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 $X$.

However, this can conflict with other meanings of ‘domain’, especially in a category like Rel; any subset of $X$ is the domain of some relation $R:X\to Y$.

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.