The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a semigroup $S$, a set $A$, and a mapping of the elements of the semigroup $S$ to functions from the set $A$ to itself. But the analogous result for inverse semigroups requires partial symmetries, i.e. partial functions instead of total functions. But if we allow partial functions, then what do we do with removable singularities?
Here comes the desire to turn all this into "somebody else's problem", by using a general definition of semigroup action in terms of category theory. We can then work in the category of sets and (total) functions, if we don't need partial functions for the semigroup action. If we need partial functions, then we can work in the category of sets and partial functions. And if we worry about removable singularities, then we can work in the appropriate category where these are removable.
Edit (because of requests for clarification): Before going into details why it is unclear to me whether this proposed "solution" will fix the issues of the apparent "incompatibility" between semigroup action and the inverse semigroup representation theorem, here are questions that I "hope" can be answered:
- How does the definition of semigroup action in terms category theory look like?
- (main question) Does this definition make sense when applied to an inverse semigroup? Does it lead to the intended representation theorem if used with the category of sets and partial function?
- Is there an analogous definition of groupoid action? Are the various connections between semigroups and groupoids compatible with these definitions of an action?
But how should such a definition look like? I'm not sure, but let's look at a (potential) similar definition for groupoid action: A groupoid action relative to a category $\mathcal C$ would be a groupoid $\mathcal G$ and a functor $\mathcal G\to \mathcal C$. In analogy to this, one could interpret a monoid as a category with a single object, a monoid action as a single object category $\mathcal M$ and a functor $\mathcal M\to \mathcal C$. A semigroup action would then be defined as a subsemigroup of a corresponding monoid action.
One problem I have is that if there were such a thing as a semigroupoid (there is: it's called a category...), it would be easy to interpret a small semigroupoid (and hence also a small groupoid) as a semigroup. (Add a new absorbing element and use it to define the result of any undefined composition from the semigroupoid.) But can one define a semigroup action in such a way that also the semigroupoid action can be interpreted appropriately in terms of semigroup action? This doesn't mean that the above definition won't work, maybe one just has to switch to a corresponding category of categories for being able to interpret it appropriately.
Side note One general issue for me related to the proposed "solution" is that I'm not too familiar with the treatment of partial homomorphisms within category theory. The category of pointed sets "hinted at" by Qiaochu Yuan makes it "crystal clear" how partial homomorphisms work in the case of sets. But is the situation for other (concrete) categories really as "straightforward" as this? Andrej Bauer suggested some references, and the topic also seems to be discussed in some category theory texts in connection with "limits and colimits", where a construction based on subobjects, equivalence classes and pullbacks is described. I will have to read and understand these.