I need to find a solution to have an inverse structure form: not classic modules over a monoid but monoids over modules. I had received this answer from here

monoid objects are the minimal structure for modules to make sense over, and modules themselves don't have enough structure to be monoid objects in general.

But is possible to use sheafification or stackification to 'feed' modules to give them the 'missing structure' to do a module an monoidal identity ? Is there an inverse way to overcome this structure missing problem for modules ?

Then it says

the categorification process can be made to continue by finding a monoid object in the category of modules, and then passing to modules over this

If solution is that, how is possible to give the missing structure for a single module using the category of modules ?

in the set-enriched case, a module over a monoid is just a set equipped with a (left) action by that monoid. You can't multiply elements in this thing together; you can only let them be “scaled” by elements in the base monoid.

Eh... this is problem: "scaled by elements": I don't want this. If I use a sheaf structure, augmented R-algebra or stackification can I have a monoid over a module ?

  • $\begingroup$ In the category of $R$-modules, the monoid objects are just the $R$-algebras. An $R$-algebra $A$ has an underlying $R$-module structure, and we can form $A$-modules. $\endgroup$ – Berci May 30 '18 at 22:08
  • $\begingroup$ 'Module' is basically a synonim for monoid (ring) action. When you have a module, it already presumes a monoid (object in a monoidal category) and a monoid action. $\endgroup$ – Berci May 30 '18 at 22:13
  • $\begingroup$ yes..but if module is a monoid (ring) action I wish to add (I try to understand what is this 'add') extra structure because I want a field structure for my module. I know that finite (integral) domain or finite division ring is (or returns as) a field so I don't find only a monoid action , but an action structure within for my module (a field structured module), maybe a monad structure, not only an external monoid action. R-modules stand for ring, but I need to have a field because I know that every field is a ring but also Every finite division ring is a field $\endgroup$ – user3520363 May 31 '18 at 12:27

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