Posted in math.CT on April 25, 2015 |
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Posted in math.CT on March 28, 2015 |
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The goal of this post is to summarize some more-or-less standard facts about projective objects. A subtlety that arises here is that in abelian categories there are several conditions equivalent to being projective (we’ll list seven of them below) which are not equivalent in general. We’ll pay more attention than might be usual to this issue.
In particular, several times below we’ll give a list of conditions and a hypothesis under which they’ll be equivalent, and these conditions won’t all be equivalent in general. In these lists we’ll adopt the following convention: whenever we give a list of conditions and prove implications between them, the list will be organized so that proofs downward are easier and require fewer hypotheses, while proofs upward are harder and require more hypotheses. We’ll also prove more implications than we strictly need in order to see this more explicitly.
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Posted in math.AG, math.AT, math.NT, tagged cohomology on November 29, 2014 |
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The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let be a commutative ring, which in the most classical case might be a number field like or the ring of integers in a number field like . Suppose we want to find solutions, over , of a system of polynomial equations
Then it’s not hard to see that this problem is equivalent to the problem of finding -algebra homomorphisms from to . This is equivalent to the problem of finding left inverses to the morphism
of commutative rings making an -algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map
of affine schemes. Allowing to be a more general scheme over can also capture more general Diophantine problems.
The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.
We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.
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In this post we’ll compute the (topological) cohomology of smooth projective (complex) hypersurfaces in . When the resulting complex surfaces give nice examples of 4-manifolds, and we’ll make use of various facts about 4-manifold topology to try to say more in this case; in particular we’ll be able to compute, in a fairly indirect way, the ring structure on cohomology. This answers a question raised by Akhil Mathew in this blog post.
Our route towards this result will turn out to pass through all of the most common types of characteristic classes: we’ll invoke, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes.
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Let be a (locally small) category. Recall that any such category naturally admits a Yoneda embedding
into its presheaf category (where we use to denote the category of functors ). The Yoneda lemma asserts in particular that is full and faithful, which justifies calling it an embedding.
When is in addition assumed to be small, the Yoneda embedding has the following elegant universal property.
Theorem: The Yoneda embedding exhibits as the free cocompletion of in the sense that for any cocomplete category , the restriction functor
from the category of cocontinuous functors to the category of functors is an equivalence. In particular, any functor extends (uniquely, up to natural isomorphism) to a cocontinuous functor , and all cocontinuous functors arise this way (up to natural isomorphism).
Colimits should be thought of as a general notion of gluing, so the above should be understood as the claim that is the category obtained by “freely gluing together” the objects of in a way dictated by the morphisms. This intuition is important when trying to understand the definition of, among other things, a simplicial set. A simplicial set is by definition a presheaf on a certain category, the simplex category, and the universal property above says that this means simplicial sets are obtained by “freely gluing together” simplices.
In this post we’ll content ourselves with meandering towards a proof of the above result. In a subsequent post we’ll give a sampling of applications.
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