = \begin{pmatrix} 0 & a_{ts}\gamma_{ts} & 1 \end{pmatrix}, \\
X_t & = \begin{pmatrix} m_{ts} & 0 & 0 \end{pmatrix}.
\end{align*}

\section{Augmented Kalman filter}

\subsection{General case}

Denote $\eta_t=F_tu_t$ for $t=0,1,\dots,n$. Then the augmented Kalman filter for $t=1,\dots,n$ is
\begin{align*}
e_t & = y_t-X_t\beta-Z_ta_t, \\
D_t & = Z_tP_tZ_t^\prime+G_tG_t^\prime, \\
K_t & = (T_tP_tZ_t^\prime+H_tG_t^\prime)D_t^{-1}, \\
a_{t+1} & = W_t\beta+T_ta_t+K_te_t, \\
P_{t+1} & = T_tP_tL_t^\prime+H_tJ_t^\prime,
\end{align*}
with values
\begin{align*}
a_1 & = W_0\beta, \\
P_1 & = H_0H_0^\prime, \\
L_t & = T_t-K_tZ_t, \\
J_t & = H_t-K_tG_t.
\end{align*}

We also consider extra equations with the decomposition $\beta=b+B\mu$, (from Nakajima, Appendix B),
\begin{align*}
f_t & = y_t-X_tb-Z_ta_t^*, \\
a_{t+1}^* & = W_tb+T_ta_t^*+K_tf_t, \\
F_t & = X_tB-Z_tA_t^*, \\
A_{t+1}^* & = -W_tB+T_tA_t^*+K_tF_t,
\end{align*}
with values
\begin{align*}
a_1^* & = W_0b, \\
A_1^* & = -W_0B.
\end{align*}
Note that $e_t = f_t-F_t\mu$ and $a_t = a_t^*-A_t^*\mu$.

Then the posterior of $\mu$ given $y$ is $\mathcal{N}(Q_{n+1}^{-1}q_{n+1},Q_{n+1}^{-1})$, where the prior of $\mu$ is $\mathcal{N}(c, C)$, and
\begin{align*}
q_{t+1} & = q_t+F_t^\prime D_t^{-1}f_t, \qquad q_1=C^{-1}c, \\
Q_{t+1} & = Q_t+F_t^\prime D_t^{-1}F_t, \qquad Q_1=C^{-1}.
\end{align*}

Then the log likelihood of $y$ is given by
\begin{equation*}
\log f(y) = -0.5\left(\sum_{t=1}^n\log\left(\lvert\text{det}(D_t)\rvert\right)+\log\left(\lvert\text{det}(Q_{t+1})\rvert\right)+\sum_{t=1}^n f_t^\prime D_t^{-1}f_t + c^\prime C^{-1}c - q_{n+1}^\prime Q_{n+1}^{-1}q_{n+1}\right) + \text{constant}.
\end{equation*}

\subsection{Our case}

We have
\begin{align*}
F_t & = H_t, \\
b & = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \\
B & = \begin{pmatrix} 0 \\ 0 \\ \mu(1-\phi) \end{pmatrix}, \\
c & = \mu_\mu, \\
C & = \sigma_\mu^2.
\end{align*}

Denote $\gamma_{ts}=d_t\rho\sigma\exp(m_{ts}/2)$ and $h_{ts}=b_{ts}v_{ts}\gamma_{ts}$. We don't calculate $e_t$ or $a_t$, we deduce them later. Then
\begin{align*}
D_t & = P_t+v_{ts}^2, \\
K_t & = \frac{\phi P_t+h_{ts}v_{ts}}{D_t}, \\
P_{t+1} & = \phi P_tL_t+h_{ts}j_{t1} + j_{t2}^2, \\
f_t & = y_t - m_{ts} - a_t^*, \\
F_t & = -A_t^*, \\
a_{t+1}^* & = a_{ts}\gamma_{ts}+\phi a_t^* + K_tf_t, \\
A_{t+1}^* & = \phi-1+\phi A_t^*+K_tF_t,
\end{align*}
with values
\begin{align*}
a_1^* & = 0, \\
A_1^* & = -1, \\
P_1 & = \frac{\sigma^2}{1-\phi^2}, \\
L_t & = \phi-K_t, \\
J_t & = \begin{pmatrix} h_{ts}-K_tv_{ts} & \sigma\sqrt{1-\rho^2} \end{pmatrix}.
\end{align*}
Don't confuse $a_t$ with $a_{ts}$, they are completely different! The latter is the constant $a$ specified for the approximating Gaussian mixture in state $s$.

About the posterior of $\mu$ given $y$,
\begin{align*}
q_{t+1} & = q_t+F_tf_t/D_t, \qquad q_1=\mu_\mu/\sigma_\mu^2, \\
Q_{t+1} & = Q_t+F_t^2/D_t, \qquad Q_1=1/\sigma_\mu^2.
\end{align*}

And the log likelihood of $y$,
\begin{equation*}
\log f(y) = -0.5\left(\sum_{t=1}^n\log\left(\lvert D_t\rvert\right)+\log\left(\lvert Q_{t+1}\rvert\right)+\sum_{t=1}^n f_t^2/D_t + \mu_\mu^2/\sigma_\mu^2 - q_{n+1}^2/Q_{n+1}\right) + \text{constant}.
\end{equation*}

\section{Simulation smoother}

\subsection{General case}

Then, for $t=n,\dots,1$, the simulation smoother is
\begin{align*}
C_t & = F_t(I-G_t^\prime D_t^{-1}G_t-J_t^\prime U_tJ_t)F_t^\prime, \\
\varepsilon_t & \sim\mathcal{N}(0, \sigma_g^2C_t), \\
V_t & = F_t(G_t^\prime D_t^{-1}Z_t+J_t^\prime U_tL_t), \\
r_{t-1} & = Z_t^\prime D_t^{-1}e_t + L_t^\prime r_t-V_t^\prime C_t^{-1}\varepsilon_t, \\
U_{t-1} & = Z_t^\prime D_t^{-1}Z_t+L_t^\prime U_tL_t+V_t^\prime C_t^{-1}V_t, \\
\eta_t & = F_t(G_t^\prime D_t^{-1}e_t+J_t^\prime r_t) + \varepsilon_t,
\end{align*}
with values
\begin{align*}
r_n &=0, \\
U_n &=0, \\
\eta_0 &= F_0H_0^\prime r_0 + \varepsilon_0, \\
\varepsilon_0 &\sim\mathcal{N}(0,\sigma_g^2C_0), \\
C_0 &= F_0(I-H_0^\prime U_0H_0)F_0^\prime.
\end{align*}

\subsection{Our case}

\begin{align*}
C_t & = (h_{ts}^2 + \sigma^2(1-\rho^2))-(h_{ts}v_{ts})^2/D_t-U_t(H_tJ_t^\prime)^2, \\
\varepsilon_t & \sim\mathcal{N}(0, C_t), \\
V_t & = h_{ts}v_{ts}/D_t+U_tL_t(h_{ts}j_{t1} + j_{t2}^2), \\
r_{t-1} & = e_t/D_t + L_tr_t-V_t\varepsilon_t/C_t, \\
U_{t-1} & = 1/D_t+U_tL_t^2+V_t^2/C_t, \\
\eta_t & = h_{ts}v_{ts}e_t/D_t+(h_{ts}j_{t1} + j_{t2}^2)r_t + \varepsilon_t,
\end{align*}
with values
\begin{align*}
r_n &=0, \\
U_n &=0, \\
\eta_0 &= \frac{\sigma^2}{1-\phi^2}r_0 + \varepsilon_0, \\
\varepsilon_0 &\sim\mathcal{N}(0,C_0), \\
C_0 &= \frac{\sigma^2}{1-\phi^2}\left(1-\frac{\sigma^2}{1-\phi^2}U_0\right).
\end{align*}

\end{document}