Is there any category whose objects are semigroup acts and morphisms are semilinear morphism between acts?

Question

I know that for a particular semigroup S all the S-acts form a category with S-act homomorphisms. My question is what happens if we do not fix the semigroup i.e. taking all semigroup acts (S,A) where S and A both varies and semilinear morphisms (μ,f):(S,A)→(T,B) playing the role of morphisms berween objects can we have a category? If yes where can I get some access to read about them?

Answer 1

$\require{AMScd}$You can define a contravariant functor ${\rm Act} : {\bf Sgrp} \to {\bf Cat}$ sending a semigroup $S$ to the category ${\rm Act}(S)$ of $S$-acts (I'll let you find how it is defined on morphisms $u : A\to T$, sending a $T$-act $B$ to the $S$-act $u^*B$), and the comma category

$$\begin{CD} {\cal E} @>>> 1 \\ @VVV\Downarrow @VV{\bf 1}V\\ {\bf Sgrp} @>>{\rm Act}> {\bf Cat} \end{CD}$$

is the category you are trying to define. (${\bf 1}$ picks the terminal category in $\bf Cat$). By definition of this comma category, $\cal E$ has

1. objects given by pairs $(S,A)$ where $A\in{\rm Act}(S)$;
2. morphisms $(S,A)\to (T,B)$ given by pairs $(u,f)$ where $u : S\to T$ is a semigroup homomorphism, and $f : A\to B$ is a function that is equivariant between the $S$-act $A$ and the $S$-act $u^*B$.

Unwinding this second condition, $f : A\to B$ is such that $f(s.a) = u(s).fa$ for every $a\in A$, $s\in S$.