I'm using the following definition:
A category $\mathcal C$ is given by the following:
- A collection of objects $A$
- A set $M(A, B)$ for any two objects $A, B \in \mathcal C$.
- A function $M(B, C) \times M(A, B) \to M(A, C)$ for each triple of objects $A, B, C \in \mathcal C$.
These terms must satisfy the following axioms
I: If $\alpha : A \to B$, $\beta : B \to C$ and $\gamma : C \to D$, then $\gamma(\beta \alpha) = (\gamma \beta)\alpha$.
II: For each $A \in \mathcal C$, there exists $i_A : A \to A$ such that if $\beta : B \to A$, $\gamma : A \to C$ then $i_A \beta = \beta$ and $\gamma i_A = \gamma$.
A category $\mathcal C$ shall be called a $K$-category where $K$ is a commutative ring with unity if it has a distinguished object 0, the zero object, and satisfies the following three axioms:
III: $M(A, B)$ is a unital $K$-module.
IV: The function $M(B, C) \times M(A, B) \to M(A, C)$ is a bilinear map of $K$-modules
V: $M(0, 0)$ is the zero module, also written 0.
The following theorem is what I am having trouble with. I can't seem to be able to get the result using the previous definition of $K$-category (From Swan's 'The Theory of Sheaves')
Prove: $M(0, A) = M(0, 0) = M(A, 0)$.
Proof:Since we know $M(A, B)$ is a set (moreover module) for any $A, B \in \mathcal C$, then it would suffice to show that $M(0, A) \subseteq M(0, 0)$ and $M(0, 0) \subseteq M(0, A)$. The latter inclusion is trivial since $M(0, 0)$ is the zero module and every module has the zero module as a submodule.
Now let $f \in M(0, A)$. We want to show that $f \in M(0, 0)$, but since $M(0, 0)$ is the zero module and it is an additive subgroup, its only element is the 'zero'-map such that the following two axioms hold: (Axioms of module)
- $f + 0 = f$ for all $f \in M(A, B)$
- $f + (-f) = 0$ for all $f \in M(A, B)$
Here is where I am stuck. I can't seem to get the result directly from using the axioms I was given. Any help?
As a sub-question (soft), does anyone have any opinions/recommendations on my choice of textbook on Sheaf Theory? This is my first introduction to the subject and I wanted a textbook that didn't have too much topology and this seemed great, but I've been finding a bit of types :(