Two morphisms in a category (or just edges in a directed graph) are parallel if they have the same source and target. Equivalently a pair of parallel morphisms in consists of an object , and object , and two morphisms .
This may be extended to a family of any number of morphisms, but the morphisms are always compared pairwise to see if they are parallel. Degenerate cases: a family of one parallel morphism is simply a morphism; a family of zero parallel morphisms is simply a pair of objects.
The above considerations can be formalized in the following definition.
The walking parallel pair category has two objects, 0 and 1, and two nonidentity arrows, .
Now functors are precisely pairs of parallel morphisms.
The limit of a pair (or family) or morphisms is called their equalizer; the colimit is their coequalizer. (Of course, these do not always exist.)
Last revised on February 29, 2024 at 11:25:28. See the history of this page for a list of all contributions to it.