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 ?