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In defining a (locally small) category, some books include the condition that if $A\neq C$ or $B\neq D$, then $\hom(A,B)$ and $\hom(C,D)$ are disjoint sets. Is it necessary?

ashpool
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    It seems strange to me to allow morphisms to have multiple domains for some reason. – Nick Jan 26 '14 at 01:25
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    You could have the definition either way, and it will not matter too much. For instance suppose that you have $f\in hom(A,B)\cap hom(C,D)$ for $A\neq C$ or $B\neq D$. Then you can define at category where the him sets are triples, $(f,A,B)$ and $(f,C,D)$. In other words you can tag the morphisms with their domains and codomains and they will be separate. – Baby Dragon Jan 26 '14 at 01:30
  • Regarding Baby Dragon's comment: Actually this is done in some (good) texts on set theory, where a function from $A$ to $B$ is really defined as a triple $(A,B,\Gamma)$ (not just $\Gamma$!), where $\Gamma \subseteq A \times B$ has the property $\forall a \in A \exists ! b \in B : (a,b) \in \Gamma$. – Martin Brandenburg Jan 26 '14 at 01:53

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Yes, this is (almost) correct. Every morphism should have a specified domain and codomain. For example, the inclusion $\mathbb{Q} \to \mathbb{R}$ should not be confused with the identity $\mathbb{Q} \to \mathbb{Q}$. The inclusion is not an epimorphism, but the identity is of course an isomorphism.

Actually the "correct" definition of a category is a tuple $(M,O,s,t,\circ,\mathrm{id})$, where $(M,O,s,t)$ is a (possibly large) directed graph, $M$ being the set of edges, $O$ the set of nodes, $s$ the map which assigns to each edge its source, and $t$ the map which assigns to each edge its target, and besides we have a map $\circ : M \times_{t,O,s} M \to M$ (composition) as as well as a map $\mathrm{id} : O \to M$ (identity) satisfying the usual conditions.

By replacing "maps" with "morphisms" here, this definition can be written down in every category with fiber products (internalization), leading to the notion of "category object" and in particular (after adding inverses) "groupoid object", which is useful in algebraic geometry as well as algebraic topology.

As you see, there are no hom-sets here. But you can define them as the pullbacks $\hom(x,y) = \{x\} \times_O M \times_O \{y\}$. For a category theorist, it isn't really sensible to ask for disjointness of these sets, since this is a question of set theory, not invariant under equivalences of categories (which has been called evil in a former nlab article). But the source and target maps $s$ and $t$ allow us to recover $x,y$ from a morphism $x \to y$.

Martin Brandenburg
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