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Posts Tagged ‘abstract nonsense’

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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Previously we saw that Cantor’s theorem, the halting problem, and Russell’s paradox all employ the same diagonalization argument, which takes the following form. Let X be a set and let

\displaystyle f : X \times X \to 2

be a function. Then we can write down a function g : X \to 2 such that g(x) \neq f(x, x). If we curry f to obtain a function

\displaystyle \text{curry}(f) : X \to 2^X

it now follows that there cannot exist x \in X such that \text{curry}(f)(x) = g, since \text{curry}(f)(x)(x) = f(x, x) \neq g(x).

Currying is a fundamental notion. In mathematics, it is constantly implicitly used to talk about function spaces. In computer science, it is how some programming languages like Haskell describe functions which take multiple arguments: such a function is modeled as taking one argument and returning a function which takes further arguments. In type theory, it reproduces function types. In logic, it reproduces material implication.

Today we will discuss the appropriate categorical setting for understanding currying, namely that of cartesian closed categories. As an application of the formalism, we will prove the Lawvere fixed point theorem, which generalizes the argument behind Cantor’s theorem to cartesian closed categories.

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Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

  • Given a Lie group G, its tangent space T_e(G) at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
  • Given a space X, its cohomology H^{\bullet}(X, \mathbb{Z}) is a priori a graded abelian group, but it ends up having the structure of a graded ring.
  • Given a space X, its cohomology H^{\bullet}(X, \mathbb{F}_p) over \mathbb{F}_p is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-p Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups \pi_n(X) of a space: that is, we will show that, in a suitable sense, each individual homotopy group \pi_n(X) is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)

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Previously we looked at several examples of n-ary operations on concrete categories (C, U). In every example except two, U was a representable functor and C had finite coproducts, which made determining the n-ary operations straightforward using the Yoneda lemma. The two examples where U was not representable were commutative Banach algebras and commutative C*-algebras, and it is possible to construct many others. Without representability we can’t apply the Yoneda lemma, so it’s unclear how to determine the operations in these cases.

However, for both commutative Banach algebras and commutative C*-algebras, and in many other cases, there is a sense in which a sequence of objects approximates what the representing object of U “ought” to be, except that it does not quite exist in the category C itself. These objects will turn out to define a pro-object in C, and when U is pro-representable in the sense that it’s described by a pro-object, we’ll attempt to describe n-ary operations U^n \to U in terms of the pro-representing object.

The machinery developed here is relevant to understanding Grothendieck’s version of Galois theory, which among other things leads to the notion of étale fundamental group; we will briefly discuss this.

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Groups are in particular sets equipped with two operations: a binary operation (the group operation) (x_1, x_2) \mapsto x_1 x_2 and a unary operation (inverse) x_1 \mapsto x_1^{-1}. Using these two operations, we can build up many other operations, such as the ternary operation (x_1, x_2, x_3) \mapsto x_1^2 x_2^{-1} x_3 x_1, and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).

When we think of groups as objects of the category \text{Grp}, where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?

It turns out that the answer is yes. The rest of this post will describe a general categorical definition of n-ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.

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A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

  • replace the study of a group G with the study of its category G\text{-Rep} of linear representations,
  • replace the study of a ring R with the study of its category R\text{-Mod} of R-modules,
  • replace the study of a topological space X with the study of its category \text{Sh}(X) of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if G is a finite group, then as a category, the only data that can be recovered from G\text{-Rep} is the number of conjugacy classes of G, which is not much information about G. We get considerably more data if we also have the monoidal structure on G\text{-Rep}, which gives us the character table of G (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of G. It turns out that to recover G we need the symmetric monoidal structure on G\text{-Rep}; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

Theorem: A group G can be recovered from its category G\text{-Set} of G-sets.

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In many familiar categories, a morphism f : a \to b admits a canonical factorization, which we will write

a \xrightarrow{e} c \xrightarrow{m} b,

as the composite of some kind of epimorphism e and some kind of monomorphism m. Here we should think of c as something like the image of f. This is most familiar, for example, in the case of \text{Set}, \text{Grp}, \text{Ring}, and other algebraic categories, where c is the set-theoretic image of f in the usual sense.

Today we will discuss some general properties of factorizations of a morphism into an epimorphism followed by a monomorphism, or epi-mono factorizations. The failure of such factorizations to be unique turns out to be closely related to the failure of epimorphisms or monomorphisms to be regular.

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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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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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Previously we discussed categories with finite biproducts, or semiadditive categories. Today, partially as a further warmup for the axioms defining an abelian category, we’ll discuss monomorphisms and epimorphisms.

Monomorphisms and epimorphisms are supposed to be a categorical generalization of the familiar notion of an injective resp. surjective structure-preserving map (such as an injective resp. surjective group homomorphism or an injective resp. surjective continuous function). This idea more or less works out for monomorphisms, but epimorphisms are somewhat infamous for behaving in unexpected ways, and even monomorphisms can behave unexpectedly sometimes.

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