Morphisms between representations of group algebras I am trying to put together a categorical understanding of group representations, and so I would like to understand the equivalence between representations of a group $G \rightarrow \operatorname{GL}(V)$ and representations of the group algebra $F[G] \rightarrow \operatorname{End}(V)$ as a functor from the category of representations of $G$ to the category or representations of $F[G]$.
The problem is, I'm not sure exactly what the morphisms should be for representations of $F[G]$. I can think of the category of representations of $G$ as the functor category from $B_G$ to $\operatorname{Vect}_F$, where $B_G$ is the one-element category $G$. Then morphisms between representations are natural transformations between functors, and this boils down to the usual definition. For representations of $F[G]$, however, I'm not sure how to see those as a category, because I don't know how to represent $F[G]$ as a category. Is there a way to see the collection of $F$-algebra homomorphisms from $F[G]$ to $\operatorname{End}(V)$ as a category of functors, similar to the case for representations of $G$?
 A: For the rest of this post, we fix a field (or more generally a commutative ring) $$.
Instead of just groups, we will consider arbitrary monoids.
To solve the problem at hand, we will consider $$-linear categories.
These will fit into the following table.




set world
$$-linear world




sets
vector spaces


monoids
algebras


categories
$$-linear category




More explicitly, we will have all the following similarities:

*

*A monoid is a set $M$ together with a multiplication map $M × M \to M$ that is associative and admits a unit.
Similarly, an algebra is a vector space $A$ together with a bilinear $A × A \to A$ that is associative and admits a unit.


*Given a category $\mathcal{C}$, we have for every two objects $X$ and $Y$ of $\mathcal{A}$ the set $\mathrm{Hom}_{\mathcal{C}}(X, Y)$.
Similarly, given a $$-linear category $\mathcal{A}$, we have for every two objects $A$ and $B$ a vector space $\mathrm{Hom}_{\mathcal{A}}(A, B)$.


*Given a category $\mathcal{C}$, we have for every object $X$ of $\mathcal{C}$ the monoid $\mathrm{End}_{\mathcal{C}}(X)$.
Similarly, given a $$-linear category $\mathcal{A}$, we have for every object $A$ of $\mathcal{A}$ the algebra $\mathrm{End}_{\mathcal{A}}(A)$.


*A monoid is the same as a one-object category.
Similarly, an algebra is the same as a one-object $$-linear category.


*We can go from the right column to the left column by forgetting the $$-linear structure:
every vector space has an underlying set, every algebra has an underlying (multiplicative) monoid, and every $$-linear category has an underlying category.


*Conversely, we can also go from the left column to the right column:
every set $X$ can be linearized into the free vector space $[X]$,
every monoid $M$ can be linearized into the monoid algebra $[M]$,
and every category $\mathcal{C}$ can be linearized into the $$-linear category $[\mathcal{C}]$.
These linearizations satisfy similar universal properties.


*For every set $X$ and vector space $V$, we can consider the vector space of maps from $X$ to $V$.
Similarly, for every category $\mathcal{C}$ and $$-linear category $\mathcal{A}$, we can consider the $$-linear category of functors from $\mathcal{C}$ to $\mathcal{A}$.
Definition of a $$-linear category

Definition.
A $$-linear category consists of

*

*a category $\mathcal{A}$, and

*for every two objects $A$ and $B$ of $\mathcal{A}$ a $$-vector space structure on $\mathrm{Hom}_{\mathcal{A}}(A, B)$,

such that for every three objects $A$, $B$ and $C$ of $\mathcal{A}$ the composition map
$$
  \mathrm{Hom}_{\mathcal{A}}(A, B) × \mathrm{Hom}_{\mathcal{A}}(B, C)
  \to
  \mathrm{Hom}_{\mathcal{A}}(A, C) \,,
  \quad
  (f, g)
  \mapsto
  g ∘ f
  $$
is $$-bilinear.

Every $$-linear category has an underlying ordinary category, which comes from forgetting the vector space structure on the morphism sets.
We can therefore say that ‘a $$-linear category is a category’.
Given a specific category $\mathcal{A}$, we can conversely ask if we can make $\mathcal{A}$ into a $$-linear category.
For this, we need to define for every two objects $A$ and $B$ of $\mathcal{A}$ a vector space structure on the set $\mathrm{Hom}_{\mathcal{A}}(A, B)$, in such a way that
\begin{alignat*}{2}
 (g_1 + g_2) ∘ f &= g_1 ∘ f + g_2 ∘ f \,,
 &\quad
 (λ g) ∘ f &= λ (g ∘ f)
 \\
 g ∘ (f_1 + f_2) &= g ∘ f_1 + g ∘ f_2 \,,
 &\quad
 g ∘ (λ f) &= λ (g ∘ f) \,,
\end{alignat*}
whenever these equations make sense.
Examples and constructions for $$-linear categories
The primordial example of a $$-linear category is the one of $$-vector spaces:

*

*Let $\mathbf{Vect}$ denote the category of $$-vector spaces.
Given two $$-vector spaces $V$ and $W$, we learn in linear algebra that “$\mathrm{Hom}_(V, W)$ is again a $$-vector space”.
This allows us to make $\mathbf{Vect}$ into a $$-linear category.

Many examples of $$-linear categories are in a similar vein:

Construction.
Let $\mathcal{A}$ be a category.
Suppose that every object $A$ of $\mathcal{A}$ has an underlying vector space structure, and that for every two objects $A$ and $B$ of $\mathcal{A}$ the set $\mathrm{Hom}_{\mathcal{A}}(A, B)$ is a linear subspace of $\mathrm{Hom}_(A, B)$.
By endowing $\mathrm{Hom}_{\mathcal{A}}(A, B)$ with the vector space structure induced from $\mathrm{Hom}_(A, B)$, we can make $\mathcal{A}$ into a $$-linear category.

Let us emphasize two special cases of this general construction:

*

*For every $$-algebra $A$ we have the category of left $A$-modules, which we shall denote by $\mathbf{Mod}(A)$.


*For every monoid $M$ we have the category of representations of $M$ over $$, which we shall denote by $\mathbf{Rep}(M)$.
Let us also give a non-example of the above construction:

*

*Let $\mathbf{Alg}$ be the category of $$-algebras.
Every $$-algebra has an underlying vector space structure.
But given two $$-algebras $A$ and $B$, the set $\mathrm{Hom}_{\mathbf{Alg}}(A, B)$ is not a linear subspace of $\mathrm{Hom}_(A, B)$.

It should also be stressed that in an arbitrary $$-linear category, the objects don’t need to have an underlying vector space structure!
Let us introduce an example for this:
functor categories.

Construction.
Let $\mathcal{C}$ be a category and let $\mathcal{A}$ be a $$-linear category.
Let $F$ and $G$ be two functors from $\mathcal{C}$ to $\mathcal{A}$.
Given two natural transformations
$$
 α, β \colon F \Longrightarrow G
$$
and a scalar $λ ∈ $, we can define new natural transformations
$$
 α + β, λ α \colon F \Longrightarrow G
$$
via the components
$$
 (α + β)_X ≔ α_X + β_X
 \qquad
 \text{for every $X ∈ \mathcal{C}$}
$$
and
$$
 (λ α)_X ≔ λ α_X
 \qquad
 \text{for every $X ∈ \mathcal{C}$.}
$$
The set of natural transformations from $F$ to $G$ becomes a $$-vector space in this way.
These vector space structures make the functor category $\mathbf{Fun}(\mathcal{C}, \mathcal{A})$ into a $$-linear category.

One should compare the above construction to a statement from linear algebra:
for a set $X$ and a vector space $V$, the set of maps from $X$ to $V$ can again be made into a vector space via pointwise addition and scalar multiplication of functions.
Functors between $$-linear categories

Definition.
Let $\mathcal{A}$ and $\mathcal{B}$ be two $$-linear categories.
A functor $F$ from $\mathcal{A}$ to $\mathcal{B}$ is $$-linear if for every two objects $A$ and $B$ the map
$$
 \mathrm{Hom}_{\mathcal{A}}(A, B)
 \xrightarrow{\enspace F \enspace}
 \mathrm{Hom}_{\mathcal{B}}(F(A), F(B))
$$
is $$-linear.
The full subcategory of $\mathbf{Fun}( \mathcal{A}, \mathcal{B} )$ whose objects are the $$-linear functors is denoted by $\mathbf{Fun}_ (\mathcal{A}, \mathcal{B} )$.

We can now talk about equivalences and isomorphisms of $$-linear categories.

*

*Let $M$ be a monoid.
The category $\mathbf{Rep}(M)$ is isomorphic to the functor category $\mathrm{Fun}( \mathrm{B}_M, \mathbf{Vect} )$ as $$-linear categories.

One-object $$-linear categories
Given an object $A$ in a $$-linear category $\mathcal{A}$, we can consider $\mathrm{End}_{\mathcal{A}}(A)$.
This is a $$-vector space, and together with composition of endomorphisms, it becomes a $$-algebra.
Suppose conversely that $A$ is a $$-algebra.
We can then define a $$-linear category $\mathrm{B}_A$ consisting of a single object $\ast$ with $\mathrm{End}_{\mathrm{B}_A}(\ast) = A$.
We can now formulate a linear version of the previous isomorphism $\mathbf{Rep}(M) ≅ \mathbf{Fun}(\mathrm{B}_M, \mathbf{Vect})$.

*

*Let $A$ be a $$-algebra.
The category $\mathbf{Mod}(A)$ is isomorphic to the functor category $\mathbf{Fun}_( \mathrm{B}_A, \mathbf{Vect} )$ as $$-linear categories.
(This is how we can regard the category of representations of $A$ as a functor category:
by using $$-linear categories and $$-linear functors instead of just ordinary categories and ordinary functors.)

Linearization of categories

Construction.
Let $\mathcal{C}$ be a category.
We can define a $$-linear category $[\mathcal{C}]$ as follows:

*

*The objects of $[\mathcal{C}]$ are the objects of $\mathcal{C}$.

*For every two objects $A$ and $B$ of $\mathcal{C}$, the vector space $\mathrm{Hom}_{[\mathcal{C}]}(A, B)$ is the free vector space on the set $\mathrm{Hom}_{\mathcal{C}}(A, B)$.

*The composition of morphism in $[\mathcal{C}]$ is the $$-bilinear extension of the composition of morphisms in $\mathcal{C}$.
The $$-linear category $[\mathcal{C}]$ is the linearization of $\mathcal{C}$.


Let us make the following observation.

*

*Let $M$ be a monoid.
Then $[\mathrm{B}_M] = \mathrm{B}_{[M]}$.

The linearization $[\mathcal{C}]$ has the following universal property:

Proposition.
Let $\mathcal{C}$ be a category and let $\mathcal{A}$ be a $$-linear category.
Every functor from $\mathcal{C}$ to $\mathcal{A}$ extends uniquely to a $$-linear functor from $[\mathcal{C}]$ to $\mathcal{A}$.
We get an isomorphism of $$-linear categories
$$
    \mathbf{Fun}_( [\mathcal{C}], \mathcal{A} )
    ≅
    \mathbf{Fun}( \mathcal{C}, \mathcal{A} ) \,.
  $$

This universal property should be compared to the correspondence between linear maps $[X] \to V$ and maps $X \to V$ for a set $X$ and vector space $V$, and the correspondence between algebra homomorphisms $[M] \to A$ and monoid homomorphisms $M \to A$ for a monoid $M$ and an algebra $A$.
Consequence
For every monoid $M$ we can now consider the chain of isomorphisms of $$-linear categories
$$
 \mathbf{Rep}(M)
 ≅
 \mathbf{Fun}( \mathrm{B}_M, \mathbf{Vect} )
 ≅
 \mathbf{Fun}_( [\mathrm{B}_M], \mathbf{Vect} )
 =
 \mathbf{Fun}_( \mathrm{B}_{[M]}, \mathbf{Vect} )
 ≅
 \mathbf{Mod}([M]) \,.
$$
