nLab parallel morphisms




Two morphisms in a category CC (or just edges in a directed graph) are parallel if they have the same source and target. Equivalently a pair of parallel morphisms in CC consists of an object xx, and object yy, and two morphisms f,g:xyf, g: x \to y.

x gf y \array{ x & \underoverset {\underset{g}{\longrightarrow}} {\overset{f}{\longrightarrow}} {} & y }

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 walking parallel pair

The above considerations can be formalized in the following definition.


The walking parallel pair category PP has two objects, 0 and 1, and two nonidentity arrows, f,g:01f,g\colon 0\to 1.

Now functors PCP\to C are precisely pairs of parallel morphisms.

Limits and colimits

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.)

shapes of free diagrams and the names of their limits/colimits

free diagramlimit/colimit
empty diagramterminal object/initial object
discrete diagramproduct/coproduct
parallel morphismsequalizer/coequalizer
span/cospanpullback,fiber product/pushout
tower/cotowersequential limit/sequential colimit

Last revised on February 29, 2024 at 11:25:28. See the history of this page for a list of all contributions to it.