Can I define a category as a monoid with partially defined multiplication?

Question

A groupoid can either be thought of as a category whose morphisms are isomorphisms, or as a generalization of a group whose multiplication is only partially defined. Can I do a similar thing with categories and monoids? More generally, if I have some algebraic structure X on a one object category, can I think of a many object generalization of that structure (an "Xoid") as a generalization of X with partially defined operations?

Answer 1

This is colloquially known as oidification for the reasons that you suspect. Interestingly, that link suggests that the 'exception to the rule' is precisely your example of oidifying a monoid.

Answer 2

I'm not an expert, but I think that a category is a special type of partial monoid with partial identity elements satisfying some axioms. Unfortunately, I don't know what is the exact list of axioms.