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Posts Tagged ‘universal properties’

Let C be a (locally small) category. Recall that any such category naturally admits a Yoneda embedding

\displaystyle Y : C \ni c \mapsto \text{Hom}(-, c) \in \widehat{C}

into its presheaf category \widehat{C} = [C^{op}, \text{Set}] (where we use [C, D] to denote the category of functors C \to D). The Yoneda lemma asserts in particular that Y is full and faithful, which justifies calling it an embedding.

When C is in addition assumed to be small, the Yoneda embedding has the following elegant universal property.

Theorem: The Yoneda embedding Y : C \to \widehat{C} exhibits \widehat{C} as the free cocompletion of C in the sense that for any cocomplete category D, the restriction functor

\displaystyle Y^{\ast} : [\widehat{C}, D]_{\text{cocont}} \to [C, D]

from the category of cocontinuous functors \widehat{C} \to D to the category of functors C \to D is an equivalence. In particular, any functor C \to D extends (uniquely, up to natural isomorphism) to a cocontinuous functor \widehat{C} \to D, and all cocontinuous functors \widehat{C} \to D 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 \widehat{C} is the category obtained by “freely gluing together” the objects of C 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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(This is an old post I never got around to finishing. It was originally going to have a second half about pointless topology; the interested reader can consult Vickers’ Topology via Logic on this theme.)

Standard presentations of propositional logic treat the Boolean operators “and,” “or,” and “not” as fundamental (e.g. these are the operators axiomatized by Boolean algebras). But from the point of view of category theory, arguably the most fundamental Boolean operator is “implies,” because it gives a collection of propositions the structure of a category, or more precisely a poset. We can endow the set of propositions with a morphism p \to q whenever p \Rightarrow q, and no morphisms otherwise. Then the identity morphisms \text{id}_p : p \to p simply reflect the fact that a proposition always implies itself, while composition of morphisms

\displaystyle (p \Rightarrow q) \wedge (q \Rightarrow r) \to (p \Rightarrow r)

is a familiar inference rule (hypothetical syllogism). Since it is possible to define “and,” “or,” and “not” in terms of “implies” in the Boolean setting, we might want to see what happens when we start from the perspective that propositional logic ought to be about certain posets and figure out how to recover the familiar operations from propositional logic by thinking about what their universal properties should be.

It turns out that when we do this, we don’t get ordinary propositional logic back in the sense that the posets we end up identifying are not just the Boolean algebras: instead we’ll get Heyting algebras, and the corresponding notion of logic we’ll get is intuitionistic logic.

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Occasionally I see mathematical questions that seem “grammatically incorrect” in some sense.

Example. “Is [0, 1] open or closed?”
Example. “Is \{ 1, 2, 3 \} a group?”
Example. “What’s the Fourier series of \sin x + \sin \pi x?”

Here are some sillier examples.

Example. “Is a rectangle prime?”
Example. “Is 17 \in 3?”
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 17 \in 3?” 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.

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What does it mean to define a mathematical object?

Roughly speaking, there are two strategies you could use. One is to construct that object from other objects. For example, \mathbb{R} can be constructed from \mathbb{Q} using Cauchy sequences or Dedekind cuts.

Another is to give a list of properties that uniquely specify the object. For example, \mathbb{R} can be defined as the unique Dedekind-complete totally ordered field.

Let’s call these strategies construction and specification. Construction has the advantage of concreteness, and it also proves that the object exists. However, it has the disadvantage that constructions are not unique in general and you might not be working with the most useful construction at a given time; to prove things about an interesting mathematical object you might have to switch between various constructions (while also proving that they all construct the same object).

Relying excessively on construction also has a curious disadvantage when talking to students about the object in question: namely, they might ask you how to show that object X satisfies property P, and you might respond “well, it’s obvious if you use construction A,” and they might respond “but I was taught that X is [construction B]!” The possibility of multiple different constructions of the same mathematical object makes construction somewhat unsatisfying in terms of describing what you even mean when you say “object X.”

This is the appeal of specification. Although specification is more abstract and harder to grasp at first, it has the advantage you don’t have to use any particular construction of an object to prove things about it: instead, you should in principle be able to prove everything from the properties alone, since those uniquely specify the object. This often leads to cleaner proofs. Specifications also usually give a better motivation for looking at an object: it’s easier to make sense of the claim that we want a mathematical object with various properties than to make sense of the claim that we want a mathematical object which is constructed in some arbitrary-looking (to the uninitiated) fashion.

Below we’ll go through a few examples of mathematical objects and some constructions and specifications of them.

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Previously we observed that although monomorphisms tended to give expected generalizations of injective function in many categories, epimorphisms sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective morphisms in familiar concrete categories, the definition of a split epimorphism is too restrictive.

In this post we will discuss two other intermediate notions of epimorphism. (These all give dual notions of monomorphisms, but their epimorphic variants are more interesting as a possible solution to the above problem.) There are yet others, but these two appear to be the most relevant in the context of abelian categories.

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Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by \text{Spec } R I mean a commutative ring R regarded as an object in the opposite category \text{CRing}^{op}.

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There are various natural questions one can ask about monomorphisms and epimorphisms all of which lead to the same answer:

  • What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
  • What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
  • A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
  • If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?

The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.

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