Reading Notes

A Graphical View of Categories (II): Monos, Epis, Zeros, Groupoids

Monomorphisms and epimorphisms are special sorts of arrows: they are analogues / generalisations of injective and surjective maps (resp.)—except now we do not speak of elements! The properties of injections and surjections that we wish to abstract involve cancellation under composition: a monomorphism is a morphism which may be cancelled on the left (in compositions); an epimorphism is one which may be cancelled on the right. In graph terms: a monomorphism induces through composition a unique arrow out of its domain for each arrow out of its codomain; an epimorphism induces through composition a unique arrow into its codomain for every arrow into its domain.

An object is terminal if there is a unique arrow from it to every other object in the diagram; it is initial if there is a unique arrow to it from every other object in the diagram. You may be reminded of sources and sinks, but the notions are not quite the same: here we do not care that terminal objects may also have outgoing arrows or that initials may also have incoming arrows, and we do care about the number of arrows between our initials / terminals and each of the other vertices. An object is a zero object if it is both terminal and initial.

The inverse of an arrow is an arrow going the other way, such that composition yields identity morphisms (on both sides.)

A category all of whose arrows are invertible is a groupoid; a groupoid with one object is a group. In other words, one may represent a group as a bouquet of circles, with one loop for each group element, corresponding to the morphism given by left-multiplication by that element.

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