nLab
domain

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Meanings

In category theory, the domain of a morphism is its source object; that is, the domain of f:XYf\colon X \to Y is XX. 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 AXA\subseteq X, there exists a relation R:XYR\colon X \to Y whose “domain” is AA under some uses of the term.

Other similar meanings of the term include:

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. Still another meaning occurs in algebra in the term integral domain.

Revised on January 2, 2015 13:41:00 by Todd Trimble (127.0.0.1)