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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?

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  • $\begingroup$ 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. $\endgroup$ – Derek Elkins Feb 24 at 7:09
  • $\begingroup$ 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 ? $\endgroup$ – Max Feb 24 at 11:06
  • $\begingroup$ @DerekElkins, what do you mean by Ab? $\endgroup$ – user56834 Feb 25 at 6:31
  • $\begingroup$ @user56834 Sorry, the category of Abelian groups and group homomorphisms. $\endgroup$ – Derek Elkins Feb 25 at 6:35

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