Martin's Blog

Cayley's Theorem and the Yoneda Lemma

Posted by Martin Orr on Sunday, 10 May 2009 at 16:07

When I wrote my first post on Cayley's theorem, I noticed that Wikipedia claims that the Yoneda lemma is "a vast generalisation of Cayley's theorem". In this post I will try to understand why, and end up concluding that this is probably false.

no comments Tags categories, groups, maths, yoneda

Groups as categories

Posted by Martin Orr on Saturday, 02 May 2009 at 17:12

This post explains how we can consider groups as categories, along with treating the G-sets and G-homomorphisms I considered in my last post on group actions as category-theoretic objects. This is preparation for talking about the Yoneda lemma. Before reading this post, you will need to know the definitions of categories, functors and natural transformations.

no comments Tags categories, groups, maths

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).

no comments Tags categories, groups, maths

Active and passive groups

Posted by Martin Orr on Thursday, 02 October 2008 at 19:13

I have been studying category theory recently, and revising my idea of what a group is. There are two ways of thinking about groups, which I shall call active and passive. I have tended to almost exclusively think about groups passively, but I have realised that treating groups as active is useful too.

3 comments Tags groups, maths

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