Martin Orr's Blog

Group actions and universality

Posted by Martin Orr on Monday, 27 April 2009 at 13:23

This continues my earlier post on groups and actions. I want to think some more about Cayley's theorem, and describe how it provides an example of a universal property. (With regard to James's comment on that post, I think universality may be a better way than injectivity of describing my concept of "active group" but I'm not sure how to do that in full).

Universal properties come up everywhere in algebra and assert that there exist unique morphisms satisfying certain conditions. They usually have approximately the following form (it is possible to use category theory to give a precise general definition of a universal property, but that's too technical for here):

A universal object is an object X (with some specified properties) such that for every A (with some properties) there is a unique morphism u : X \to Y (or maybe u : Y \to X) where u, Y in some way relate to A.

That was probably incomprehensible, so here's an example:

A "vector space on a basis S" (S is part of the given data) is a vector space X with a function e : S \to X such that every time you have a vector space A and function a : S \to A, there is a unique linear map u : X \to Y such that a = ue.

Now where do groups and actions come in?

Fix a (passive) group G and let a "G-set" be a set X together with an action of G on X. A "G-homomorphism" is a function u : X \to Y, with X and Y G-sets, such that u(g.x) = g.u(x) for all x \in X, g \in G.

Now (the proof of) Cayley's theorem says that G is itself a G-set, using what's called the "regular action".

But there's more to it than that: the regular action is in some sense the simplest possible action of G which preserves the full structure of G. This ought to be described by a universal property.

We get part way by observing that, given any G-set X, if you pick some element e of X then you get a G-homomorphism u : G \to X given by u(g) = g.e. But this doesn't give us the uniqueness needed for a universal property: pick different e and you get a different u.

To resolve this, we make the choice of e part of the data:

A "pointed G-set" (X, e) is a G-set X together with a specified element e of X. A "pointed G-homomorphism" is a a function u : X \to Y, with (X, e) and (Y, e') pointed G-sets, such that u is a G-homomorphism and u(e) = e'.

To make G with the regular action into a pointed G-set, we fall back to the method we used in the proof of Cayley's theorem: there is only one possible element of G you can choose in complete generality, and that is the identity element.

So now we can characterise the regular action of G as the "universal pointed G-set": for every pointed G-set (X, e) there is a unique pointed G-homomorphism (G, 1) \to (X, e). (Exercise: prove this. I've essentially done it for you, but you should check uniqueness.)

Tags categories, groups, maths

Trackbacks

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