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.

Reading Notes

A Graphical View of Categories (I)

A category is [may be thought of as] a directed graph in the sense of Serre—so an ordered 4-tuple (V, E, \iota, \tau) whose members are, resp, the vertices (objects), the edges or arrows (morphisms), the initial vertex (domain) and terminal vertex (codomain) functions—with

  • loops at all vertices (these being the identity morphisms)
  • f \to g \to h implies g \to h (composition of morphisms)

There are some subtleties involved here in trying to apply this, in full generality, to actual collections of objects of mathematical interest, due to things of a set-theoretic nature related to Russell’s paradox, but we shall brush over them here.

A functor is a morphism of categories, i.e. a map between categories which respects the structure restrictions on the edges / arrows. From Mac Lane’s Categories for Working Mathematicians (13): ‘Functors were first explicitly recognized in algebraic topology, where they arise naturally when geometric properties are described by means of algebraic invariants … The leading idea in the use of functors in topology is that H_n or \pi_n gives an algebraic picture or image not just of the topological spaces, but also of all the continuous maps between them.’

A functor is full if every arrow in the target category comes from an arrow in the source category, i.e. a surjective morphism of categories; a functor is faithful if it is an embedding of directed graphs, i.e. an injective morphism of categories.

A natural transformation between two functors S, T: C \to B, sometimes called a morphism of functors, is a function from the objects of C to arrows in B which is equivariant under the arrows in C, in other words

‘As Eilenberg-Mac Lane first observed, “category’ has been defined in order to be able to define “functor” and “functor” has been defined in order to be able to define “natural transformation”.’ (Mac Lane 18)