Confusion with notions of reducibility of representations

Question

I'm taking a first course on linear groups. I've never read anything about group representations before.

Consider the following three conditions on an object of a pointed category.

1. Simple - has only the identity and the terminal arrow as quotients.
2. Indecomposable - is not a product.
3. Semi-simple - is a product of simple objects.

If the category in question is abelian, the bijection between subobjects and quotients, along with the existence of biproducts, give simpler characterizations of these conditions.

I learned from the nlab that these conditions are studied in representation theory. However, the lecturer gave seemingly different definitions in class. I don't feel as comfortable with these.

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Definition 1a. Let $V$ be an $\mathbb F$-vector space and $G\leq \mathrm{GL}(V)$ a linear group.

1. A linear subspace $W\leq V$ is a $G$-invariant subspace if $gW=W$ for every $g\in G$. A minimal $G$-invariant subspace is one which does not contain a non-trivial $G$-invariant subspace.
2. $G$ is called reducible if $V$ admits a non-trivial proper $G$-invariant subspace $W$. $G$ is called irreducible if it isn't reducible.
3. $G$ is completely reducible if there exist minimal $G$-invariant subspaces $W_1,\dots,W_k$ such that $V=W_1\oplus\cdots\oplus W_k$.

Definition 1b. Let $\rho:G\to \mathrm{GL}(V)$ be a representation of $G$. A linear subspace $W\leq V$ is called a $G$-invariant subspace if it's an $\mathrm{Im}\rho$-invariant subspace. Analogously for reducibility, irreducibility, and complete reducibility.

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Definition 1a feels "wrong" to me. It feels $V$ shouldn't be mentioned at all, and that we should look at the category of all vector spaces equipped with $G$-actions (respecting the linear structure). Then I'm somehow hoping that "minimal" can be replaced with "simple" in a corresponding category. Then, definition 1b will be much more reminiscent of simplicity and semisimplicity.

Moreover, definition 1b simply looks different to me from the definition at the nlab (as a simple object in the category of representations). Why are these definitions equivalent? In particular, why is "complete reducibility" equivalent to being a semisimple object in the category of representations?

The minimal $G$-invariant subspaces are minimal elements of the subposet $\mathrm{Fix}(G\curvearrowright \mathrm{Sub}(V))$ of $G$-fixed points of the action of $G$ on the lattice of subspaces of $V$. I don't understand how this coincides with simple objects in some category not mentioning $V$ explicitly..

Answer 1

As you alluded to, in an abelian category, "simple" is equivalent to "the only subobjects (up to isomorphism) are the identity and the initial arrow". In the case of these definitions, the relevant category is (as you guessed) the category of vector spaces equipped with $G$-actions, which is an abelian category. A monomorphism $f:W\to V$ in this category is just an injective linear map preserving the action of $G$. The image of such a monomorphism $f$ is a $G$-invariant subspace of $V$ which is isomorphic to $W$ (and conversely if $W$ is $G$-invariant subspace then the inclusion $W\to V$ is a monomorphism), and so up to isomorphism a monomorphism $W\to V$ is the same thing as a $G$-invariant subspace of $V$. So a simple object in this category is just a vector space $V$ with an action of $G$ such that the only $G$-invariant subspaces are $V$ and $0$.

That is, simple objects are the same as irreducible representations. Moreover, a $G$-invariant subspace $W$ of $V$ is minimal iff the action of $G$ on $W$ is irreducible. That is, minimal subspaces of $V$ are the same thing as subobjects of $V$ which happen to be simple objects as well. So "completely reducible" just says that $V$ is an (internal) finite direct sum of simple subobjects, which is equivalent to being an (external/categorical) finite product of simple objects.

(You mentioned indecomposability as well, but that does not correspond to any of your lecturer's definitions. Both "minimal" and "irreducible" correspond to "simple", not "indecomposable".)

Answer 2

You can, of course, avoid to mention anything concrete and say that $Rep(G)$ is a category $Fun(G, Vect_{\mathbb k})$ where $G$ is one-object category. More generally, $Fun(G, R$-mod$)$ is always equivalent to category of left modules over the group algebra $A[G]$-mod even for noncommutative basic ring. Simplicity, semisimplicity and irresucibility work in it tautologically. But it's often very convenient to have definitions wich work "on objects and elements" and not "on arrows" because if you try, for example, to prove 5-lemma abstractly without referring to elements of any kind you'll have hard time.

Functors between one object groupoids are obviously homomorphisms, and given $\phi: G \to G'$ you have pullback $\phi*$ and its left and right adjoints which play a role in studiyng group cohomology. If you want to go further, there's a functor $\mathcal R: Grp^{op} \to Cat$ sending a group to it's representation category from which you can construct the "category of all representations over $\mathbb k$" via Grothendieck constuction.