Category $\text{Matr}$ with $\text{Obj(Matr)} = \mathbb{Z}_{\geq 0}$

Question

Let $R$ be a commutative ring. I want to define a category $\text{Matr}$ with

1. $\text{Obj(Matr)} = \mathbb{Z}_{\geq 0}$
2. If $n$ and $m$ are two objects, then $\text{Hom}(n,m)$ is the set of all matrices $m \times n$ with entries from $R$.
3. Composition is the usual matrix multiplication.

The problem is that I don't really understand what to do when $n$ or $m$ is $0$. If I set $\text{Hom}(n,m) = \emptyset$ when $n$ xor $m$ is $0$ then what should I do with $\text{Hom}(0,0)$?

Answer 1

You shouldn't set, say, $\mathrm{Hom}(5,0) = \emptyset$, as the matrices in this set should represent the $R$-linear mappings $R^5 \to R^0 = \{0\}$. This set should contain a single mapping, namely the 0 map.

Generally, whenever $n = 0 \lor m = 0$, you want $\mathrm{Hom}(n,m)$ to contain a unique element: the unique $n \times m$ matrix (without any entries), which represents the 0 mapping.

In particular, this makes the number 0 a so-called 0 object.

Answer 2

You dont need to make any distinction when $n$ or $m$ equals $0$. Just recall that a $n\times m$-matrix in $R$ is a map $$ \{1,2,\ldots,n\}\times \{1,2,\ldots,m\}\to R. $$ When either $n$ or $m$ equals $0$, the set $\{1,2,\ldots,n\}\times \{1,2,\ldots,m\}$ is empty. In such case, there exists a unique $n\times m$-matrix in $R$, given by the empty map to $R$.