Since a great many folk with even moderate mathematical education know what groups are, I have chosen to use them to illustrate plaintext notation by describing groups in terms of them. Where available, I shall include links to proofs of relevant statements, but I have striven to make the illustrations intelligible without reference to these.

# Groups

Here are some definitions of what a group is.

A group is a category of mutually composable isomorphisms.

This says that: you can combine any two morphisms (members of the group), in either order, with a member of the group as the composite; the identities before and after each morphism must all be the same (as they're all mutually composable), so there's a unique identity; and each member has an inverse (because it's iso).

A group is an associative, cancellable, complete binary operator.

The associative binary operator is the category's composition with completeness following from each morphism being factorisable and cancellability from the fact that each of the morphisms used is iso, hence both epic (left-cancellable) and monic (right-cancellable).

We define the category Group to be the collection of functions (G:f:H) between groups which respect the group structure, in the sense that composing, in H, the results of applying f to two members of G gives the same answer as applying f to the composite, in G, of its two given members. That is, for any b,d in G: we have bd in G so f(bd) in H; and f(b), f(d) in H combine to give f(b)f(d) = f(bd). The morphisms of Group are commonly called group homomorphisms.

In the category Set (whose morphisms are called functions), we can look at the collection of isomorphisms from a given set, S, to itself: this is called iso(S,S). This forms a sub-category of Set and its members are mutually composable, so it's a group. For any group G with binary operator +, we can define a function left = (G: g -> (G: h->g+h :G) :hom(G,G)), observe that the left(g) is inverse to left(-g) and so conclude that we have embedded G in iso(G,G).

Written by Eddy.