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## Connected objects and a reconstruction theorem

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.

## Epi-mono factorizations

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.

## Regular and effective monomorphisms and epimorphisms

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.

## Split epimorphisms and split monomorphisms

• 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.

## Monomorphisms and epimorphisms

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.

## A meditation on semiadditive categories

The goal of today’s post is to introduce and discuss semiadditive categories. Roughly speaking, these are categories in which one can add both objects and morphisms. Prominent examples include the abelian categories appearing in homological algebra, such as categories of sheaves and modules and categories of chain complexes.

Semiadditive categories display some interesting categorical features, such as the prominence of pairs of universal properties and the surprising ways in which commutative monoid structures arise, which seem to be underemphasized in textbook treatments and which I would like to emphasize here. I would also like to emphasize that their most important properties are unrelated to the ability to subtract morphisms which is provided in an additive category.

In this post, for convenience all categories will be locally small (that is, $\text{Set}$-enriched).

## Hilbert spaces (and dagger categories)

Hilbert spaces are a particularly nice class of Banach spaces. They axiomatize ideas from Euclidean geometry such as orthogonality, projection, and the Pythagorean theorem, but the ideas apply to many infinite-dimensional spaces of functions of interest to various branches of mathematics. Hilbert spaces are also fundamental to quantum mechanics, as vectors in Hilbert spaces (up to phase) describe (pure) states of quantum systems.

Today we’ll develop and discuss some of the basic theory of Hilbert spaces. As with the theory of Banach spaces, there are (at least) two types of morphisms we might want to talk about (unitary operators and bounded operators), and we will discuss an elegant formalism that allows us to talk about both. Things written by John Baez will be cited excessively.

## Banach spaces (and Lawvere metrics, and closed categories)

One annoying feature of the abstract theory of vector spaces, and one that often trips up beginners, is that it is not possible to make sense of an infinite sum of vectors in general. If we want to make sense of infinite sums, we should probably define them as limits of finite sums, so rather than work with bare vector spaces we need to work with topological vector spaces over a topological field, usually $\mathbb{R}$ or $\mathbb{C}$ (but sometimes fields like $\mathbb{Q}_p$ are also considered, e.g. in number theory). Common and important examples include spaces of continuous or differentiable functions.

Today we’ll discuss a class of topological vector spaces which is convenient to work with but which still covers many examples of interest, namely Banach spaces. The material in the first half of this post is completely standard and can be found in any text on functional analysis.

In the second half of the post we discuss a category of Banach spaces such that two Banach spaces are isomorphic in this category if and only if they are isometrically isomorphic but which still allows us to talk about bounded linear operators between Banach spaces, and to do this we briefly discuss Lawvere metrics; this material can be found on the nLab.

## ab, ba, and the spectrum

Let $a, b$ be two $n \times n$ matrices. If $a, b$ don’t commute, then $ab \neq ba$; however, the two share several properties. If either $a$ or $b$ is invertible, then $ab$ is conjugate to $ba$, so in particular they have the same characteristic polynomial.

What if neither $a$ nor $b$ are invertible? As it turns out, $ab$ and $ba$ still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have $ab = 0$ but $ba$ nonzero). There are several ways of proving this result, which implies in particular that $ab$ and $ba$ have the same eigenvalues.

What if $a, b$ are linear transformations on an infinite-dimensional vector space? Do $ab$ and $ba$ still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”

For two categories $C, D$ let $D^C$ denote the functor category, whose objects are functors $C \to D$ and whose morphisms are natural transformations. For $C$ a locally small category, the Yoneda embedding is the functor $C \to \text{Set}^{C^{op}}$ sending an object $x \in C$ to the contravariant functor $\text{Hom}(-, x)$ and sending a morphism $x \to y$ to the natural transformation $\text{Hom}(-, x) \to \text{Hom}(-, y)$ given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.