Contents

category theory

# Contents

## 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.)

Last revised on January 31, 2021 at 16:55:35. See the history of this page for a list of all contributions to it.