# Are the “compatibility axioms” of vector space axioms derivable from a general “compatibility principle”

I have seen a couple of times that concepts specific to particular structures that seem "sensible" but still somewhat arbitrary can be derived from a general canonical principle at a higher level of abstraction. E.g. if I'm not mistaken, the "compatibility" of the group structure of a Lie group with its topological structure, can be derived from projecting functorially the axioms of group theory onto the category of topological spaces.

A vector space $$(V,F,v_0,+_v,\cdot_v)$$ can be seen as a commutative group $$(V,v_0,+_v)$$, together with a field $$F$$ and a scalar product $$\cdot_v$$ such that "the group $$V$$ is 'structurally compatible' with the the field $$F$$".

Is there a way to state this idea of "structurally compatible" at a higher level of abstraction, such as category theory or mathematical logic, and then derive the specific axioms of vector spaces from that?

• An $R$-module (which will be a vector space when $R$ is a field) can be viewed as an $\mathbf{Ab}$-enriched functor from a ring viewed as a one-object $\mathbf{Ab}$-enriched category into $\mathbf{Ab}$. This is basically the $\mathbf{Ab}$-enriched version of a monoid action being a (plain) functor from a one-object category into $\mathbf{Set}$. I don't know if this is the kind of thing you want though. – Derek Elkins Feb 24 at 7:09
• To any group you can associate a type of algebraic structure that is the same type as a field, and such that a vector space structure is precisely a morphism from the field to that algebraic structure : I'm talking about the endomorphism ring of the group. From there the compatibility follows. Were you looking for something like this ? – Max Feb 24 at 11:06
• @DerekElkins, what do you mean by Ab? – user56834 Feb 25 at 6:31
• @user56834 Sorry, the category of Abelian groups and group homomorphisms. – Derek Elkins Feb 25 at 6:35