Definition for the action of a category on a set.

Question

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the disjoint union of the $\rho(c)$ for all objects $c$ of $C$, and the action of a morphism $f$ of $C$ on an element $s$ of $S$ is $\rho(f)(s)$ if the domain corresponds. However, it seems to me that this defines only a partial action from $S$ to $S$, i.e it is not always defined for all elements of $S$. Can someone help with this definition ?

Answer 1

The action of $f$ does indeed define a partial function from $S$ to itself. But that's intentional.

Category actions are equivalent to diagrams. The "picture" you should have in mind is that $S$ is the disjoint union of all of the objects in the diagram, so that you really do want the action of $f$ to be a partial operation, defined only on those objects of $S$ that come from the vertex of the diagram associated to the domain of $f$.

---

A concrete example of a category action is to let $\mathcal{C}$ be the opposite poset of poset of all subsets of $X$, and let $S$ be the set of all partial functions on $X$. The morphisms of $\mathcal{C}$ are the inclusions $\rho_{A,B}$ where $A \subseteq B$. If $f$ is a partial function with domain $B$, then $\rho_{A,B} \cdot f$ is the restriction of $f$ to $A$.