nLab parallel morphisms

Idea

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

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

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.

Definition

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

Now functors $P\to C$ are pecisely 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 diagram	limit/colimit
empty diagram	terminal object/initial object
discrete diagram	product/coproduct
parallel morphisms	equalizer/coequalizer
span/cospan	pullback,fiber product/pushout
tower/cotower	sequential limit/sequential colimit