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.
Read Full Post »
Occasionally I see mathematical questions that seem “grammatically incorrect” in some sense.
Example. “Is open or closed?”
Example. “Is a group?”
Example. “What’s the Fourier series of ?”
Here are some sillier examples.
Example. “Is a rectangle prime?”
Example. “Is ?”
Example. “What’s the Fourier series of the empty set?”
What all of these examples have in common is that they are type errors: they are attempts to apply some mathematical process to a kind of mathematical object it was never intended to take as input. If you tried to write a program in some highly mathematical programming language to answer these questions, it (hopefully!) wouldn’t compile.
Mathematical objects are usually not explicitly thought of as having types in the same way that objects in a programming language with a type system has types. Ordinary mathematics is supposed to be formalizable within Zermelo-Fraenkel (ZF) set theory, possibly with the axiom of choice, and in ZF every mathematical object is constructed as a set. In that sense they all have the same type. (In particular, the question “is ?” is perfectly meaningful in ZF! This is one reason not to like ZF as a foundation of mathematics.) However, I think that in practice mathematical objects are implicitly thought of as having types, and that this is a mental habit mathematicians pick up but don’t often talk about.
Instead of thinking in terms of set theory, thinking of mathematical objects as having types allows us to import various useful concepts into mathematics, such as the notions of type safety, typecasting, subtyping, and overloading, that help us make more precise what we mean by a mathematical sentence being “grammatically incorrect.” The rest of this post will be a leisurely discussion of these and other type-based concepts as applied to mathematics in general. There are various categorical ideas here, but for the sake of accessibility we will restrict them to parenthetical remarks.
Read Full Post »
We continue our exploration of ultrafilters. Today we’ll discuss the infinite Ramsey theorem, which is the following classical result:
Theorem: Suppose the complete graph on countably many vertices has its edges colored in one of colors. Then there is a monochromatic (i.e. an infinite subgraph all of whose edges are the same color).
The finite Ramsey theorem implies that there is a monochromatic for every positive integer , but this is a strictly stronger result; it implies not only the finite Ramsey theorem but the “strengthened” finite Ramsey theorem, and by the Paris-Harrington theorem this is independent of Peano arithmetic (although Peano arithmetic can prove the finite Ramsey theorem). Indeed, while the standard proof of the finite Ramsey theorem uses the finite pigeonhole principle, the standard proof of the infinite Ramsey theorem uses the infinite pigeonhole principle, which is stronger; this is part of the subject of a post by Terence Tao which is quite enlightening.
Given a non-principal ultrafilter on , any partition of into finitely many disjoint subsets (that is, any coloring) has the property that exactly one of the subsets is in (that is, has “full measure”), and this subset must be infinite. This subset can, in turn, be colored (partitioned), and exactly one of the blocks of the partition is in , and it must again be infinite, and so forth. It follows that a non-principal ultrafilter lets us use the infinite pigeonhole principle repeatedly (in fact this is in some sense what a non-principal ultrafilter is), and since this is exactly what is needed to prove the infinite Ramsey theorem we might expect that we could use a non-principal ultrafilter to prove the infinite Ramsey theorem. Today we’ll describe this proof, and then describe how the infinite Ramsey theorem implies the finite Ramsey theorem, which involves a different use of a non-principal ultrafilter on .
Read Full Post »
Remark: To forestall empty set difficulties, whenever I talk about arbitrary sets in this post they will be non-empty.
We continue our exploration of ultrafilters from the previous post. Recall that a (proper) filter on a poset is a non-empty subset such that
- For every , there is some such that .
- For every , if then .
- is not the whole set .
If has finite infima (meets), then the first condition, given the second, can be replaced with the condition that if then . (This holds in particular if is the poset structure on a Boolean ring, in which case .) A filter is an ultrafilter if in addition it is maximal under inclusion among (proper) filters. For Boolean rings, an equivalent condition is that for every either or , but not both. Recall that this condition tells us that ultrafilters are precisely complements of maximal ideals, and can be identified with morphisms in . If for some set , then we will sometimes call an ultrafilter on an ultrafilter on (for example, this is what people usually mean by “an ultrafilter on “).
Today we will meander towards an ultrafilter point of view on topology. This point of view provides, among other things, a short, elegant proof of Tychonoff’s theorem.
Read Full Post »