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I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the usual sense.

Let me explain by giving example:

i) a group action is a group acting on a set (associativity, stabilization by neutral elements),

ii) a vector space is a field acting on a commutative group (associativity, distributivity, stabilization by neutral elements),

iii) a module is a ring acting on a commutative group (associativity, distributivity, stabilization by neutral elements),

iv) an algebra is a field acting on a "pseudo ring" (non associative ring),

degenerated case: v) a ring is a monoid $M$ biacting on a commutative group $A$, where $M = A$ as objects.

Now, what I would like to do is taking the categories associated to the sketches of a group and a set (say), and produce a new category that is sketching the notion of "action of group on a set". The previous example would become

i) the category associated to the sketch of a group object acting on the terminal category is the sketch of a group object action,

ii) the category associated to the sketch of a field object acting on the category associated to the sketch of a commutative group object is a vector space object,

iii) the ring object acting on the commutative group object is a module object, etc.

Does anyone has ever heard of such concept? Do you guys think that it could also be formalized for general sketches, thus creating the general notion of "category acting on a category" in the previous sense?

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    $\begingroup$ These are all essentially examples of monoid actions for a monoid in a monoidal category. $\endgroup$
    – Zhen Lin
    Commented Sep 2, 2015 at 11:29
  • $\begingroup$ What I want is to take two categories and consider that the resulting action of one category on the other is another category as I said. I don't see how what you say is related to what I want, mind you expliciting? $\endgroup$
    – sure
    Commented Sep 2, 2015 at 12:14
  • $\begingroup$ It is not as general as you ask for. I am just pointing out that there is something that captures the examples you mention. $\endgroup$
    – Zhen Lin
    Commented Sep 2, 2015 at 12:28
  • $\begingroup$ Actually to come up with the correct notion of action is not entirely trivial. There is something called actor or split extension classifier which gives endomorphisms for monoids, automorphisms for groups, derivations for Lie algebras, bimultiplications for associative algebras, biderivations for Leibniz algebras... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Sep 2, 2015 at 13:02
  • $\begingroup$ @მამუკაჯიბლაძე: do you have any references where this concept is introduced? I can't find it easily using google and arxiv. Thanks. $\endgroup$
    – sure
    Commented Sep 2, 2015 at 14:22

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