Action representatives

Let $S$ be a set and $R \subseteq S^2$ a preorder on $S$. Then the preorder $R$ defines the direction in which the elements of $S$ can move over time, but it says nothing about the driving forces responsible for moving the elements of the preorder in the direction they are going in. Take spacetime for example, then we can partially order events in spacetime by causality, but by itself a causal set does nothing to explain the fundamental forces responsible for motion or what makes one event cause another.

In general, given a preorder we want to describe in more detail the sort of actions that move the elements of the order along. Abstract algebra allows us to do this by using structures like transformation monoids. This leads to the idea of action representatives. Let $S$ be a set, $T$ a monoid acting on $S$, and $R$ the increasing action preorder of the transformation semigroup $T$. Then the action preorder $R$ is a set of ordered pairs, if we select an ordered pair $(x,y) \in R$ then a representative action $t \in A(x,y)$ is an element $t$ of $T$ that transforms $a$ into $b$. \[ A(x,y) = \{ t \in T: t(x) = y \} \] A representative action for an ordered pair only exists if it is in the preorder $R$. At the same time, for a transformation monoid representative actions always exist for any ordered pair in the action preorder. To see this note that if $x \subseteq y \subseteq z$ then the composition of the actions that transform $x$ to $y$ and $y$ to $z$ transform $x$ to $z$. The identity is an action representative for any loop. The composition of actions generalizes transitivity, and so representative actions are always ensured to exist.

The interesting thing, and the purpose of this discussion, is the analogy with category theory. We can treat a category $C$ as a set of morphisms acting on a set of objects. In that case, the representative actions of $C$ of an ordered pair $(x,y)$ correspond to hom classes $Mor_C(x,y)$. As in the case of transformation monoids, representative actions exist if and only if an element precedes another one in the corresponding morphic preordering. In this way, a category is one way of extending a preorder with actions that describe the ways that one element can be transformed to another.

The difference between the monoid action picture and the categorical picture, is that given any morphic action of a category $C$ it corresponds to only one ordered pair. On the other hand, a transformation of a set is the representative of as many actions as the cardinality of the set itself. By restricting to describing only the invidual actions that move elements along, categories most readily provide for structural descriptions of preorders.