I am studying representation theory of categories, and I am getting stuck on the following definition given in Sam & Snowden’s paper on Noetherian Categories.
Define a representation $P_x$ of $C$ By $P_x(y)=k[Hom(x,y)]$, i.e. $P_x(y)$ is the free left $k$-module with basis Hom(x,y).
Now, I understand the concept of this definition, but in practice it does not make sense to me for any general small category. For instance, if we consider the poset category $C$ with Ob$(C)=\mathbb{N}$ and Mor$(C)=\{n\to m \text{ if } n\leq m\}$. Then say we look at $P_1(2)=k[Hom(1,2)]=k[1\to 2]$. What does this actually look like? $[1\to 2]$ should become a free abelian group, but I do not see or understand how.
It's not $[1 \to 2]$ which is an abelian group, it's $k[1 \to 2]$. By definition, $k[\hom(x,y)]$ is the set of formal linear combinations of elements of $\hom(x,y)$ with coefficients in $k$. In other words, an element of $k[\hom(x,y)]$ is (by definition) an expression of the type: $$\sum_{i=1}^n \lambda_i f_i$$ where $\lambda_i \in k$ and $f_i \in \hom(x,y)$. Then algebra in $k[\hom(x,y)]$ is done by manipulating these expressions, adding and multiplying by scalars term-wise. In yet other words, $P_x(y)$ is a free $k$-module, with a basis given by the set $\hom(x,y)$.
It does not matter that $\hom(x,y)$ is a set of morphisms of some categories. You are not really adding functions, you are adding expressions. If for example $1 \to 2$ is a morphism of your poset category $C$, then $(1 \to 2) + (1 \to 2)$ is simply the expression $2 \cdot (1 \to 2)$, nothing else.