# 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.

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## 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.

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## 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).

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## 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.