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Vector spaces can be thought of as a sort of canonical field action on an Abelian group, likewise modules can be thought of as the action of a ring on a group.

1. How far can this be generalized?

For example, can a ring be thought of as the action of a group on its set endowed with a different group structure? Can we think of free vector spaces as the action of a field on a set? I know that a group is the group action of itself on itself (I think).

Perhaps better phrased:

(Rephrased) Which algebraic objects can not be thought of as the action of one type of algebraic structure on another algebraic structure?

This question might be too broad or general as written, in which case an answer to the question:

(Alternate) Is there a formal/rigorous notion of "field action" generalizing "group action" such that vector spaces are exactly the action of a field on an Abelian group?

would suffice.

Note: This question seems related: 'Free Vector Space' and 'Vector Space'

When I say "action" I guess I mean just a functions from one space onto another. I might also have something in mind like the notion of group object from category theory, but I am not aware of a notion of "ring objects" or "field objects" in category theory, so such a characterization might be lacking.

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Chill2Macht
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    A group action is specifically a group homomorphism into a symmetric group. What do you think would be the "ringification" of the symmetric group? – AJY Sep 03 '16 at 01:42
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    I would look into monoidal categories. Specifically, a monoid in a monoidal category (for example, a ring is a monoid in the category of abelian groups, a monoid is a monoid in the category of sets) has an associated category of modules. – Andrew Dudzik Sep 03 '16 at 01:48
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    I'd say that your question is pretty meaningless. What can or cannot be thought of as an action is a psychological matter. It may be the case that you have some concrete situation that you want to know if it is an action in some sense or not, and that would be (possibly) an interesting question. But what you think cannot be viewed as an action may well be very much an action to somebody else. – Mariano Suárez-Álvarez Sep 03 '16 at 02:22
  • @Slade This looks like a good recommendation, since it seems to cover most of the examples I have in mind and is similar to the notion of a group object -- plus it is not so complicated as to not be found in MacLane's book. I appreciate it! – Chill2Macht Sep 03 '16 at 03:04

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Here's a pretty wide generalization:

Any functor $\mathcal{C} \to \mathcal{D}$ can be thought of as an action of $\mathcal{C}$ on $\mathcal{D}$-objects.

For example, if $\mathcal{C}$ is the category corresponding to the group $G$ (i.e. one object, and $\hom(\bullet, \bullet) = G$), then functors $\mathcal{C} \to \mathbf{Set}$ are precisely the left actions of $G$ on sets, functors $\mathcal{C} \to \mathbf{Ab}$ are left $G$-modules, and functors $\mathcal{C} \to \mathbf{Vect}$ are the representations of $G$.

If $R$ is a ring and $\mathcal{C}$ is the corresponding preadditive category (i.e. one object, and $\hom(\bullet, \bullet) = R$), then additive functors $\mathcal{C} \to \mathbf{Ab}$ are the left $R$-modules

If $R$ is itself a field, then $R$-modules are, in fact, $R$-vector spaces.

As an interesting example of this, consider the "Cayley representation" of a small category $\mathcal{C}$ acting on itself. The corresponding functor $F : \mathcal{C} \to \mathbf{Set}$ is

$$ F(X) = \{ f \in \mathrm{Arr(\mathcal{C})} \mid \mathrm{codom}(f) = X \} $$ $$ F(X \xrightarrow{f} Y) : F(X) \to F(Y) : g \mapsto f \circ g $$

(this is closely related to the Yoneda embedding)

Another interesting example I've seen that applies outside of category theory is an account I read in the first place I saw this done with preadditive categories generalizing rings.

The text asserted that there had been several competing notions for the notion of a "module over a graded ring $S$". When it was realized that you could encode a graded ring as a category with:

  • One object for each grading
  • $\hom(m, n)$ is the set of all triples $(m,s,n)$ where $s$ is a degree $n-m$ element of $S$
  • Composition and addition are the multiplication and addition of $S$

then the idea of defining a module to be a functor from this category to $\mathbf{Ab}$ settled the issue of which definition was the "right" one.

  • I like this -- this is a new way to think of functors which I did not know before, plus ties into the Yoneda embedding – Chill2Macht Sep 03 '16 at 23:39