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I am reading an article and it is written there: If A is a k-linear category (possibly without direct sums) we can embed it in the additive category A × N, where a morphism (x,m) → (y, n) is an n × m matrix with entries in A(x, y) = HomA(x, y). Of course if A is additive then A ≈ A × N. Could anyone explain it? Why is it additive?

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Well, you haven't got the objects quite right. A better description of the objects of the free additive category $\oplus A$ over $A$ is: $n$-tuples of objects $x = (x_{1}, \ldots, x_{n})$ of $A$ where $n \in \mathbb{N}$. Now a morphism in $\text{Hom}\,(x,y)$ is a matrix $(f_{ij})$, where $f_{ij} \in \text{Hom}_A\,(x_{j},y_{i})$ and composition is given by the familiar formula for matrix multiplication.

Can you see why this category is additive?

Also show that it deserves its name and think about the following:

  1. every additive functor $F: A \to B$ to an additive category $B$ extends uniquely to an additive functor $\oplus F : \oplus A \to B$.
  2. If $A$ is already additive then $A$ and $\oplus A$ are equivalent.
  3. What is the free $k$-linear category over $A$?
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