nLab source

This entry is about the basic notion of “source” of a morphism in category theory. For another use in category theory see at sink and for the use in physics see at source field. There is also source forms in variational calculus.

This entry is about the basic notion of “domain” of a morphism in category theory. For other uses, see at domain.

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Definition

The source object, or simply source, of a morphism $f: x \to y$ in some category $C$ is the object $x$. The source of $f$ is also called its domain, although that can be confusing in categories of partial functions.

Given a small category $C$ with set of objects $C_0$ and set of morphisms $C_1$, the source function of $C$ is the function $s: C_1 \to C_0$ that maps each morphism in $C_1$ to its source object in $C_0$.

Generalising this, given an internal category $C$ with object of objects $C_0$ and object of morphisms $C_1$, the source morphism of $C$ is the morphism $s: C_1 \to C_0$ that is part of the definition of internal category.

Warning: there is another meaning of ‘source’ in category theory; see sink.

Related concepts

• target, codomain

• simplicial identities