Given a category C and a function $ \Theta : Mor(\textbf{C}) \times Mor(\textbf{C}) \longrightarrow Mor(\textbf{Rel}) $ and suppose that the relation, with $(r,s)\in\Theta(u,\bar{u})$ and so forth, is such that: $\require{AMScd}$ $$ \begin{CD} A @>u>> B@>v>>C\\ @VrV V \Theta@VsV V\Theta @VVtV\\ \bar{A} @>>\bar{u}> \bar{B}@>>\bar{v}> \bar{C} \end{CD} \quad\Rightarrow\quad \begin{CD} A @>vu>> C\\ @VrV V\Theta @VVtV\\ \bar{A} @>>\bar{v}\bar{u}> \bar{C} \end{CD} $$ Then $\Theta$ give arise to a category with arrows as objects and pairs of arrows as morphisms. If $(r,s)\in \Theta(u,\bar{u})\Leftrightarrow s\circ u = \bar{u}\circ r$, then $\Theta$ is the commuting relation.
Since the commuting relation is so commonly used for universal definitions, one might be blind for other kinds of constructions, especially of concrete categories?
In a study of the character of the relation between constructs and their morphisms, I have in Rel discovered that the relation $(r,s)\in \Theta(u,\bar{u})\Leftrightarrow [(a,b)\in u\wedge (\bar{a},\bar{b})\in \bar{u}\wedge (a,\bar{a})\in r \Rightarrow (b,\bar{b})\in s]$, seems to satisfy my needs (even if it perhaps not seems so sexy). But when I tried to present this idea, an obviously skillful mathematician drew the conclusion that I must have meant the relation $(r,s)\in \Theta(u,\bar{u})\Leftrightarrow s=\bar{u}\circ r\circ u^{op}$, since this latter $\Theta$ implied the the previous original $\Theta$.
- Are the two definitions equal?
- Does successful ideas have too heavy impact on our thinking?
- Is there room for alternative conditions on diagrams in category theory?
(I admit that it took years before I realized that it perhaps shouldn't be commutative diagrams, neither with nor without reversed arrows...)