Functions reminds about cause and effect while relation are more like interaction and more diffuse and complex. But one reason why relations not is more common as morphisms, could depend on the fact that functions as objects has commutative squares as canonical morphisms, which does not works for relations with the ordinary composition of relations. However, there is a canonical extension of commutativity that might be used instead: $$ \begin{array}{rcl} A & \overset{\displaystyle u}{\longrightarrow} & \! B \\ r \downarrow & \looparrowright & \downarrow s\\ \bar{A} & \; \underset{\displaystyle \bar{u}} {\longrightarrow} &\! \bar{B} \end{array} $$ where the $\looparrowright$ stands for a relation between the four binary relations $ r,s,u,\bar{u} $ that equals to commutativity if the relations are functions:

$ a \underline{u} b \wedge \bar{a}\bar{\underline{u}} \bar{b} \Rightarrow (a \underline{r}\bar{a} \Rightarrow b \underline{s}\bar{b}) $, for all $ a \in A $, $ \bar{a}\in \bar{A} $, $ b \in B $, $ \bar{b} \in \bar{B} $.

Now it holds that: $$ \begin{array}{lcccrcrcl} \:\: A &\overset{\displaystyle u}{\longrightarrow} &B &\overset{\displaystyle v}{\longrightarrow} &C\:\: & & A & \overset{\displaystyle vu}{\longrightarrow} & \! C \\ r \downarrow & \looparrowright & \:\:\: \downarrow s &\looparrowright &\downarrow t & \Longrightarrow & r \downarrow & \looparrowright & \downarrow t \\ \:\: \bar{A}&\underset{\displaystyle \bar{u}}{\longrightarrow} & \bar{B} &\underset {\displaystyle \bar{v}}{\longrightarrow}& \bar{C} \:\: & & \!\bar{A} & \; \underset{\displaystyle \bar{v}\bar{u}}{\longrightarrow} &\! \bar{C} \\ \end{array} $$ This can for example be used to define categories of mathematical structures as:

$\star$ Monoid structure $ M\times M \overset{r}\longrightarrow M $, where $r$ is the multiplication

$\star$ Topological structur $ \mathcal{P}(X)\overset{r}\longrightarrow X $ where $r$ is closeness/nearness: $ M\underline{r}x\Leftrightarrow x\in \bar{M} $

$\star$ Metric spaces $ \mathbb{R}\times X\overset{r}\longrightarrow X $ where $r$ is $ (\xi,x)\underline{r}x'\Leftrightarrow d(x,x')\leq\xi $

The structure categories are richer than the constructs. For example, several structures for the same metric can be defined, giving arise to different morphisms (which trivially transform to the morphisms in the corresponding construct). The main idea is to describe mathematical structures as relations

$ F(X) \overset{r}\longrightarrow X $ for some functor $F$ on Rel, where $$ \begin{array}{ccc} F(X)\!\!\!\! & \overset{F(f)}\longrightarrow & \!\!\!F(Y) \\ r \downarrow & \looparrowright & \downarrow s \\ X\!\!\!\!\! & \underset{ f} \longrightarrow & \!\!\!\!\! Y \end{array} $$ gives the condition on the function $f$ for the diagram to be a morphism. Note that $F(f)$ in general is a relation.

Functions reminds about cause and effect while relation are more like interaction and more diffuse and complex. But one reason why relations not is more common as morphisms, could depend on the fact that functions as objects has commutative squares as canonical morphisms, which does not works so good for relations since commutation do not preserve the structure of relations (compare preserving of graph structures). For functions commutation preserve this structure on the function. However, there is a canonical extension of commutation that might be used instead: $\require{AMScd}$ \begin{CD} A @>u>> B\\ @VrV\quad\;\;\,\displaystyle \looparrowright V @VVsV\\ \bar{A} @>>\bar{u}> \bar{B} \end{CD} where the $\looparrowright$ stands for a relation between the four binary relations $ r,s,u,\bar{u} $ that equals to commutativity if the relations are functions:

$ a \underline{u} b \wedge \bar{a}\bar{\underline{u}} \bar{b} \Rightarrow (a \underline{r}\bar{a} \Rightarrow b \underline{s}\bar{b}) $, for all $ a \in A $, $ \bar{a}\in \bar{A} $, $ b \in B $, $ \bar{b} \in \bar{B} $.

Now it holds that: $$ \begin{CD} A @>u>> B@>v>>C\\ @VrV\quad\;\;\,\displaystyle \looparrowright V @VsV\quad\;\;\,\,\displaystyle \looparrowright V @VVtV\\ \bar{A} @>>\bar{u}> \bar{B}@>>\bar{v}> \bar{C} \end{CD} \qquad\Rightarrow\qquad \begin{CD} A @>vu>> C\\ @VrV\quad\;\;\,\displaystyle \looparrowright V @VVtV\\ \bar{A} @>>\bar{v}\bar{u}> \bar{C} \end{CD} $$ This can for example be used to define categories of mathematical structures as:

$\star$ Monoid structure $ M\times M \overset{r}\longrightarrow M $, where $r$ is the multiplication

$\star$ Topological structur $ \mathcal{P}(X)\overset{r}\longrightarrow X $ where $r$ is closeness/nearness: $ M\underline{r}x\Leftrightarrow x\in \bar{M} $

$\star$ Metric spaces $ \mathbb{R}\times X\overset{r}\longrightarrow X $ where $r$ is $ (\xi,x)\underline{r}x'\Leftrightarrow d(x,x')\leq\xi $

The structure categories are richer than the constructs. For example, several structures for the same metric can be defined, giving arise to different morphisms (which trivially transform to the morphisms in the corresponding construct). The main idea is to describe mathematical structures as relations

$ F(X) \overset{r}\longrightarrow X^{I} $ for some functor $F$ on Rel, where \begin{CD} F(X) @>F(f)>> F(Y)\\ @VrV\qquad \;\;\displaystyle \looparrowright V @VVsV\\ X^{I} @>>f^{I}> Y^{I} \end{CD} gives the condition on the function $f$ for the diagram to be a morphism. Note that $F(f)$ in general is a relation.

Functions reminds about cause and effect while relation are more like interaction and more diffuse and complex. But one reason that relations not is more common as morphisms, could depend on the fact that functions as objects has commutative squares as canonical morphisms, which does not works for relations with the ordinary composition of relations. However, there is a canonical extension of commutativity that might be used instead: $$ \begin{array}{rcl} A & \overset{\displaystyle u}{\longrightarrow} & \! B \\ r \downarrow & \looparrowright & \downarrow s\\ \bar{A} & \; \underset{\displaystyle \bar{u}} {\longrightarrow} &\! \bar{B} \end{array} $$ where the $\looparrowright$ stands for a relation between the four binary relations $ r,s,u,\bar{u} $ that equals to commutativity if the relations are functions:

$ a \underline{u} b \wedge \bar{a}\bar{\underline{u}} \bar{b} \Rightarrow (a \underline{r}\bar{a} \Rightarrow b \underline{s}\bar{b}) $, for all $ a \in A $, $ \bar{a}\in \bar{A} $, $ b \in B $, $ \bar{b} \in \bar{B} $.

Now it holds that: $$ \begin{array}{lcccrcrcl} \:\: A &\overset{\displaystyle u}{\longrightarrow} &B &\overset{\displaystyle v}{\longrightarrow} &C\:\: & & A & \overset{\displaystyle vu}{\longrightarrow} & \! C \\ r \downarrow & \looparrowright & \:\:\: \downarrow s &\looparrowright &\downarrow t & \Longrightarrow & r \downarrow & \looparrowright & \downarrow t \\ \:\: \bar{A}&\underset{\displaystyle \bar{u}}{\longrightarrow} & \bar{B} &\underset {\displaystyle \bar{v}}{\longrightarrow}& \bar{C} \:\: & & \!\bar{A} & \; \underset{\displaystyle \bar{v}\bar{u}}{\longrightarrow} &\! \bar{C} \\ \end{array} $$ This can for example be used to define categories of mathematical structures as:

$\star$ Monoid structure $ M\times M \overset{r}\longrightarrow M $, where $r$ is the multiplication

$\star$ Topological structur $ \mathcal{P}(X)\overset{r}\longrightarrow X $ where $r$ is closeness/nearness: $ M\underline{r}x\Leftrightarrow x\in \bar{M} $

$\star$ Metric spaces $ \mathbb{R}\times X\overset{r}\longrightarrow X $ where $r$ is $ (\xi,x)\underline{r}x'\Leftrightarrow d(x,x')\leq\xi $

The structure categories are richer than the constructs. For example, several structures for the same metric can be defined, giving arise to different morphisms (which trivially transform to the morphisms in the corresponding construct). The main idea is to describe mathematical structures as relations

$ F(X) \overset{r}\longrightarrow X $ for some functor $F$ on Rel, where $$ \begin{array}{ccc} F(X)\!\!\!\! & \overset{F(f)}\longrightarrow & \!\!\!F(Y) \\ r \downarrow & \looparrowright & \downarrow s \\ X\!\!\!\!\! & \underset{ f} \longrightarrow & \!\!\!\!\! Y \end{array} $$ gives the condition on the function $f$ for the diagram to be a morphism. Note that $F(f)$ in general is a relation.

Functions reminds about cause and effect while relation are more like interaction and more diffuse and complex. But one reason why relations not is more common as morphisms, could depend on the fact that functions as objects has commutative squares as canonical morphisms, which does not works for relations with the ordinary composition of relations. However, there is a canonical extension of commutativity that might be used instead: $$ \begin{array}{rcl} A & \overset{\displaystyle u}{\longrightarrow} & \! B \\ r \downarrow & \looparrowright & \downarrow s\\ \bar{A} & \; \underset{\displaystyle \bar{u}} {\longrightarrow} &\! \bar{B} \end{array} $$ where the $\looparrowright$ stands for a relation between the four binary relations $ r,s,u,\bar{u} $ that equals to commutativity if the relations are functions:

$ a \underline{u} b \wedge \bar{a}\bar{\underline{u}} \bar{b} \Rightarrow (a \underline{r}\bar{a} \Rightarrow b \underline{s}\bar{b}) $, for all $ a \in A $, $ \bar{a}\in \bar{A} $, $ b \in B $, $ \bar{b} \in \bar{B} $.

Now it holds that: $$ \begin{array}{lcccrcrcl} \:\: A &\overset{\displaystyle u}{\longrightarrow} &B &\overset{\displaystyle v}{\longrightarrow} &C\:\: & & A & \overset{\displaystyle vu}{\longrightarrow} & \! C \\ r \downarrow & \looparrowright & \:\:\: \downarrow s &\looparrowright &\downarrow t & \Longrightarrow & r \downarrow & \looparrowright & \downarrow t \\ \:\: \bar{A}&\underset{\displaystyle \bar{u}}{\longrightarrow} & \bar{B} &\underset {\displaystyle \bar{v}}{\longrightarrow}& \bar{C} \:\: & & \!\bar{A} & \; \underset{\displaystyle \bar{v}\bar{u}}{\longrightarrow} &\! \bar{C} \\ \end{array} $$ This can for example be used to define categories of mathematical structures as:

$\star$ Monoid structure $ M\times M \overset{r}\longrightarrow M $, where $r$ is the multiplication

$\star$ Topological structur $ \mathcal{P}(X)\overset{r}\longrightarrow X $ where $r$ is closeness/nearness: $ M\underline{r}x\Leftrightarrow x\in \bar{M} $

$\star$ Metric spaces $ \mathbb{R}\times X\overset{r}\longrightarrow X $ where $r$ is $ (\xi,x)\underline{r}x'\Leftrightarrow d(x,x')\leq\xi $

The structure categories are richer than the constructs. For example, several structures for the same metric can be defined, giving arise to different morphisms (which trivially transform to the morphisms in the corresponding construct). The main idea is to describe mathematical structures as relations

$ F(X) \overset{r}\longrightarrow X $ for some functor $F$ on Rel, where $$ \begin{array}{ccc} F(X)\!\!\!\! & \overset{F(f)}\longrightarrow & \!\!\!F(Y) \\ r \downarrow & \looparrowright & \downarrow s \\ X\!\!\!\!\! & \underset{ f} \longrightarrow & \!\!\!\!\! Y \end{array} $$ gives the condition on the function $f$ for the diagram to be a morphism. Note that $F(f)$ in general is a relation.