nLab fork

Idea

A fork is a diagram of the form

$$A\underset{\quad e \quad}{\to}B\underoverset{\quad g \quad}{f}{\rightrightarrows}C$$

such that $f e=g e$. An example of a special type of a fork is an equalizer. Another example is a reflexive fork, where $C = A$ and $f e = 1_A$.

Note that, technically speaking, the diagram above may not commute, since $f$ and $g$ themselves are not necessarily equal. (But some authors, especially when talking about equalizers, loosely use the term “commutative diagram” even in this case.)

A dual notion is also called a cofork, although some references do not distinguish forks and coforks.

Related concepts

• parallel pair
• monadicity theorem