nLab
domain

In category theory, the domain of a morphism is generally the same as its source; that is, the domain of $f:X\to Y$ is $X$. In particular, this is the case for a function: its domain is the set of elements to which it can be applied.

However, this can conflict with other meanings of ‘domain’, especially in a category like Rel. For instance, for any subset $A\subseteq X$, there exists a relation $R:X\to Y$ whose “domain” is $A$ under some uses of the term.

Other similar meanings of the term include:

• the domain of discourse of a predicate,
• an open subset of a manifold, especially in complex analysis.

A separate meaning of ‘domain’ occurs in domain theory, which is at the interface of logic and theoretical computer science. There a domain is a particular type of poset.