Martin Orr's Blog

Morphisms and functors of points

Posted by Martin Orr on Thursday, 01 October 2009 at 15:45

This post will discuss the fact that A-points of an affine k-scheme X (and more general objects) are the same as morphisms \mathop{\mathrm{Spec}_k} A \to X. James already brought this up in his comment last time. As well as proving this in the affine k-scheme case, I shall attempt to give an intuitive explanation of this fact, although I don't find this entirely satisfying.

no comments Tags alg-geom, maths, points-func, yoneda Read more...

Functors, affine varieties and Yoneda

Posted by Martin Orr on Wednesday, 02 September 2009 at 22:51

In this article, I will examine in more detail the functor of points of an affine variety, which I defined in the last article. I shall show that this functor is the same as a Hom-functor on the category of k-algebras, and that morphisms of varieties correspond to natural transformations of functors.

no comments Tags alg-geom, maths, points-func, yoneda Read more...

Naturality in the Yoneda lemma for groups

Posted by Martin Orr on Saturday, 16 May 2009 at 19:54

In my last post on the Yoneda lemma for groups, I ignored the naturality part of the lemma. I want to work in detail what this means once - it is a lot of fiddly composing of morphisms and I probably won't do it again (at least in public). If you're not in the mood for following such details, then there is little point in reading this, although you could skip to the last paragraph.

no comments Tags categories, groups, maths, yoneda Read more...

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 Read more...

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