nLab
parallel morphisms

Contents

Idea

Two morphisms in a category CC 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.

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

Revised on May 6, 2017 04:46:20 by Urs Schreiber (92.218.150.85)