# Generalised geometry #1: Generalised tangent bundle

1. Introduction

The motivation for generalised geometry as first formulated in [Hitc03], [Hitc05], and [Gual04] was to combine complex and symplectic manifolds into a single, common framework. In the sense of Hitchin’s formulation, which follows Courant and Dorfman, generalised geometry has deep application in physics since emphasis is placed on adapting description of the physical motion of extended objects (i.e., strings). In this way, one can view generalised geometry as analogous to how traditional geometry is adapted to the physical motion of point-particles. There are also more general forms of generalised geometries, which can be thought of as further extended and adapted geometries to describe higher dimensional objects such as membranes (and hence also M-theory). These notions of geometry, which we can organise under the conceptual umbrella of extended geometries, correlate closely with the study of extended field theories that captures both Double Field Theory (DFT) and Exceptional Field Theory (EFT).

In these notes, interest in generalised geometry begins with the way in which generalised and extended geometry makes manifest hidden symmetries in string / M-theory. In particular, emphasis is on obtaining a deeper understanding and sense of mathematical intuition for the structure of generalised diffeomorphisms and gauge symmetries. The purpose was to then extend this emphasis to a study of the gauge structure of DFT, which is well known to be closely related with generalised geometry but in fact extends beyond it. We won’t get into this last concern in these notes; it is merely stated to make clear the original motivation for reviewing the topics.

Given that generalised geometry inspired the seminal formulations of DFT, it is no coincidence that what we observe in a detailed review of generalised geometry is the way in which the metric and p-form potentials are explicitly combined into a single object that acts on an enlarged space. This enables a description of diffeomorphisms and gauge transformations of the graviton and Kalb-Ramond B-field in a combined way. In fact, one of Hitchin’s motivations for the introduction of generalised geometry was to give a natural geometric meaning to the B-field. As will become clear in a later note, a key observation in this regard is that the automorphism group of the Courant algebroid ${TM \oplus T^{\star}M}$ is the semidirect product of the group of diffeomorphisms and B-field transformations. We will then study the structure of this group.

Remark 1 (Generalised geometry, branes, and SUGRA) Although not a focus of these notes, it is worth mentioning that generalised geometry in the sense of Hitchin is an important framework for studying branes and also T-dualities, including mirror symmetry. It also offers a powerful collection of tools to study Calabi-Yau manifolds, particularly generalised Calabi-Yau, proving important in the search for more realistic flux compactifications.

2. Generalised tangent bundle

The main objects to study on generalised geometry are Courant algebroids. But before we reach this stage, there are two fundamental structures of generalised geometry that we must first define: 1) the generalised tangent bundle and, 2) the Courant bracket. In this note, we introduce the generalised tangent bundle. Then in the following notes we explore the properties of this structure and the related extension of linear algebra to generalised linear algebra. This brings us to finally study the Courant bracket, its properties and symmetries, before we study Courant algebroids and generalised diffeomorphisms.

Definition 1 (Generalised bundle) The generalised tangent bundle is obtained by replacing the standard tangent bundle ${T}$ of a D-dimensional manifold ${M}$ with the following generalised analogue

$\displaystyle E \cong TM \oplus T^{\star}M. \ \ \ \ \ (1)$

The generalised tangent bundle ${E}$ is therefore a direct sum of the tangent bundle ${TM}$ and co-tangent bundle ${T^{\star}M}$. As we will learn, the bundle ${E}$ has a natural symmetric form with respect to which both ${TM}$ and ${T^{\star}M}$ are maximally isotropic.

Remark 2 (Notation) Often in these notes we will use ${E}$ and ${TM \oplus T^{\star}M}$ interchangeably, which should be clear in the given context.

The generalised bundle (1) fits the following exact short sequence

$\displaystyle 0 \longrightarrow T^{\star}M \hookrightarrow E \overset{\rho}{\longrightarrow} TM \longrightarrow 0, \ \ \ \ \ (2)$

which, later on, we’ll see is the sort of sequence that describes an exact Courant algebroid.

Remark 3 (Early comment on Courant algebroids) As we will study in a later entry, it is the view afforded by generalised geometry that the bundle ${E}$ is in fact an extension of ${TM}$ by ${T^{\star}M}$, and so it is a direct example of a Courant algebroid such that, in the exact sequence (2), the Courant algebroid has a symmetric form plus other structure (e.g., the Courant bracket) that makes it isomorphic to ${E}$. This is true for suitable isotropic splittings of the exact sequence, an example of which is called a Dirac structure.

The sections of ${E}$ are non-trivial sections of ${TM \oplus T^{\star}M}$. This means that, unlike in standard geometry and how we typically consider vector fields as sections of ${TM}$ only, we now consider elements of the non-trivial sections

$\displaystyle X = x + \xi, Y = y + \varepsilon, \ x,y \in \Gamma(TM), \ \xi, \varepsilon \in \Gamma(T^{\star}M), \ \ \ \ \ (3)$

where ${x, y}$ are vector parts and ${\xi, \varepsilon}$ 1-form parts.

The set of smooth sections ${C^{\infty}(M)}$ of the bundle ${E}$ are denoted by ${\Gamma(E)}$ such that the set of smooth vector fields is denoted by ${\Gamma(TM)}$ and the set of smooth 1-forms by ${\Gamma(T^{\star}M)}$.

Remark 4 (Sequence and string background fields) For the sequence (2), note that in the map ${\rho}$ there exist sections ${\sigma}$ that are given by rank 2 tensors, which can then be split into symmetric and antisymmetric parts, ${\sigma_{\mu \nu} = g_{\mu \nu} + b_{\mu \nu}}$. The sections of ${E}$ describe infinitesimal symmetries of these fields, as they are encoded in a generalised vector field ${X}$ capturing infinitesimal diffeomorphisms and a 1-form ${\xi}$ describing the b-field gauge symmetry.

References

[Gual04] M. Gualtieri. Generalized complex geometry [PhD thesis]. arXiv: 0401221[math.DG]. [Gua11] Marco Gualtieri. Generalized complex geometry. Ann. of Math. (2), 174(1):75–123, 2011. url: https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p03-s.pdf. [Hitc03] N. Hitchin. Generalized Calabi–Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281. arXiv: 0209099 [math.DG]. [Hitc05] N. Hitchin. Brackets, forms and invariant functionals. arXiv: 0508618 [math.DG]. [Hitc10] N. Hitchin. Lectures on generalized geometry. arXiv: 1008.0973 [math.DG]. [Rub18] R. Rubio. Generalised geometry: An introduction [lecture notes]. url: https://mat.uab.cat/~rubio/gengeo/Rubio-GenGeo.pdf

# Mathematical language of duality

As we’ve discussed at various times on this blog, many of the most important recent developments in string / M-theory are based on duality relations. Physical insight is quite ahead of mathematics in this regard. But, in the last decade or two, mathematics has started to properly formulate a language of duality that, on first look, seems incredibly simple but is ultimately very powerful: namely, the language of categories. In foundational mathematical terms, category theory provides tools to express structures – often very general structures – and their duals in a way that comes out naturally through the concept of a categorical product and coproduct. Below is a very brief summary.

## Definition of a category

Let us quickly recall the definition of a category $\mathcal{C}$. As mentioned in a past post, a category can be constructed for essentially any mathematical object. We can think of a category as a quintessential representation of structure.

Definition 1. A category $\mathcal{C}$ consists of a class of objects, and, for every pair of objects $A,B \in \mathcal{C}$, a class of morphisms $hom(A,B)$ satisfying the properties:

• Each morphism has specified domain and codomain objects. If f is a morphism with domain A and codomain B we write $f: A \rightarrow B$.
• For each $A \in \mathcal{C}$, there is an identity morphism $id_A \in \text{hom}(A,A)$ such that for every $B \in \mathcal{C}$ we have left-right unit laws:

$f \circ id_A = f \text{for all} f \in \text{hom}(A,B),$

$id_A \circ f = f \text{for all} f \in \text{hom}(B,A).$

• For any pair of morphisms f,g with codomain of f equal to codomain of g, there exists a composite morphism $g \circ f$. The domain of the composite morphism is equal to the domain of f and the codomain is equal to the codomain of g.

In simple terms, a category is just a collection of objects (metric spaces, topological spaces, or whatever) and structure preserving maps between those objects. It is, in a sense, like a deeper generalisation of set theory, except that we can have categories of sets. A simple illustration of a category is as follows

There are two axioms that must be satisfied in the defining a category:

• For any $f: A \rightarrow B$, the composites $1_B f$ and $f1_A$ are equal to f.
• Composition is associative and unital. For all $A, B,C,D \in \mathcal{C}$, $f \in \text{hom}(A,B)$, $g \in \text{hom}(B,C)$, and $h \in \text{hom}(C, D)$, we have $f \circ (h \circ g) = (g \circ f) \circ h$.

## Functors

We can also define a functor, which maps between categories. We define the notion of a functor as corresponding to a mapping that sends the objects and arrows of one category to the objects and arrows in another category in a structure preserving way.

Definition 2. A functor $F$ from $C$ to $D$ is a structure preserving map between categories such that for each object $A$ of $C$, we have $F(A)$ in $D$.

For each arrow (morphism) $f: A \rightarrow B$ in $C$, we have $F(f): F(A) \rightarrow F(B)$ such that $F(g) \circ F(f) = F(g \circ f)$ and $F(Id_A) = Id_{F(A)}$.

Suppose $f: A \rightarrow C$ is a functor between categories $A$ and $C$. For purposes of illustration, we’ll call $A$ an indexing category, and let’s suppose it’s a simple one with objects $a_1, a_2, \ \text{and} \ a_3$:

A functor f out of this category $A$ is simply the choice of three objects and three arrows in the category $C$ such that

where $f(a_1) = c_1$, $f(a_2) = c_2$, and $f(a_3) = c_3$. The image of the arrows in $A$ are the arrows g, k, and h in $C$ where $g = h \circ k$.

## Categorical products

What is very neat and exciting is that we can also define the notion of a categorical product (e.g., a product of two categories). For a long time, it was thought that taking a product between two sets was one of the most fundamental operations in mathematics. But, it turns out, from the definition of a categorical product we can still drill deeper and therefore also capture the essence behind the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

This topic is again quite technical but, in short, a simple definition of a categorical product is as follows:

Definition 3. For any categories $C$ and $D$, there is a category $C \times D$, their product, whose

• objects are ordered pairs $c,d$, where c is an object of $C$ and d is an object of $D$,
• morphisms are ordered pairs with $\pi_1 : C \times D \rightarrow C$, $\pi_2 : C \times D \rightarrow D$ such that for the other candidate $X$ we define the maps $f: X \rightarrow A$, $g: X \rightarrow B$ for every unique $h: C \times D$, and $\pi_1 \circ h = f$ and $\pi_2 \circ h = g$,
• and in which composition and identities are defined componentwise.

### A first glimpse at duality

Now, what is absolutely amazing is how, from the notion of a product of categories (which is like a generalisation of the Cartesian product of ordered sets), the first glimpse of a fundamental mathematical description of duality naturally emerges in the definition of a categorical coproduct.

Let us return to the definition of a categorical product and its diagram in the previous section. We want to think of its coproduct (i.e., the product in the opposite category). We will have the same picture, except all of the arrows will be reversed which is the same as exchanging domain and codomain.

Definition 4. The co-product $C + D$, $p_1 : C \rightarrow C + B$, $p_2 : D \rightarrow C +D$ is such that for each $X$, $f: C \rightarrow X$, $g: D \rightarrow X$ there exists a unique $h: C + D \rightarrow X$ that makes the diagram commute $h \circ p_1 = f$ and $h \circ p_2 = g$.

The coproduct naturally takes the form of the category-theoretic dual notion to the categorical product. We can think of this in terms of a mapping from $C$ to $C^{\text{op}}$.

Definition 5. Let $C$ be any category. The opposite category $C^{\text{op}}$ has

• the same objects as in $C$, and
• a morphism $f^{\text{op}}$ in $C^{\text{op}}$ for each a morphism $f \in C$ so that the domain of $f^{\text{op}}$ is defined to be the codomain of f and the codomain of $f^{\text{op}}$ is defined to be the domain of f: i.e., $f^{\text{op}}: X \rightarrow Y \in C^{\text{op}} \leftrightarrow f: Y \rightarrow X \in C$.

What this means is that, given $C^{\text{op}}$ has the same objects and morphisms as $C$, the notion of duality in category theory is defined by a reversal of arrows: i.e., each morphism in $C^{\text{op}}$ is pointing in the opposite direction.

The dual of each of the axioms for a category is also an axiom, while the dual of the dual returns the original statement. This is the duality principle in a nutshell.

[1] E. Riehl, Category theory in context. Dover Publications, 2016. [online].

[2] S. Mac Lane, Category theory for the working mathematician. Springer, 1978. [online].

[3] P. Smith, Category theory: A gentle introduction. [online].

[4] J. Baez, Category theory course. [online].

# The language of morphisms and the notion of a diagram

In category theory, different types of notation are common. Ubiquitous and important in the formalism is diagram notation. I like to think of it as follows: the diagram finds natural expression in category theory because, as emphasised in the first entry of my notes, in approaching the idea of a category $\mathcal{C}$ we may take the view that what we are defining is the language of morphisms. Indeed, it is the primacy of morphisms, and not the objects of a category, that is emphasised.

Recall that morphisms – what we have also described as structure-preserving maps – are represented graphically as arrows. From this, the next important idea is composition – that is, a view of all of the ways in which we may compose arrows. As discussed, composition is built into the definition of a category, and one observation that may be proffered is that, as a language of morphisms, when working toward category theory we may extend our view that it is furthermore a minimalist language of composition. (Eventually, once we discuss categorical products and functors (a kind of morphism of categories), we will extend this view of category theory as fundamentally the language of duality). In a sense, it is as though we are building the ideas in stages. Putting these two things together, the primacy of morphisms and the rule of composition, we also arrive at the notion of how we may express the equality of arrows using diagrams.

Categorical diagrams are powerful for many reasons. One reason has to do with how a new proof technique is devised: i.e., what is called the diagram chase [1]. For example, commutative diagrams are utilised as a new technique of proof in homology theory. In this note, as a matter of introduction, we will think about diagrams and morphisms in a basic way. Saunders Mac Lane [2] motivates it concisely, ‘Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows’. Furthermore, in these notes, while ‘[m]any properties of mathematical constructions may be represented by universal properties of diagrams’, eventually the motivation is to study physical objects like branes.

## The notion of a diagram

What follows is a very brief introduction to the notion of a diagram. In particular, we are talking here about commutative diagrams. We will be somewhat informal in definition. The reason is as follows: there are two ways to approach the concept of a diagram and to give it definition. One way is through the notion of a functor, which is the way I prefer, but it is less intuitive and the topic of functors has been saved for more advanced discussion. Another way comes from a more direct and perhaps intuitive representation, namely how we may approach the commutative diagram as a directed graph of morphisms in a category.

Let’s start with something basic and familiar, an example of structured sets. A lot of objects in mathematics may be thought of as structured sets, and we can begin with a basic illustration of a structured graph with arrows from one vertex to another. This can be thought of as extra structure, or, indeed, a set with structure. A simple graphical representation is given below.

Here we have a vertex set of some graph mapped to vertex set of another graph. The arrows represent a function that preserves the graph structure. That is, there is a graph (homo)morphism such that, if vertex $A$ is linked to $B$ then $f(A)$ is linked to $f(B)$. That is to say, if $A$ and $B$ are linked in one graph, then the image of $A$ is linked to the image of $B$. (In fact, if we were working in category theory it would be said that this forms another category).

Now, let us advance this idea. Consider a set $A$ and a set $B$. Define an arrow $f: A \rightarrow B$ representing a function from $A$ to $B$, following the rule that $a \rightarrow f(A)$ assigns to each element $a \in A$ an element $f(a) \in B$. A diagram of sets and functions may be written as below.

Here we have a commutative triangle, which we may define as a category $\mathcal{C}$ (suppressing identity arrows, etc. in the diagram) in which the hypotenuse $h$ equals the composite $g \circ f$. In other words, this diagram is commutative when $h = g \circ f$, where $g \circ f : A \rightarrow C$ is the composite function $a \rightarrow g(f a)$. Generally, a diagram commutes if any two paths of composable arrows in the directed graph with common source and target have the same composite. The vertices in the graph are labelled by objects in the category, and the edges are labelled by morphisms (arrows) in the category.

Definition 1. A category diagram is commutative if for every pair of objects, all directed paths from one to the other give rise to the same morphism, i.e., the composite arrow along the first path is equal to the composite arrow along the second path.

As a matter of convention, unless explicitly stated otherwise, in these notes (and in many texts) a diagram is commutative.

## The language of morphisms

It is beneficial to think a bit more about different kinds of arrows and how they interact with other arrows in the relevant category. (This is a very brief review, and more detail can be found in [1, 2, 3, 4]).

In set-theoretic language, when we speak of sets we can of course speak of elements of those sets. In category-theoretic language, however, when speaking of the category Set it is no longer possible to speak of the elements of sets as each set is simply an object in Set. More pointedly, objects in categories have nothing ‘inside’ of them in the way we may think of elements of a set. In this sense, it is nice to think of the idea of a category as a representation of structure. One may then be inclined to ask: where is the information? How does one know how many elements there are in these sets? Is there a loss of information? It turns out that we don’t really lose information. In the example of Set, we can recover information about the sets just from knowing the object that represents the sets and arrows between the objects that represent the functions. The main tool at our disposal, again, is the morphism. What categorical language and logic gives us is not so much a disadvantage as a valuable shift in perspective.

As Emily Riehl writes, ‘A category provides a context in which to answer the question, “When is one thing the same as another thing?”. Almost universally in mathematics, one regards two objects of the same category to be “the same” when they are isomorphic’.

## Isomorphism

Take a simple set theoretical example of an isomorphism.

Here we have a one-to-one map. The property of having an inverse $g$ is what makes this map an isomorphism. Notice, if we compose $f$ and $g$ we obtain an identity $id_A$. Similarly, if we compose $g$ and $f$ we obtain an identity $id_B$. Now, compare this with a category-theoretic view of an isomorphism.

Here we have an arrow $f: A \rightarrow B$, which is an isomorphism if there exists $g: B \rightarrow A$ such that $g \circ f = id_A$ and $f \circ g = id_B$.

Definition 2. In a category $\mathcal{C}$, two objects $A,B \in \mathcal{C}$ are isomorphic (to each other) if there are $f \in \mathcal{C}(A,B)$, $g \in \mathcal{C}(B,A)$ such that $g f = id_A$ and $f g = id_B$. In this case, we say that $f$ and $g$ are isomorphisms, and write $A \equiv B$.

If $f$ is an isomorphism, the morphism $g$ such that $g f = f g = id$ is uniquely determined. We write $g = f^-1$, and say that $g$ is the inverse of $f.$

Different categories possess different notions of isomorphism. Standard examples include: in the category Set, isomorphisms are bijective functions; in Grp they are bijective group (homo)morphisms; in the category of metric spaces Met (with non-expansive maps), they are isometries; in Vect they are invertible linear transformations; and in the category of topological spaces Top, they are (homeo)morphisms (i.e. continuous bijections with continuous inverses). In the language of category theory, it is generally not the case that one might distinguish between isomorphic objects. Instead, objects are determined ‘up to isomorphism’, which is a pertinent point that will be highlighted again when we start discussing categorical homotopy theory among other things.

## Monomorphisms

Definition 3. A morphism $f \in \mathcal{C}(x,y)$ is a monomorphism if it satisfies the property: for every $w \in \mathcal{C}$, $g_1, g_2 \in \mathcal{C}(w,x)$, $f_{g_1} = f_{g_2} \implies g_1 = g_2$. This property is called left cancellation.

If $f$ is a monomorphism, this is conventionally denoted $\hookrightarrow$ or $\mapsto$. In adjectival form, an monomorphism is mono.

Example. Consider the category Top of locally path-connected, pointed topological spaces (a pointed space is a space together with a choice of basepoint). Let $X$ be such a space and $\pi : \tilde{X} \rightarrow X$ a covering map (which may or may not be injective). This covering map $\pi$ is a monomorphism. Having restricted to the case where we can lift maps to $X$ back up to the covering space $\tilde{X}$, any map $f$ which can be factored through $\tilde{X}$ as some map $\pi \circ \tilde{f}$ can be lifted. Since the basepoint is fixed, it can be lifted uniquely back to $f$, picking out arrows $g,h$ from some space $Y$ to $\tilde{X}$. Since these maps are both lifts of the maps $\pi \circ g = \pi \circ h$, uniqueness of lifts gives $g = h$.

## Epimorphism

Definition 4. A morphism $f \in \mathcal{C}(x,y)$ is an epimorphism if it satisfies the property: for every $z \in \mathcal{C}$, $h_1, h_2 \in \mathcal{C}(y,z)$, $h_1 f, h_2 f \implies h_1 = h_2$. This property is called right cancellation.

When $f$ is an epimorphism, it is convention to represent it as $\twoheadrightarrow$. In adjectival form, an epimorphism is epic.

Example. Consider the category TopHaus of Hausdorff topological spaces. Let $\mathcal{C}$ be the category of TopHaus, and $i : Q \rightarrow R$ be the standard inclusion. If $X$ is another Hausdorff topological space, $g_1, g_2 : R \rightarrow X$ are continuous functions with $g_1 i = g_2 i$. Then it can be argued $g_1 = g_2$ using the fact that in Hausdorff spaces, convergent sequences have at most one limit, and continuous functions preserve limits: for any $x \in R$, $x = \lim_{n \rightarrow \infty} q_n$ where $q_n \in \mathbb{Q}$. It follows,

$g_1(x) = g_1 (\lim_{n \rightarrow \infty} x_n)$

$= \lim_{n \rightarrow \infty} g_1(x_n)$

$= \lim_{n \rightarrow \infty} g_2(x_n)$

$= g_2(\lim_{n \rightarrow \infty} x_n)$

$= g_2(x).$

The injection from $Q$ to $R$ is not a surjection, even though it is an epimorphism.

## Sections and retractions

Definition 5. Let $\mathcal C$ be a category. A section is a morphism $f \in \mathcal{C}(x,y)$ such that there is some morphism $g \in \mathcal{C}(y,x)$ with $gf = id_x$. Sometimes it is said that $f$ is a section of $g$, or a right inverse to $g$. Given $g$, if such an $f$ exists we say that $g$ admits a section. All sections are monomorphisms.

Definition 6. A retraction is a morphism $g \in \mathcal{C}(x,y)$ such that there is some morphism $f \in \mathcal{C}(y,x)$ with $gf = id_x$. Sometimes it is said $f$ is a retraction of, or left inverse to, $g$. Given $g$, if such an $f$ exists we say that $g$ admits a retraction. All retractions are epimorphisms.

If a morphism is both a section and a retraction, then it is an isomorphism.

Monomorphisms and epimorphisms should be regarded as categorical analogues of the notions of injective and surjective functions (see Section 1.2 in [1]).

Though we have yet to study duality in a category-theoretic context, it is interesting that notions of monomorphism and epimorphism are dual, which means their abstract categorical properties are also dual (see Lemma 1.2.11. in [1]).

It is fairly straightforward, I think, to see that identity arrows are always monic. Dually, they are thus always epic as well. A number of theorems also follows, which, for intuitive explanation, can be reviewed in Section 5.1. in [3].

References

[1] E. Riehl, Category theory in context. Dover Publications, 2016. [online].

[2] S. Mac Lane, Category theory for the working mathematician. Springer, 1978. [online].

[3] P. Smith, Category theory: A gentle introduction [online].

[4] D. Epelbaum and A. Trisal, Introduction to category theory. [Lecture notes].

# Introduction to category theory

This is the first entry in my notes on category theory, higher category theory, and, finally, higher structures. The main focus of my notes, especially as the discussion advances, is application in string / M-theory, concluding with an introduction to the study of higher structures in M-theory. We start with basic category theory roughly following the book ‘Category Theory in Context’ by Emily Riehl (online version here), as well as the perspective of a selection of other texts and lectures cited throughout. For the engaged reader, I recommend reviewing the respective pages on nLab for further references.

## Introduction

There is a line by Wilfrid Sellars: ‘The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term’. The things we must come to know ‘in the broadest possible sense’ – at its most abstract, a type of conceptual modelling – must in some way be classified such that we may distinguish the type of thing, the relation between thing of similar and dissimilar class, and its particular properties or attributes. For example, think of basic biological nomenclature going back to Aristotle. Another example would be the standard model of particle physics. (For the time being, we will put aside philosophical issues going back to Hegel, Russell, and others, as well as broader debates having to do with process vs. substance metaphysics, and so on).

From a mathematics and physics point of view, if we take Sellars’ statement seriously, then, at the highest level in the conceptual hierarchy what we begin to contemplate is a way to think about what Peter Smith describes in his notes on category theory as, ‘structured families of structures’. That is to say, we naturally come upon the need for some systematic framework for the study of abstract structures, how we may define a family of such structures, and their interrelation. We take as a starting point in these notes motivation from both foundational mathematics and fundamental physics.

A simple example of a structure is a topological space. Simpler still, take an example from group theory. Any group may be described as a structure, which comprises a number of objects equipped with a binary operation defined on them that obeys well-defined axioms. Now, what of a family of groups? We can of course also define a family of groups with structure-preserving homomorphisms between them (for a review of groups and sets leading up to the basic ideas of category theory, see Chapter 2 in the above notes by Smith). This gives an example of a structured family. This reference to groups is apt, because as we will see later in these notes: classically, a group is a monoid in which every element has an inverse (necessarily unique). A monoid, as we will review in a future entry, is one of the most basic algebraic examples of a category.

More generally, when looking at a family of structures along with the structure-preserving maps between them, our goal will be to reach an even higher level of abstraction that takes the form of a further structure: i.e., a structure-of-structures. We can then continue this game and ask, what is the interrelation of this structure-of-structures? From this question we will look to climb to another level and speak of operations that map one functor to another in a way that preserves their functorial properties.

When I think of the idea of a category, this increasing picture of generality and of climbing levels of abstraction is often what I like to picture. To use the words of Emily Riehl [1], ‘the purview of category theory is mathematical analogy’. While some give it the description, however affectionately, of ‘abstract nonsense’, I prefer to think of category theory – and, more broadly, the category theoretic perspective – as very much akin to the geologist constructing a topological map containing only vital information. This notion of climbing levels of abstraction, is, in many ways, simplifying abstraction. What use would it be to perform analysis within the framework of these increasing levels of simplifying abstraction? In foundational mathematics, the motivation is quite clear. In fundamental physics, on the other hand, it may at first seem less obvious. But as we will discuss in these notes, particularly in the context of quantum field theory and string / M-theory, there is quite a lot of motivation to think systematically about structured families of mathematical structures.

## What is a category?

One way to approach the idea of a category is to emphasise the primacy of morphisms. In the paradigm view, in contrast to set theory, category theory focuses not on elements but on the relations between objects (i.e., the (homo)morphisms between objects). In this sense, we may approach category theory as a language of composition.

Let us build toward this emphasis on composition in a simple way. Consider some collection of objects $A, B, C, D$ with a structure preserving morphism $f$ from $A$ to $B$, another structure preserving morphism $g$ from $B$ to $C$ , and, finally, a structure preserving morphism $h$ from $C$ to $D$. (In a handwavy way, this is how we motivated the idea of a category in a previous post). In diagrammatic notation we have,

$\displaystyle A \ \xrightarrow[]{f} \ B \ \xrightarrow[]{g} \ C \ \xrightarrow[]{h} \ D$.

It is fairly intuitive that we should be able to define a composition of these maps. All we need, as an axiom, is associativity. For example, we may compose $f$ and $g$ such that we obtain a map from $A$ to $C$ . We may write such a composition as $g \circ f$. Similarly for all the other ways we may compose the maps $f, g$, and $h$. This means that we ought to be able to then also compose a map for the entire journey from $A$ to $D$. Diagrammatically, this means we obtain:

One sees that we can apply the structure preserving map $f$ followed by the composite g-followed-by-h. Alternatively, we may just as well apply the composite f-followed-by-g and then afterwards apply the map $h$. This very basic picture of a collection of objects $A,B,C,D$, the maps between them, and how we may invoke the principle of composition for these maps already goes some way toward how we shall formally define a category. One will notice below that we need a bit more than associativity as an axiom, and along with the objects of a category we will speak of morphisms simply as arrows. From now on, if $A \in \text{Ob}(\mathcal{C})$ we write $A \in \mathcal{C}$.

Definition 1. A category $\mathcal{C}$ consists of a class of objects, and, for every pair of objects $A,B \in \mathcal{C}$, a class of morphisms, $\text{hom}(A,B)$, satisfying the properties:

• Each morphism has specified domain and codomain objects. If $f$ is a morphism with domain $A$ and codomain $B$ we write $f: A \rightarrow B$.
• For each $A \in \mathcal{C}$, there is an identity morphism $id_A \in \text{hom}(A,A)$ such that for every $B \in \mathcal{C}$ we have left-right unit laws:
1. $\displaystyle f \circ id_A = f \text{for all} f \in \text{hom}(A,B)$
2. $\displaystyle id_A \circ f = f \text{for all} f \in \text{hom}(B,A)$
• For any pair of morphisms $f,g$ with codomain of $f$ equal to codomain of $g$, there exists a composite morphism $g \circ f$. The domain of the composite morphism is equal to the domain of $f$ and the codomain is equal to the codomain of $g$.

Two axioms must be satisfied:

• For any $f: A \rightarrow B$, the composites $1_B f$ and $f1_A$ are equal to $f$.
• Composition is associative and unital. For all $A,B,C,D \in \mathcal{C}$, $f \in \text{hom}(A,B)$ , $g \in \text{hom}(B,C)$, and $h \in \text{hom}(C, D)$, we have $f \circ (h \circ g) = (g \circ f) \circ h$ .

Further remarks may be reviewed in [1, 2, 3]. We emphasise that for any mathematical object there exists a category with objects of that kind and morphisms – i.e., structure-preserving maps denoted as arrows – between them. The objects and arrows of a category are called the data. The objects of a category can be formal entities like functions or relations. In many examples of a category, the arrows represent functions, but not all cases of an arrow represents a morphism. These subtleties will be saved for future discussion.

An important notational point is that one should keep close attention on morphisms. Often categories with the same class of objects – e.g., a category of topological spaces compared with another category of topological spaces – may be distinguished by their different classes of morphisms. It is helpful to denote the category as $\text{hom}_{\mathcal{C}}(A,B)$ or $\mathcal{C}(A,B)$ to denote morphisms from $A$ to $B$ in the category $\mathcal{C}$.

Importantly, to avoid confusion, we speak of ‘classes’ or ‘collections’ of objects and morphisms rather than ‘sets’. One motivation is to avoid confusion when speaking of $\text{Set}$, which is the the category of all sets with morphisms (as functions) between sets. If a set of objects were required, instead of a class, then we would require a set of all sets. As it will be made clear when we reach the discussion on how to consider categories of categories, we may speak of sets of sets but, as Russell’s Paradox implies, there is no set whose elements are ‘all sets’. So we cannot speak of a set of all sets or a category of all sets. Likewise, it is conventional when we consider categories of categories to avoid the notion of a category of all categories (see Remark 1.1.5. in [1]). Instead, we speak of a limit in the form of a universe of sets and, in more advanced discussion, we will come to consider categories as universes.

Related to this concern about set-theoretical issues, it is important to note that we work with an extension of the standard Zermelo–Fraenkel axioms of set theory, allowing ‘small’ and ‘large’ sets to be discussed. In category theoretic language, we invoke similar terminology:

Definition 2. A category $\mathcal{C}$ is finite iff it has overall only a finite number of arrows.

A category $\mathcal{C}$ is small iff it has overall only a ‘set’s worth’ of arrows – i.e. the class of objects is a set such that the arrows of $\mathcal{C}$ can be put into one-one correspondence with the members of the set.

A category $\mathcal{C}$ is locally small iff for every pair of $\mathcal{C}$ – objects $A,B$ there is only a ‘set’s worth’ of arrows from $A$ to $B$, i.e. those arrows can be put into one-one correspondence with the members of some set.

## Examples of categories

What follows are a few examples illustrating the variety of mathematical objects that assemble into a category:

• Set, the category of sets where morphisms are given by ordinary functions, with specified domain and codomain. There is a subtlety here in that the view of Set as the category of all sets becomes paradoxical, so, typically, we limit to a universe of sets (more on this in a separate entry).

Example. In this category the objects are sets, morphisms are functions between sets, and the associativity of the composition law is the associativity of composition of functions.

We may define the category Set (The category of sets): $\mathcal{O}$(Set) is the class of all sets, and, for any two sets $A,B \in \mathcal{O}$(Set) define $\text{hom}(A,B) = f: A \rightarrow B$ as the set of functions from $A$ to $B$. The composition law is given by the usual composition of functions. Since composition of functions is associative, and there is always an identity function, Set is a category. This ends the example.

Other categories of note:

• Grp, the category of groups where morphisms are given by group homomorphisms.
• Vect_k, the category of vector spaces over some fixed field $k$, where morphisms are given by linear transformations.
• Ring, the category with rings as objects and ring homomorphisms as morphisms
• Top, the category of topological spaces where morphisms are given by continuous maps
• Met, is the category with metric spaces as objects and continuous maps as morphisms.
• Meas, is the category with measurable spaces as objects and measurable maps as morphisms.
• Graph, the category of graphs as objects and graph morphisms (functions carrying vertices to vertices and edges to edges, preserving incidence relations) as morphisms. In the variant DirGraph, objects are directed graphs, whose edges are now depicted as arrows, and morphisms are directed graph morphisms, which must preserve sources and targets.
• Man, the category of smooth (i.e., infinitely differentiable) manifolds as objects and smooth maps as morphisms.

All of the above examples are concrete categories, whose objects have underlying sets and whose morphisms are functions between these underlying sets (what we have called ‘structure-preserving’ morphisms). We will speak more about concrete categories, including formal definition, in a later note. For the sake of introduction, it is also worth noting that there are also \textit{abstract categories}. One example is as follows:

BG, the category defined by the group $G$ (or what we will describe as a monoid in the next entry) with a single object. The elements of $G$ are morphisms, with each group element representing a distinct endomorphism of the single object. Here composition is given by multiplication. There is an identity element $e \in G$ that acts as the identity morphism.

In the next post, we will review some other category definitions, review diagrammatic notation, and discuss in more detail the important role and subtlety of morphisms. In a closely followed entry, we will then finally turn our attention to monoids, groupoids, pre-ordered collections, and other related concepts, as well as start discussing examples in string theory.

## References

These notes primarily follow a selection of lectures and texts:

[1] E. Riehl, Category theory in context. Dover Publications, 2016. [online]

[2] S. Mac Lane, Category theory for the working mathematician. Springer, 1978. [online].

[3] P. Smith, Category theory: A gentle introduction [online].

[4] J. Baez, Category theory course [lecture notes].

# Jensen Polynomials, the Riemann-zeta Function, and SYK

A new paper by Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier appears to have a made some intriguing steps when it comes to the Riemann Hypothesis (RH). The paper is titled, ‘Jensen polynomials for the Riemann zeta function and other sequences’. The preprint originally appeared in arXiv [arXiv:1902.07321 [math.NT]] in February 2019. It was one of a long list of papers that I wanted to read over the summer. And with the final version now in the Proceedings of the National Academy of Sciences (PNAS), I would like to discuss a bit about the author’s work and one way in which it relates to my own research.

First, the regular reader will recall that in a past post on string mass and the Riemann-zeta function, we discussed the RH very briefly, including the late Sir Michael Atiyah’s claim to have solved it, and finally the separate idea of a stringy proof. The status of Atiyah’s claim still seems unclear, though I mentioned previously that it doesn’t look like it will hold. The idea of a stringy proof also remains a distant dream. But we may at least recall from this earlier post some basic properties of the RH.

What is very interesting about the Griffin et al paper is that it returns to a rather old approach to the RH, based on George Pólya’s research in 1927. The authors also build on the work of Johan Jensen. The connection is as follows. It was the former, Pólya, a Hungarian mathematician, who proved that, for the Riemann-zeta function $\zeta{s}$ at its point of symmetry, the RH is equivalent to the hyperbolicity of Jensen polynomials. For the inquisitive reader, as an entry I recommend this 1990 article in the Journal of Mathematical Analysis and Applications by George Csordas, Rirchard S. Varga, and Istvan Vincze titled, ‘Jensen polynomials with applications to the Riemann zeta-function’.

Pólya’s work is generally very interesting, something I have been familiarising myself with in relation to the Sachdev-Ye-Kitaev model (more on this later) and quantum gravity. When it comes to the RH, his approach was left mostly abandoned for decades. But Griffin et al formulate what is basically a new general framework, leveraging Pólya’s insights, and in the process proving a few new theorems and even proving criterion pertaining to the RH.

1. Hyperbolicity of Polynomials

I won’t discuss their paper in length, instead focusing on a particular section of the work. But as a short entry to their study, Griffin et al pick up from the work of Pólya, summarising his result about how the RH is equivalent to the hyperbolicity of all Jensen polynomials associated with a particular sequence of Taylor coefficients,

$\displaystyle (-1 + 4z^{2}) \Lambda(\frac{1}{2} + z) = \sum_{n=0}^{\infty} \frac{\gamma (n)}{n!} \cdot z^{2n} \ \ (1)$

Where ${\Lambda(s) = \pi^{-s/2} \Gamma (s/2)\zeta{s} = \Lambda (1 - s)}$, as stated in the paper. Now, if I am not mistaken, the sequence of Taylor coefficients belongs to what is called the Laguerre-Pólya class, in which case if there is some function ${f(x)}$ that belongs to this class, the function satisfies the Laguerre inequalities.

Additionally,  the Jensen polynomial can be seen in (1). Written generally, a Jensen polynomial is of the form ${g_{n}(t) := \sum_{k = 0}^{n} {n \choose k} \gamma_{k}t^{k}}$, where ${\gamma_{k}}$‘s are positive and they satisfy the Turán inequalities ${\gamma_{k}^{2} - \gamma_{k - 1} \gamma_{k + 1} \geq 0}$.

Now, given that a polynomial with real coefficients is hyperbolic if all of its zeros are real, where read in Griffin et al how the Jensen polynomial of degree ${d}$ and shift ${n}$ in the arbitrary sequence of real numbers ${\{ \alpha (0), \alpha (1), ... \}}$ is the following polynomial,

$\displaystyle J_{\alpha}^{d,n} (X) := \sum_{j = 0}^{d} {d \choose j} \alpha (n + j)X^{j} \ \ (2)$

Where ${n}$ and ${d}$ are the non-negative integers and where, I think, ${J_{\alpha}^{d,n} (X)}$ is the hyperbolicity of polynomials. Now, recall that we have our previous Taylor coefficients ${\gamma}$. From the above result, the following statement is given that the RH is equivalent to ${J_{\gamma}^{d,n}(X)}$ – the hyperbolicity of polynomials – for all non-negative integers. What is very curious, and what I would like to look into a bit more, is how this conditions holds under differentiation. In any case, as the authors point out, one can prove the RH by showing hyperbolicity for ${J_{\alpha}^{d,n} (X)}$; but proving the RH is of course notoriously difficult!

Alternatively, another path may be chosen. My understanding is that Griffin-Ono-Rolen-Zagier use shifts in ${n}$ for small ${d}$, because, from what I understand about hyperbolic polynomials, one wants to limit the hyperbolicity in the ${d}$ direction. Then the idea, should I not be corrected, is to study the asymptotic behaviour of ${\gamma(n)}$.

This is the general entry, from which the authors then go on to consider a number of theorems. I won’t go through all of the theorems. One can just as well read the paper and the proofs. What I want to do is focus particularly on Theorem 3.

2. Theorem 3

Aside from the more general considerations and potential breakthroughs with respect to the RH, one of my interests triggered in the Griffin-Ono-Rolen-Zagier paper has to do with my ongoing studies concerning Gaussian Unitary Ensembles (GUE) and Random Matrix Theory (RMT) in the context of the Sachdev-Ye-Kitaev (SYK) model (plus similar models) and quantum gravity. Moreover, RMT has become an interest in relation to chaos and complexity, not least because in SYK and similar models we consider late-time behaviour of quantum black holes in relation to theories of quantum chaos and random matrices.

But for now, one thing that is quite fascinating about Jensen polynomials for the Riemann-zeta function is the proof in Griffin et al of the GUE random matrix model prediction. That is, the derivative aspect GUE random matrix model prediction for the zeros of Jensen polynomials. One of the claims here is that the GUE and the RH are satisfied by the symmetric version of the zeta function. To quote in length,

‘To make this precise, recall that Dyson, Montgomery, and Odlyzko [9, 10, 11] conjecture that the nontrivial zeros of the Riemann zeta function are distributed like the eigenvalues of random Hermitian matrices. These eigenvalues satisfy Wigner’s Semicircular Law, as do the roots of the Hermite polynomials ${H_{d}(X)}$, when suitably normalized, as ${d \rightarrow +\infty}$ (see Chapter 3 of [12]). The roots of ${J){\gamma}^{d,0} (X)}$, as ${d \rightarrow +\infty}$ approximate the zeros of ${\Lambda (\frac{1}{2} + z)}$ (see [1] or Lemma 2.2 of [13]), and so GUE predicts that these roots also obey the Semicircular Law. Since the derivatives of ${\Lambda (\frac{1}{2} + z)}$ are also predicted to satisfy GUE, it is natural to consider the limiting behavior of ${J_{\gamma}^{d,n}(X)}$ as ${n \rightarrow +\infty}$. The work here proves that these derivative aspect limits are the Hermite polynomials ${H_{d}(X)}$, which, as mentioned above, satisfy GUE in degree aspect.’

I think Theorem 3 raises some very interesting, albeit searching questions. I also think it possibly raises or inspires (even if naively) some course of thought about the connection of insights being made in SYK and SYK-like models, RMT more generally, and even studies of the zeros of the Riemann-zeta function in relation to quantum black holes. In my own mind, I also immediately think of the Hilbert-Polya hypothesis and the Jensen polynomials in this context, as well as ideas pertaining to the eigenvalues of Hamiltonians in different random matrix models of quantum chaos. There is connection and certainly also an interesting analogy here. To what degree? It is not entirely clear, from my current vantage. There are also some differences that need to be considered in all of these areas. But it may not be naive to ask, in relation to some developing inclinations in SYK and other tensor models, about how GUE random matrices and local Riemann zeros are or may be connected.

Perhaps I should save such considerations for a separate article.