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.

Statement \Sigma  Dual statement \Sigma^{\star}
f : a \rightarrow b  f : b \rightarrow a
a = \text{dom} f  a = \text{cod} f
i = 1_a  i = 1_a
h = g \circ f  h = f \circ g
f  is monicf  is epis
u  is a right inverse of h  u  is a left inverse of h
f  is invertiblef  is invertible
t is a terminal objectt is an initial object

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.

Reading

[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].

Stringification as categorisation

In quantum field theory one is typically taught to use perturbation theory when the equations of motion for the fields are nonlinear and weakly interacting. For example, in \phi^4  theory one can use a formal series as described by Rosly and Selivanov [1]. Perturbative theory is about mastering series expansions. The basic idea, upon constructing some correlation function in the full nonlinear model, is to expand in powers of \alpha  , namely the interaction strength. In the language of perturbative physics, Feynman diagrams give a representation of each term in the expansion such that we use them to illustrate linear operators. This ultimately enables us to obtain a good approximation to the exact solution. Needless to say, there is a real power and usefulness about perturbative methods and the sum of Feynman diagrams.

When computing amplitudes with Feynman diagrams, the amplitudes depend on various topological properties (i.e., vertices, loops, and so on). Although not always made explicit in the perturbative view, from the Fenynman diagrams of 0-dimensional points with 1-dimensional graphs (to use the language of p-branes, which we’ll get to in a moment), we have topologies that describe linear operators: i.e., what Feynman diagrams start to make explicit is the deeper role of topology in physics [2]. This was summarised wonderfully in a lovely article by Atiyah, Dijkgraaf, and Hitchin [3]. Mathematically, and from the perspective of geometry, the main idea is that a linear operator behaves very much like an n-dimensional manifold going between manifolds of one dimension less, which we may define as a cobordism (i.e., think of a stringy ‘trousers’ diagram) [2,4].

Now, consider the story of p-branes, in particular the perspective as we pass from standard quantum field theory to string theory. The language of p-branes as first described by Duff et al [5] may be reviewed in any introductory string theory textbook. We can, from first-principles, motivate string theory thusly: in a special, if not unique way, we may generalise the point-like 0-dimensional particle to the 1-dimensional string, which is made explicit when we generalise the action for a relativistic particle to the Nambu-Goto action for the relativistic string. In the language of p-branes, which are p-dimensional objects moving through a D(D \geq p)  dimensional space-time, a 0-brane is a (0-dimensional) point particle that that traces out a (0+1)-dimensional worldline. The generalisation of the point particle action S_0 = -m \int ds to a p-brane action in a D(\geq p) -dimensional space-time background is given by S_p = -T_p \int d\mu_p . Here T_p is the p-brane tension with units mass/vol, and d\mu_p is the (p + 1)-dimensional volume element. For the special case where p=1 such that we have 1-brane, we obtain the string action which sweeps out a (1+1)-dimensional surface that is the string worldsheet propagating through space-time. We can also go on to speak of higher-dimensional objects, such as those that govern M-theory. For instance, a 2-brane is a membrane. Historically, these were considered as 2-dimensional particles. There are also 3-branes, 4-branes, and so on.

This generalising process, if we can describe it that way, is what I like to think of as stringification. For the case where p=1 , Feynman diagrams of ordinary quantum field theory with 2-dimensional cobordisms represent world-sheets traced out by strings. The generalising picture, or stringification, show these 2-dimensional cobordisms equipped with extra structure give a powerful mathematical language (describing the relation between physics and topology, as string diagrams enable us to sum over the various topologies and provide a valuable mathematical tool for thinking about composition). But of course this picture can still be extended. Not only does the important analogy between operators and cobordisms come directly into focus, it is also, in some sense, where stringification meets categorification. That is, from the maths side, we arrive at the logic of higher-dimensional algebra and the arrows of monoidal and higher categories. In each, physical processes are describe by morphisms or functors (functors are like morphisms between categories). This generalising picture toward higher geometry, higher algebra, and, indeed, higher structures is called ‘categorifying’ or ‘homotopifying’ (my notes on which I have started to upload to this blog). In this post, I want to think a bit about this idea of stringification as categorification.

***

There is a view of M-theory, and I suppose of fundamental physics as whole, that I find fascinating and compelling: stringification as the categorisation of physics. The notion of stringification is not formal, but captures if nothing else an intuition about a certain generalising process or abstract story, or at least that is how I presently see it. It is a term I have picked up that used to float around in different contexts a couple of decades ago. As described through the language of p-branes, the story begins with the generalisation or stringification of point particle theory (and all that it implies) toward the existence of the string and eventually other extended objects in fundamental physics. Meanwhile, the notion of categorification is certainly formal, signalling, at its origin, the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories. This process, when iterated, gives definition to the notion of n-category theory, where we also replace functions with functors, and equations between functions by natural isomorphisms between functors [6]. As Schreiber pointed out in 2004, there is a sort of harmony between these two processes – stringification and categorification – which has certainly started to clarify over the last decade or more.

As one example, the observation that Schreiber describes in the linked post refers to boundaries of membranes attached to stacks of 5-branes, which conceptually appear as a higher-dimensional generalisation of how boundaries of strings appear.

To understand this think, firstly, of the simple example of the existence of D-branes (Dirichlet membranes) and how the endpoints of open strings can end on these extended objects. In fact, an introductory string textbook will guide one to see why the equations of motion of string theory require that the endpoints of an open string satisfy one of two types of boundary conditions (Dirichlet or Neumann) ending on a brane. If the endpoint is confined to the condition that it may move within some p-dimensional hyperplane, one then obtains a first description of Dp-branes. (I think this was one of the first things I calculated when learning strings!). For the sake of saving space I won’t go into the arrangement of D-branes or other related topics. The main point that I am driving at, the technicalities of which we could review in another post, is how these branes are dynamic and as such they may influence the dynamics of a string (i.e., how an open string might move and vibrate). Thus, the arrangement of branes (e.g., we can have parallel branes or ‘stacks’) will also impact or control the types of particles in our theory. It is truly a beautiful picture.

In p-brane language, if you take the Nambu–Goto action and for the quantum theory study the spectrum of particles, you will see that it exhibits what we may describe as the photon, which of course is the fundamental quantum of the electromagnetic field. Now, what is nice about this is that, the resemblance of the photon is actually a p-dimensional version of the electromagnetic field, so it is in fact a p-dimensional analogue of Maxwell’s equations.

What Schreiber is highlighting in his post is not just that in string theory, the points of the string ending on a Dp-brane give rise to ordinary gauge theory. (One could even take the view that string theory predicts electromagnitism such that string theory predicts the existence of D-branes. It is by their nature that these extended objects all carry an electromagnetic field on their volume, i.e., what we call the brane volume). The point made is that, given there is reason to extend the picture further – the picture of stringification so to speak – to higher-dimensional generalisations, we can then replace strings with membranes, and so on. From the maths side, it was realised that from the perspective of categories, something analogous is happening: replacing points with arrows (i.e., morphisms) one finds the gauge string may be described by the structure of nonabelian gerbes (a gerbe is just a generalised analogue of a fibre bundle), and so on.

When I first learned strings, the picture of stringification was in my mind but I didn’t yet have a word for it. I also didn’t possess category theoretic language at the time; it was really only a vague sense of a picture, perhaps emphasised in the way I learned string theory. So when I discovered and read last year about the idea of stringification as categorisation [7] in Schreiber’s thesis, I was excited.

A nice illustration comes from the first pages of this work. Take some ordinary point-particle, which traces out a worldline over time t . The thrust of the idea is that, given some charge, there is a connection in some bundle (yet unspecified) such that, locally, a group element g \in G is associated to the path. Diagrammatically this may be represented as,

Now consider some time t^{\prime} , where t^{\prime} > t . The particle has travelled a bit further,

We can of course compose these paths. The composition is associative and the operation is multiplication. In fact, what we’re doing is multiplying the group elements. We can also define an inverse g^{-1} . The punchline is that, from the theory of fibre bundles with connection, we can consider how this local picture may fit globally. If g is an element in a non-abelian group, the particle we are generalising is non-abelian. Generalise from a point-particle to a string, and the diagrammatic representation of the world-sheet takes the form

Ultimately, we can continue to play this game and develop the theory of non-abelian strings (and on to higher-dimensional branes), which, it turns out, corresponds with a 2-category theory [7,8]. Sparing details, in n-category theory a 2-category is a special type of category wherein, besides morphisms between objects, it possesses morphisms between morphisms. What is interesting about this example is how we can go on to show the idea of SUSY quantum mechanics on loop space relates to ideas in higher gauge theory, particularly in the sense of categorifying standard gauge theory. For example, John Baez’s paper on higher Yang-Mills [9]. But even before all of that, from the view of perturbative string theory being the categorification of supersymmetric quantum mechanics, we can play the same game such that the generalisation of the membranes of M-theory are a categorification of the supersymmetric string, and so on. The intriguing and, perhaps, grand idea, is that this process of stringification as categorification can be utilised to describe the whole of physics, or, so, it is suspected.

***

I’ve been thinking about this picture quite a bit recently, perhaps spurred by all of my ongoing studies in M-theory. The view to be encircled, as the notion of categorisation enters the stringy picture, also marks for me the beginning of the story about higher structures in fundamental physics (in terms of the view of category theory and higher category theory). In a sense, as much as I currently understand it (as I am very much in the process of studying and forming my thoughts on the matter) we are encircling not much more than an abstract story; but it is one in which many tantalising hints exist about a potentially foundational view.

The history of this higher structure view is rich with examples [10, 11], and, for many reasons, it leads us directly to a study of the plausible existence of M-theory. From the use of braided monoidal categories in the context of string diagrams through to knot theory (See Witten’s many famous lectures); the notion of quantum groups; Segal’s famous work on the axioms of conformal field theory (described in terms of monoidal functors and the category 2Cob_{\mathbb{C}} whose morphisms are string world-sheets such that we can compose the morphisms, and so on); and of course the work of Atiyah in topological quantum field theory (TQFT) followed by Dijkgraaf’s thesis on 2d TQFTs in terms of Frobenius algebras – the list is far to big to summarise in a single paragraph. All of this indicates, in some general sense, a very abstract story from basic quantum mechanics through to string theory and, I would say, as a natural consequence M-theory.

It is a fascinating perspective. There is so much to be said about this developing view, including why higher geometry and algebra seem to hold the important clues of M-theory as a fundamental theory of physics. What is also interesting, as I am beginning to understand, is that in the higher structure picture, a striking consequence from a geometric persective is that the geometry of fundamental physics (higher geometry and supergeometry) may not be described by spaces with sets of points. And, in fact, we start to see this for each value of p  . Instead of a traditional notion space associated with the definition of topological spaces or differentiable manifolds, the geometric observation is that what we’re dealing with is functorial geometry of the sort described by Grothendieck, or synthetic differential geometry of the sort described by Lawvere, or a variation of them both.

Anyway, this is just a short note of me thinking aloud.

References

[1] Rosly, A.A., and Selivanov, K.G., On amplitudes in self-dual sector of Yang-Mills theory. [arXiv:9611101 [hep-th]].

[2] Baez, J., and Stay, M., Physics, Topology, Logic and Computation: A Rosetta Stone. [arXiv:0903.0340 [quant-ph]].

[3] Atiyah, M., Dijkgraaf, R., and Hitchin, N., Geometry and physics. Phil. Trans. R. Soc., (2010), A.368, 913–926. [http://doi.org/10.1098/rsta.2009.0227].

[4] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.].

[5] M. J. Duff, T. Inami, C. N. Pope, E. Sezgin [de], and K. S. Stelle, Semiclassical quantization of the supermembrane. Nucl. Phys. B297 (1988), 515.

[6] Baez, J., and Dolan, J., Categorification. (1998). [arXiv: 9802029 [math.QA]].

[7] Schreiber, U., From Loop Space Mechanics to Nonabelian Strings [thesis]. (2005). [hep-th/0509163].

[8] Baez, J. et al., Categorified Symplectic Geometry and the Classical String. (2008). [math-ph/0808.0246v1].

[9] Baez, J., \textit{Higher Yang–Mills theory}. (2002). [hep-th/0206130].

[10] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.]

[11] Jurco, B. et al., \textit{Higher structures in M-theory}. (2019). [arXiv:1903.02807v2].

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.

Comments

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.

Closing comments

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].

Mathematical physics and M-theory: The study of higher structures

In recent posts we’ve begun to discuss some ideas at the foundation of the duality symmetric approach to M-theory. As we started to review in the last entry, one of the first goals is to formulate and study a general field theory in which T-duality is a manifest symmetry. It was discussed how this was the first-principle goal of double field theory, and it was similarly featured as a motivation in our introductory review of double sigma models. There is a lot to be discussed about the duality symmetric approach moving forward, including the effective theory for this doubled string prior to ultimately looking at lifting to M-theory, where, instead of double field theory we will be working with what is known as exceptional field theory. What also remains an important question has to do with obtaining a global formulation of such duality symmetric actions. What is clear is that higher geometry and algebra are important to achieving such a formulation, and there is much ground to cover on this topic.

Meanwhile, in the present entry I would like to share what I have been studying and learning about as it relates to the other side of my PhD research: the higher structure approach to M-theory. If the duality symmetric approach is a sort of bottom-up way to attack the M-theory proposal, particularly insofar that we are building from the field theory point of view, the higher structure approach can be looked at here as a sort of top-down way to access the question of string theory’s non-perturbative completion. Although this language is a bit schematic, as there is a lot of overlap between the two approaches and their machinery, it does lend some intuition to the different perspectives being undertaken.

***

In William Thurston’s 1994 essay, ‘On proof and progress in mathematics‘ [1], it was argued that progress in mathematics is driven not only by proof of new theorems. Progress is also made by aiding in human beings ways to think about and understand mathematics. Emily Riehl made this a point of emphasis at the beginning of her notes on categorical homotopy theory [2], including on the usefulness of qualitative insights, and I think a similar emphasis may be made here in the context of our focus in mathematical physics and particularly M-theory. A further point of philosophical emphasis in this essay is Eugene Wigner’s article on the unreasonable effectiveness of mathematics in physics and, finally, the more recent presentation by Robbert Dijkgraaf on the unreasonable effectiveness of string theory in mathematics. In my view, M-theory represents one of a few research topics at the frontier of mathematical physics. What parametrises the boundaries of this frontier is the interface between foundational maths and fundamental physics. Indeed, I take this as Dijkgraaf’s point in his presentation at String Math 2020, namely both the need for this engagement and how, historically, progress is often made when the two sides (mathematics and physics) interact. For myself, I almost joined the maths school prior to deciding my future was in mathematical physics, and I find great interest in working at this interface, where, furthermore, when thinking of M-theory Thurston’s notion of progress appears particularly apt.

The motivation may be stated thusly [3]: there presently exists many interconnected hints in support of the proposed existence of M-theory. But a systematic formulation of the full theory – i.e., string theory’s non-perturbative completion – remains an important open problem. A key issue here ultimately concerns the lack of clarity about the underlying principles of M-theory (there are many references on this point, but as one example see [4]). I look at the current situation as a puzzle or as a patchwork quilt. There are pieces of the total picture that we can identify and start to fill in. There are others that remain unknown, leaving empty spaces in our picture of M-theory. And then, finally, how all of the pieces relate or connect is another question that we need to answer but cannot currently access.

To advance the problem, there is ample reason to suggest and to argue that what is needed is new mathematical machinery. As a new researcher, this need was something that I started thinking about a year or more ago. Let me put it this way: our world is best described by quantum field theory. If M-theory is the correct description of fundamental physics, we should end up with a quantum field theoretic description. But it seems unlikely that M-theory will be captured or defined by some Lagrangian, or some S-matrix, or other traditional approaches [3]. Indeed, the tools we need are more than just fibre bundles, standard topology, or differential geometry. Although much of modern physics is built using tools and approaches that deal with local, approximate, perturbative descriptions of reality, in investigating the M-theory problem we need to find ways of dealing with the global and non-perturbative structure of physical fields, and thus we are dealing with the difficulty of employing non-perturbative methods. Entering into my PhD, this is the challenge that I see. I also see this challenge, from the perspective of fundamental physics, as being similar to the situations that have historically arisen many times. A large part of the history of fundamental physics is described by the search for new mathematical language required to aid the modelling of physical phenomena. Hisham Sati and Urs Schreiber [5] presented the argument well, describing the situation explicitly, when discussing the motivation for pursuing a rigorous mathematical foundation for quantum field theory and perturbative string theory. As an example, they cited the identification of semi-Riemannian differential geometry as the underlying structure of gravity. Or, think of the use of representation theory in particle physics. In truth, there are many examples and, to Dijkgraaf’s point, we should embrace this history.

I think this is why, as I prepare to start my formal PhD years, the 2018 Durham Symposium seems momentous, particularly as I begin to generate my thoughts on M-theory and what sort of research I might find meaningful. Although it was slightly before my time, as I was only a first-year undergraduate when the Durham symposium had taken place, I was already developing an interest in non-perturbative theory and I remember learning of the symposium with enthusiasm. It gave me confidence and, I suppose, assurance that my thoughts are moving in the right direction. I’ve also taken confidence from many other important conferences, such as the 2015 conference organised around the theme of new spaces in mathematics and physics. But, for me, the Durham symposium has become a tremendous reference, because the culmination of this search for new mathematical language is apparent, organised under the study of higher structures, and I find this programme of research immensely stimulating.

Similar to the situation in QFT where, over the past decade or more, progress has been made to understand its fundamental nature – for instance, efforts to define QFT on arbitrary corbordism – higher structures provides a concise language of gauge physics and duality that has seemed, in recent years, to open pathways to rigorously attack the M-theory question. Indeed, efforts toward an axiomatic formulation of QFT (for instance, see recent developments in the area of algebraic and topological QFT [6]) and those toward string theory’s full non-perturbative completion to M-theory have a lot in common. Furthermore, an important motivation for the study of higher structures (and higher differential geometry, higher gauge theory and symmetry algebras, and so on) comes directly from decisive hints about the inner workings of M-theory. Hence, the title of the Durham Symposium and its guiding document, ‘Higher structures in M-theory‘.

To give some immediate examples and sketch a few more introductory thoughts, the higher algebraic structures we know to govern closed string field theory is something I started to investigate as related to my recent MRes thesis. But the most basic example of a higher structure in string theory arguably goes back to the first quantisation of the bosonic string. Indeed, as I described in a past note (I think from my first-year undergrad), if I were to teach strings one day my opening lecture would be on generalising point particle theory and emphasising the motivation on why we want to do this. From this approach, I think one can show in a wonderfully pedagogical way that, when generalising from 0-dimensional point particle theory to the 1-dimensional string (and so on), higher dimensionality is a natural consequence and is essentially forced upon us. (As an aside, I remember reading a comment by Schreiber about this very same point of introduction. I recommend reading Schreiber’s many notes over the years. For instance, here is a forethoughtful contribution from 2004 that begins to motivate some of the concepts we will discuss below. A helpful online resource is also ncatlab that covers many of the topics we will be discussing on this blog, along with appropriate references). And, it turns out, this is one way we might also motivate in fundamental physics the study of higher structures; because, in this picture, the Kalb-Ramond 2-form can be seen as an example of a higher structure as it is generalised from the gauge potential 1-form [3]. Of course, since the mid-1990s, a growing body of evidence urged the string theory community to study extended objects of dimension > 1  , and around the same time attempts were already developing to use category theory (more on categories in a moment) to study string diagrams [7], as one can certainly see that string diagrams possess a powerful logic when it comes to composition.

***

So what do we mean by higher structures? From my current vantage, I would describe a higher structure as a categorified mathematical structure, which I also take to mean higher homotopy theory. But we can perhaps begin to build toward the idea by reviewing briefly two main ingredients: category theory and homotopy theory. As a matter of correspondence between mathematics and physics, category theory is the mathematical language of duality and homotopy theory is the mathematical language of the gauge principle.

We may think of category theory as being positioned at the foundations of modern mathematics [8], but, in many ways, it is quite elementary. Similar to the use of a venn diagram when teaching basic set theory, we can build the idea of a category in a fairly intuitive way.

A category {\mathcal{C}} consists of the following data [9]:

* A collection of mathematical objects. If {X} is an object of {\mathcal{C}} , then we write {X \in \mathcal{C}} .

* Every pair of objects {X, Y \in \mathcal{C}} , we may define a set of morphisms {X \rightarrow Y} denoted as {\text{Hom}\mathcal{C}(X,Y)} .

* For every {X \in \mathcal{C}} , there needs to exist an identity morphism {Id_{X} \in \mathcal{C}(X,X)} .

* For every triple {X,Y,Z \in \mathcal{C}} , we may define a composition map {\circ : \mathcal{C}(X,Y) \times \mathcal{C}(Y, C) \rightarrow C(X, Z)} .

* Composition is associative and unital.

If category theory is the mathematics of mathematics, I would currently emphasise in a physics context [10] the approach to category theory as the language that describes composition. Think of the trivial example of moving in some space (let’s not get too stuck on definitions at this point). We can compose the journey from points A to B to C to D in the following way,

\displaystyle  A \rightarrow B \rightarrow C \rightarrow D \ (1).

We can also compose the same journey in terms of pairs of vertices or what we are presently calling points such that

\displaystyle  A \rightarrow C, B \rightarrow D \ (2)  ,

and then we may write the entire journey as {A \rightarrow D} giving the same description in (1).

The idea of a category can be constructed using similar logic. Given a collection of objects {A,B,C,D} , paths {A \rightarrow B \rightarrow C \rightarrow D} denoted by the arrows may be defined as the relation amongst the objects in terms of structure preserving maps {f,g,h} called morphisms.

M-theory

So at its most basic, a category is a collection of objects and arrows between those objects. It is, in some sense, a relational set, which must follow the conditions stated above.

Example. The category of sets, denoted by Set. The category of R-modules, denoted by RMod. A morphism {f : X \rightarrow Y} is said to be an isomorphism if there exists {g : Y \rightarrow X} such that {g \circ f = Id_{X}} and {f \circ g = Id_{Y}} . In the category Set, isomorophisms are bijections.

The concept of functors is of deep importance in this language. In short, a functor is a morphism between categories. If {\mathcal{C}} and {\mathcal{D}} are categories, we may define a functor {F : \mathcal{C} \rightarrow \mathcal{D}} such that it assigns an object {FX \in \mathcal{D}} for any {X \in \mathcal{C}} , and a morphism {Ff : FX \rightarrow FY} for any {f : X \rightarrow Y} , where associativity and unitality are preserved. So, for instance, if {f : X \rightarrow Y} , {g : Y \rightarrow X} , associativity is preserved such that

\displaystyle  Fg \circ Ff = F(g \circ f) \ (3).

We may also define the notion of a natural transformation as a morphism between functors. If {F,G :  \mathcal{C} \rightarrow \mathcal{D}} define two functors, then a natural transformation {F \implies D} assigns any {X \in \mathcal{C}} a morphism {FX \rightarrow GX} .

There is a lot to be said about functors, categorical products, and also the important role duality plays in category theory. In the next entries, we will formally define these ideas as well as many others. For now, I am simply trying to provide some sense of an early introduction into some of the machinery used when we speak of higher structures, such as by giving an intuitive example of a category, with a mind toward formal definition in a following post. The same can be said for all ideas presented here, as, in the present entry, we are simply encircling concepts and sketching a bit of land, similar as a geoscientist would do when first preparing to sketch a topological map.

What one will find, on further inspection, is that category theory is deeply interesting for a number of reasons. At its deepest, there is something to be said about it as a foundational framework. One of the most inspiring realisations about category theory comes from something that seems incredibly basic: the idea in set theory of taking the product of two sets. Indeed, one may have seen this notion of a product as fundamental. But what we observe is that this most basic concept of taking a product of two sets is not fundamental in the way we may have been used to thinking, because one of the amazing things about the story of category theory is how the idea of products is more deeply defined in terms of a categorical product. The reward for this realisation, aside from shear inspiration, is technically immense.

Indeed, a category can contain essentially any mathematical object, like sets, topological spaces, modules, and so on. In many constructions, one will seek to study very generally the products of these objects – so, for example, the product of topological spaces – and the concept of a product in category theoretic language can capture all such instances and constructions. In later discussions we will see how this language allows us to look at mathematics at a large scale, which is to say that, in the abstract, we can take any collection of mathematical objects and study the relations between them. So if the goal is a completely general view, using category theory we are able to strip back a lot of inessential detail so as to drill fundamentally into things.

Additionally, there is a deep relationship between category theory and homotopy theory, which, in this post, I would like to highlight on the way to offering a gentle introduction to the concept of a higher structure. Down the road we will discuss quite a bit about higher-dimensional algebra, such as n-categories and operads, which are algebraic structures with geometric content, as we drive toward a survey of the connection between higher categorical structures and homotopy theory. In physics, there is also connection here with things like topological quantum field theory. Needless to say, there is much to cover, but when thinking of homotopy theory at its most basic, it is appropriate to go all the way back to algebraic topology.

The philosophical motivation is this: there are many cases in which we are interested in solving a geometrical problem of global nature, and, in algebraic topology, the method is generally to rework the problem into a homotopy theoretic one, and thus to reduce the original geometric problem to an algebraic problem. Let me emphasise the key point: it is a fundamental achievement of algebraic topology to enable us to reduce global topological problems into homotopy theory problems. One may motivate the study of homotopy theory thusly: if we want to think about general topological spaces – for example, arbitrary spaces that are not Hausdorff or even locally contractible – what this amounts to is that we relax our interest in the notion of equivalence under homeomorphism (i.e., topological equivalence) and instead work up to homotopy equivalence.

Definition 1 Given maps {f_0,f_1: X \rightarrow Y} , we may write {f_0 \simeq f_1} , which means {f_0} is homotopic to {f_1} , if there exists a continuous map {F : X \times I \rightarrow Y} , called a homotopy, such that {F(x,0) = f_0(x)} and { F(x,1) = f_1(x)} . We may also write {F: f_0 \implies f_1} to denote the homotopy.

As suggested a moment ago, a homotopy relation {\simeq} is an equivalence relation. This is true if {F_{01} : f_0 \implies f_1} and {F_{12} : f_1 \implies f_2} for the family of maps {f_i : X \rightarrow Y} , then

F_{02} (t,x) =  \begin{cases} F_{01}(2t,x)  :  0 \leq t \leq 1/2 \\ F_{01}(2t-1,x)  :  1/2 \leq t \leq 1 \\          \end{cases} \ (4)

gives a homotopy {F_{02} : f_0 \implies f_2} .

As an aside, what is both lovely and interesting is how, from a physics perspective, we may think of homotopy theory and ask how it might relate to the path integral; because, on first look, it would seem intuitive to ask this question. There is a long and detailed way to show it to be true, but, for simplicity, the argument goes something as follows. Think, for starters, of what we’re saying in the definition of homotopy. Given some {X} , which for now we’ll define as a set but later understand as a homotopy type, let us define two elements {x,y \in X} such that we may issue the following simple proposition {x = y} . The essential point, here, is that there may be more than one way that {x} is equal to {y} , or, in other words, there may be more than one reason or more than one path. Hence, we can construct a homotopy {\gamma} such that x \xrightarrow[]{\gamma} y  is a homotopy from {x} to {y} and then an identity map {Id_{X}(x,y)} for the set of homotopies from {x} to {y} in {X} . One can then proceed to follow the same reasoning and construct a higher homotopy by defining a homotopy of homotopy and so on.

The analogy I am drawing is that, in the path integral formalism, given some simply-connected topological space, recall that we can continuously deform the path {x(t)} to {x(y)} . In this deformfation, {\phi[x(t)]} approaches {\phi[y(t)]} continuously such that, taking the limit, we have

\displaystyle  \phi[y(t)]=\lim\phi[x(t)]=e^{iS[y(t)]}, \ \text{as }x(t)\rightarrow y(t) \text{continuously}. \ (5)

The principle of the superposition of quantum states, or, the sum of many paths, in a simply-connected space can be constructed as a single path integral; because, when all of the dust settles, the paths in this space can be shown to contribute to the total amplitude with the same phase (this is something we can lay out rigorously in another post). The result is that we end up with the Feynman path integral.

In homotopy theory, on the other hand, the analogous is true in that paths in the same homotopy class contribute to the total amplitude with the same phase. So, if one defines the appropriate propagator and constrains appropriately to the homotopy class, an equivalent expression for the path integral may be found. And really, one can probably already start to suspect this in the basic example of homotopy theory of topological spaces. Typically, given a topological space {X} and two continuous functions from this space to another topological space {Y} such that

\displaystyle f,g : X \rightarrow Y \ (6)  ,

it is straightforward to define, with two points in the mapping space, {f,g \in \text{Maps}(X,Y)} a homotopy {\eta}

\displaystyle  f \xrightarrow[]{\eta} g \ (7).

This is just a collection of continuous paths between the points.

But I digress. The focus here is to build up to the idea of higher structures.

The reason that a brief introduction to homotopy theory aids this purpose is because, if we think of a higher structure as a categorified mathematical structure, what we are referring to is a phenomenon in which natural algebraic identities hold up to homotopy. In other words, we’re speaking of mathematical structure in homotopy theory and thus of higher algebra, higher geometry, and so forth. Higher algebra consists of algebraic structure within higher category theory [11, 12]. As we discussed earlier, categories have a set of morphisms between objects, and, so, in the example of the category of sets, elements of a set may or may not be equal. Higher categories, much like higher algebra, are a generalisation of these sort of constructions we see in ordinary category theory. In the higher case we now have homotopy types of morphisms, which are called mapping spaces. And so, unless we are working with discrete objects, we must deal with homotopy as an equivalence relation should two so-called elements of a homotopy type, typically represented by vertices, be connected in a suitable way.

When we speak of higher structures as mathematical structures in homotopy theory, this is more specifically a mathematical structure in {(\infty, 1)-\text{category theory}} . This is a special category such that, from within the collection of all {(n, r)-\text{categories}} , which is defined to be an {\infty-\text{category}} satisfying a number of conditions, we find an {(\infty, 1)-\text{category theory}} to be a weak {\infty-\text{category}} in which all n-morphisms for {n \geq 2} are equivalences. I also think of a higher structure almost as a generalisation of a Bourbaki mathematical structure. But perhaps this comment should be reserved for another time.

In summary, if as motivation it is the case that we often want to study homotopy theory of homotopy theories, for instance what is called a Quillen model category, what we find is a hierarchy of interesting structures, which is described in terms of the homotopy theoretic approach to higher categories. And it is from this perspective that homotopy theories are just {(\infty, 1)-\text{category theory}} , where {\infty} denotes structure with higher morphisms (of all levels) and the 1 refers to how all the 1-morphisms and higher morphisms are weakly invertible. Hence, too, in higher category theory we may begin to speak of {(\infty, n)-\text{categories}} , which may be described as:

1. An n-category up to homotopy (satisfying the coherence laws, more on this in a later post);

2. An {(r, n)-\text{categories}} for {r = \infty} ;

3. A weak {\infty-\text{category}} or {\omega-\text{category}} where all k-morphisms are equivalences satisfying the condition {k > n} .

There are different ways to define {(\infty, n)-\text{categories}} , and their use can be found in such places as modern topological field theory. If category theory is a powerful language to study the relation between objects, n-categories enables us to then go on and study the relations between relations, and so on. As an example, consider the category of all small categories. For two categories {\mathcal{C}} , {\mathcal{D}} , whose morphisms are functors, the set or collection of all morphisms hom-set {\text{Fun}(C, D)} are then functors from {\mathcal{C}} to {\mathcal{D}} . This forms a functor category in which all morphisms are natural transformations, given that the natural transformations are morphisms between morphisms (functors). Hence, in this way, we scratch the surface of the idea of higher categories, because, taking from what was mentioned above, these are categories equipped with higher {n} -morphisms between {(n-1)}-morphisms for all {n \in \mathbb{N}} .

Moreover, if in ordinary category theory there are objects and morphisms between those objects, from the higher category view these are seen as 1-morphisms. Then, we may define a 2-category, which is just a generalisation that includes 2-morphisms between the 1-morphisms. And we can therefore continue this game giving definition to {n} -category theory. We will eventually get into more detail about the idea of {n} -categories, including things like weak {n} -categories where associativity and identity conditions are no longer given by equalities (i.e., they are no longer strict), instead satisfied up to an isomorphism of the next level. But for now, in thinking of the basic example of a composition of paths and this notion of generalising to 2-morphisms between the 1-morphism, the emphasis here is on the idea that the two conditions of associativity and identity must hold up to reparameterisation (the topic of reprematerisation being a whole other issue) – hence, up to homotopy – and what this amounts to is a 2-isomorphism for a 2-category. If none of this is clear, hopefully more focused future notes will help spell it all out with greater lucidity.

***

In using the language of higher structures in M-theory, there have been many promising developments. For instance, it can be seen how core structures of string/M-theory emerge as higher structures in super homotopy theory [4, 13], leading to a view of M-theory beginning from the superpoint in super Minkowski spacetime going up to 11-dimensions. An interesting part of this work was the use of Elmendorf’s theorem on equivariant homotopy theory. It has led to exciting new developments in our picture of brane physics, with an updated brane bouquet.

Of course, the higher structures programme is far-reaching. From double and exceptional field theory and the global formulation of such actions to the study of homotopy algebras in string field theory, M-branes, sigma models on gerbes, and even modern views on anomalies in which field theories are treated as functors – this merely scratches the surface. Some nice lecture notes on higher structures in M-theory, focusing for example on M5-brane systems and higher gauge theory were recently offered by Christian Saemann [14]. Hopefully we will be able to cover many of these ideas (and others) moving forward. Additionally, I am currently enjoying reading many older works, such as Duiliu-Emanuel Diaconescu’s paper on enhanced D-brane categories in string field theory [15], and I’ve been working through Eric Sharpe’s 1999 paper [16], which was the first to explicitly draw the correspondence between derived categories and Dp-branes in his study of Grothendieck groups of coherent sheaves. These and others will be fun papers to write about in time.

To conclude, we’ve begun to introduce, even if only schematically, some important ideas at their most basic when it comes to studying higher structures in M-theory. In the next entries, we can deepen our discussion with more detailed notes and definitions, perhaps beginning with a formal discussion on category theory and then homotopy theory, and then a more rigorous treatment of the idea of a higher structure.

References

[1] William P. Thurston. On Proof and Progress in Mathematics, pages 37–55. Springer New York, New York, NY, 2006.

[2] Emily Riehl. Categorical Homotopy Theory. New Mathematical Monographs. Cambridge University Press, 2014.

[3] Branislav Jurco, Christian Saemann, Urs Schreiber, and Martin Wolf. Higher structures in m-theory, 2019.

[4] Domenico Fiorenza, Hisham Sati, and Urs Schreiber. The rational higher structure of m-theory. Fortschritte der Physik, 67(8-9):1910017, May 2019.

[5] Hisham Sati and Urs Schreiber. Survey of mathematical foundations of qft and perturbative string theory, 2012.

[6] J. Baez and J. Dolan. Higher dimensional algebra and topological quantum field theory. Journal of Mathematical Physics, 36:6073–6105, 1995.

[7] Daniel Marsden. Category theory using string diagrams, 2014.

[8] Birgit Richter. From Categories to Homotopy Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2020.

[9] Carlos T. Simpson. Homotopy theory of higher categories, 2010.

[10] Bob Coecke and Eric Oliver Paquette. Categories for the practising physicist, 2009.

[11] T. Leinster. Topology and higher-dimensional category theory: the rough idea. arXiv: Category Theory, 2001.

[12] J. Baez. An introduction to n-categories. In Category Theory and Computer Science, 1997.

[13] John Huerta, Hisham Sati, and Urs Schreiber. Real ade-equivariant (co)homotopy and super m-branes. Communications in Mathematical Physics, 371(2):425–524, May 2019.

[14] Christian Saemann. Lectures on higher structures in m-theory, 2016.

[15] Duiliu-Emanuel Diaconescu. Enhanced d-brane categories from string field theory. arXiv: High Energy Physics – Theory, 2001.

[16] E. Sharpe. D-branes, derived categories, and grothendieck groups. Nuclear Physics, 561:433–450, 1999.

*Image: ‘Homotopy theory harnessing higher structures’, Newton Institute.

*Edited for spelling, grammar, and syntax.