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Let object of the category be integers and for objects $a$ and $b$, define: $$ \operatorname{hom}\left(a, b\right) = \left\{ f \in \mathbb{Z} | a - f = b\right\} $$ Then composition of $f \in \operatorname{hom} \left(a,b\right)$ and $g \in \operatorname{hom} \left(b,c\right)$ is addition which is associative. Also for all integers $x$, $ \exists 0 \in \operatorname{hom}\left(x,x\right)$ which acts like an identity, but then obviously all objects have the same identity. Does this prevent us from considering $\mathbb{Z}$ with subtraction as a category?

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    Yes, by (my usual) definition of category one should be able to recover $a$ and $b$ from $f\in \operatorname{hom}(a,b)$. But iIf one repairs this problem for your category it becomes simply a category with countably many objects and a unique morphism $a\to b$ for any pair of objects.2012-10-06
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    It is rather curious to have the same values (integers) both as objects and as morphisms. And it is irrelevant what morphisms are called (the only point of your definition is that all homsets are singletons), but you cannot freely call two different morphisms by the same name (for general reasons that have little to do with category theory). Some violation of this rule is accepted (for zero morphisms in an Abelian category for instance) as long as it is understood that having the same name does not imply being the same thing.2012-10-06

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