[[Graded vector space]]
# Closed category of graded vector spaces

Let $(\mathfrak{A}, +)$ be a [[monoid]] and $\mathbb{K}$ be a [[field]].
The **closed category of $\mathfrak{A}$-graded vector spaces** $\cat{gr}_{\mathfrak{A}}\Vect_{\mathbb{K}}$ over $\mathbb{K}$ is the [[category]] 
where an object is a $\mathfrak{A}$-[[graded vector space]]
and a morphism is a $\mathbb{K}$-[[linear map]], #m/def/linalg 
[[Monoidal closed category|closed monoidal]] so that $\cat{gr}_{\mathfrak{A}}\Vect_{\mathbb{K}}(V,U)$ has a gradation given by [[Homomorphism of graded vector spaces#^homogenous]] maps.
In particular, the [[endomorphism ring]] $\End_{\mathbb{K}} V$ is a [[graded algebra]].

> [!missing]- Proof
> #missing/proof

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#state/tidy  | #lang/en | #SemBr