\[\gdef\id{\mathrm{id}} \gdef\tr{\operatorname{tr}} \gdef\Hom{\operatorname{Hom}} \gdef\End{\operatorname{End}} \gdef\1{\mathbf{1}} \gdef\cat#1{\mathcal{#1}} \gdef\Cat#1{\textsf{\textbf{#1}}} \gdef\qtext#1{\enspace\text{#1}\enspace} \gdef\ldual#1{ {}^\vee\mkern{-2mu}#1}\]

Category theory formalizes many mathematical structures, especially the mapping between “objects”. A modular tensor category is a category with some extra structures and can provide the mathematical foundation of some physical concepts, especially the topological orders.

In the first part of this note, we will introduce some basic ideas of category theory. For people with programming backgrounds (or just interested in it), we will use Haskell as an example. Then some more structures will be added into the “bare” category, such as tensor product, braiding and dual. These will be helpful when we deal with modular tensor category in the next part, although some physicists may complain that they are too tedious.

Basic knowledge of categories

Category

A category $\cat{C}$ consists of

  • Objects: $x\in\cat{C}$
  • Morphisms: $f\colon x\to y$ or more precisely $f\in\Hom_\cat{C}(x,y)$. We call $\Hom_\cat{C}(x,y)$ the hom-set and $x$ and $y$ domain and codomain respectively
  • If we have two morphisms $f\colon x\to y$ and $g\colon y\to z$, then there exists the composition of morphisms $g\circ f\colon x\to z$

and the following axioms should hold:

  • Associativity: for $f\colon x\to y$, $g\colon y\to z$ and $h\colon z\to w$, their composition should satisfy

    \[(h\circ g)\circ f = h\circ(g\circ f)\]
  • Identity: $\forall x\in\cat{C}$, there exists an identity morphism $\id_x\colon x\to x$ such that

    \[f\circ \id_x = \id_y\circ f = f, \quad \forall f\colon x\to y\]

In a pictorial representation, the morphism in a category can be visualized as a “black box”:

morphism

For identity morphism $\id_x$, the box can be omitted:

id-morphism

For the composition of two morphisms $f$ and $g$, we use two consecutive boxes:

composition

In our convention, these diagrams should be read from top to bottom, as the arrows indicate. The direction will be significant when we meet the dual of objects (see below).

There are some special cases of morphisms. Let $f\colon x\to y$, then

  • $f$ is a monomorphism if for all morphisms $g_1, g_2\colon w\to x$, $f\circ g_1=f\circ g_2\implies g_1=g_2$ (left cancellable)
  • $f$ is an epimorphism if for all morphisms $h_1, h_2\colon y\to z$, $h_1\circ f=h_2\circ f\implies h_1=h_2$ (right cancellable)
  • $f$ is an isomorphism if there exists $f^{-1}\colon y\to x$ such that $f^{-1}\circ f=\id_x$ and $f\circ f^{-1}=\id_y$. Here, $f^{-1}$ is called the inverse of $f$
  • If the domain and codomain of $f$ coincide, i.e. $f\in\Hom_\cat{C}(x,x)\coloneqq\End_{\cat{C}}(x)$, then $f$ is an endomorphism
  • If $f$ is both an endomorphism and an isomorphism, then it’s called an automorphism

To better understand the abstract idea of category, objects and morphisms, we give some concrete examples:

Categories Objects Morphisms
Set sets functions
Grp groups group homomorphisms
Top topological spaces continuous maps
Vec vector spaces linear maps
Hask Haskell types pure functions

Functor

A functor $F\colon \cat{C}\to\cat{D}$ is a map between two categories such that

  • Object $x\in\cat{C}$ maps to object $F(x)\in\cat{D}$
  • Morphism $f\in\Hom_\cat{C}(x,y)$ maps to $F_f\in\Hom_\cat{D}(F(x),F(y))$

where $F$ preserves

  • Identity:

    \[F_{\id_x} = \id_{F(x)} \in \Hom_\cat{D}(F(x),F(x)), \quad \forall x\in\cat{C}\]
  • Composition:

    \[F_{g\circ f} = F_g\circ F_f \in \Hom_\cat{D}(F(x),F(z)), \quad \forall f\in\Hom_\cat{C}(x,y), \, g\in\Hom_\cat{C}(y,z)\]

In Haskell, a functor is defined as a type class:

class Functor f where
    fmap :: (a -> b) -> f a -> f b

The two laws that a functor should obey are then:

fmap id      == id               -- Preserve identities
fmap (f . g) == fmap f . fmap g  -- Preserve composition

Let’s see an example. First, we have a function that converts an integer to a string:

show :: Int -> String

Then we have the List (or []) functor, which maps the Hask category to Lst (a subcategory of Hask that contains all the list types):

instance Functor [] where
    fmap = ...

Therefore fmap show is a function (i.e. morphism) in Lst that converts a list of integers to a list of strings.

Natural transformation

Given two functors $F\colon \cat{C}\to\cat{D}$ and $G\colon \cat{C}\to\cat{D}$, a natural transformation (aka. functorial morphism) $\tau\colon F\Rightarrow G$ is defined with its component $\tau_x$ (a morphism in $\cat{D}$):

\[\tau_x\colon F(x)\to G(x), \quad \forall x\in\cat{C}\]

such that

\[\tau_y\circ F_f = G_f\circ\tau_x, \quad \forall f\in\Hom_\cat{C}(x,y)\]

In a pictorial representation, that is to say the following diagram is commutative, i.e. the two paths from $F(x)$ to $G(y)$ are equivalent:

natural-transformation-components-1

or more complicated:

natural-transformation-components-2

Natural transformation itself is usually denoted as a double arrow:

natural-transformation

A natural transformation $\tau$ with every component $\tau_x$ invertible is called a natural isomorphism. In such case, we can automatically define the natural transformation $\tau^{-1}$ with components $\tau^{-1}_x\colon G(x)\to F(x)$ such that

\[\tau^{-1}_x\circ\tau_x = \id_{F(x)}, \quad \tau_x\circ\tau^{-1}_x = \id_{G(x)}\]

In Haskell, natural transformation is given by parametric polymorphic functions, such as

natTrans :: forall a . F a -> G a

where a is the type parameter, while F and G are two functors (i.e. type classes which are instances of Functor).

To give a concrete example, we will implement a “safe” version of head. In Haskell Prelude, head is a function that returns the first item of a list:

head :: [a] -> a
head (x:xs) = x

So clearly head [1,2,3] gives 1. But it will throw an exception when applied on an empty list:

> head []
*** Exception: Prelude.head: empty list

We can use Maybe (similar to std::option<T> in C++ and Option<T> in Rust) as a wrapper to avoid such exception:

safeHead :: [a] -> Maybe a
safeHead []     = Nothing
safeHead (x:xs) = Just x

Here safeHead is a polymorphic function, or natural transformation in category theory’s language, that maps from [] functor to Maybe functor. Note that, however, the safeHead function may break the free theorem for fmap.

Tensor category

Roughly speaking, a tensor category (or monoidal category) is a category with a “tensor product”. A basic example is the vector space (or the category Vec), where the tensor product is defined as the combination of two vector spaces, as well as the linear maps over them.

The formal definition requires some other concepts:

  • For categories $\cat{C}$ and $\cat{C}^\prime$, their Cartesian product $\cat{C}\times\cat{C}^\prime$ is also a category where

    • Object is a pair $(x,x^\prime)$
    • Morphism is $(f,f^\prime)\colon(x,x^\prime)\to(y,y^\prime)$
    • Identity is $\id_{(x,x^\prime)}\coloneqq(\id_x,\id_{x^\prime})\colon(x,x^\prime)\to(x,x^\prime)$
    • Composition is $(g,g^\prime)\circ(f,f^\prime)\coloneqq(g\circ f,g^\prime\circ f^\prime)$
  • A bifunctor $F\colon\cat{C}\times\cat{C}^\prime\to\cat{D}$ is a map such that

    • Object $(x,x^\prime)\in\cat{C}\times\cat{C}^\prime$ maps to object $F(x,x^\prime)\in\cat{D}$
    • Morphism $(f,f^\prime)\in\Hom_{\cat{C}\times\cat{C}^\prime}\bigl((x,x^\prime),(y,y^\prime)\bigr)$ maps to $F_{(f,f^\prime)}\in\Hom_\cat{D}\bigl(F(x,x^\prime),F(y,y^\prime)\bigr)$
    • The identity and composition are preserved just as in a normal functor

Now we can define the tensor category $\cat{C}$ with

  • Tensor product, which is a bifunctor $\otimes\colon\cat{C}\times\cat{C}\to\cat{C}$
  • A unit object $\1\in\cat{C}$
  • Associator $\alpha$, which is a natural isomorphism:

    \[\alpha_{x,y,z} \colon (x\otimes y)\otimes z \overset\sim\to x\otimes(y\otimes z), \quad \forall x,y,z \in \cat{C}\]
  • Left unitor $\lambda$ and right unitor $\rho$, which are natural isomorphisms with

    \[\lambda_x \colon \1\otimes x \overset\sim\to x, \quad \rho_x \colon x\otimes\1 \overset\sim\to x, \quad \forall x \in \cat{C}\]

such that the following two diagrams commute:

  • Triangle equation:

    triangle-equation

  • Pentagon equation:

    pentagon-equation

In the above definition, we use “$\overset\sim\to$” to denote the natural isomorphism. If “$\overset\sim\to$” becomes “$=$”, then we call the tensor category strict. In such case, $\alpha_x$, $\lambda_x$ and $\rho_x$ become identity isomorphisms.

MacLane gives the following important result:

Coherence theorem:
Every tensor category is (tensor) equivalent to a strict one.

The name tensor category is very intuitive, as we have just equipped the category with a tensor product. The alternate name monoidal category is not so straight forward, but it indicates a remarkable fact: a strict monoidal category is indeed a monoid (i.e. a “group” without invertibility):

  • Tensor product $\otimes$ corresponds to the multiplication in the monoid with the associativity axiom
  • Unit object $\1$ corresponds to the identity element in the monoid

Braided tensor category

In physics, we use “exchange” to describe the process of moving two systems around each other, such as the exchange of two fermions/bosons. In category theory, we call it “braiding”.

A braided tensor category (or braided tensor category, BTC) $\cat{C}$ is a tensor category with a natural isomorphism

\[\sigma_{x,y} \colon x\otimes y \overset\sim\to y\otimes x, \quad \forall x,y \in \cat{C}\]

such that the following diagrams (hexagon equations) commute:

hexagon-equation

In the graphical notation, $\sigma$ and its inverse can be visualized as

sigma

Then it can be easily shown that $\sigma_{x,y}^{-1}\circ\sigma_{x,y}=\sigma_{y,x}\circ\sigma_{y,x}^{-1}=\id_{x\otimes y}$, as they are topologically equivalent:

sigma-composition

The hexagon equations become apparent as well:

hexagon-equation-string

Furthermore, we can find the following important identities via the graphical calculus:

  • $\sigma_{x^\prime,y^\prime}\circ(f\otimes g) = (g\otimes f)\circ\sigma_{x,y}, \quad \forall f\in\Hom_\cat{C}(x,x^\prime), \, g\in\Hom_\cat{C}(y,y^\prime)$

    btc-identity

  • $(\sigma_{y,z}\otimes\id_x) \circ (\id_y\otimes\sigma_{x,z}) \circ (\sigma_{x,y}\otimes\id_z) = (\id_z\otimes\sigma_{x,y}) \circ (\sigma_{x,z}\otimes\id_y) \circ (\id_x\otimes\sigma_{y,z})$, which is called the Yang–Baxter equation

    yang-baxter-equation

A braided tensor category is called symmetric, if $\sigma_{y,x}\circ\sigma_{x,y}=\id_{x\otimes y}$, or equivalently $\sigma_{x,y}=\sigma_{y,x}^{-1}$, just as the following diagram:

symmetric-category

In braided tensor categories, a compatible functor requires some more constraints. Let $\cat{C}_1$ and $\cat{C}_2$ be braided tensor categories, then a tensor functor from $\cat{C}_1$ to $\cat{C}_2$ is defined as a pair $(F,\mu)$ where

  • $F\colon\cat{C}_1\to\cat{C}_2$ is a functor
  • $\mu_{x,y}\colon F(x\otimes y)\overset\sim\to F(x)\otimes F(y)$ is a natural isomorphism in $\cat{C}_2$, with $x,y\in\cat{C}_1$

such that the following conditions are satisfied:

  • For associativity isomorphisms

    \[\begin{aligned} \alpha_{1;\,x,y,z} &\colon (x\otimes y)\otimes z \overset\sim\to x\otimes(y\otimes z) \\ \alpha_{2;\,F(x),F(y),F(z)} &\colon [F(x)\otimes F(y)]\otimes F(z) \overset\sim\to F(x)\otimes[F(y)\otimes F(z)], \quad \forall x,y,z \in \cat{C}_1 \end{aligned}\]

    in $\cat{C}_1$ and $\cat{C}_2$ respectively, we have

    \[F_{\alpha_{1;\,x,y,z}} = \alpha_{2;\,F(x),F(y),F(z)} \qtext{or simply} F_{\alpha_1}=\alpha_2, \quad \forall x,y,z \in \cat{C}_1\]

    by using natural isomorphism $\mu$;

  • Similar to associativity, but for braiding isomorphisms

    \[\begin{aligned} \sigma_{1;\,x,y} &\colon x\otimes y \overset\sim\to y\otimes x \\ \sigma_{2;\,F(x),F(y)} &\colon F(x)\otimes F(y) \overset\sim\to F(y)\otimes F(x), \quad \forall x,y \in \cat{C}_1 \end{aligned}\]

    in $\cat{C}_1$ and $\cat{C}_2$ respectively, we have

    \[F_{\sigma_{1;\,x,y}} = \sigma_{2;\,F(x),F(y)} \qtext{or simply} F_{\sigma_1}=\sigma_2, \quad \forall x,y \in \cat{C}_1\]

    still by using $\mu$.

For a tensor category $\cat{C}$ (not necessarily braided), its Drinfeld center $\cat{Z}(\cat{C})$ is defined by another tensor category whose

  • Object is a pair $(x,\phi^{(x)})$, where $x\in\cat{C}$ is an object and $\phi^{(x)}$ is a natural isomorphism:

    \[\phi^{(x)}_y \colon x\otimes y \overset\sim\to y\otimes x\]

    such that

    \[\phi^{(x)}_{y\otimes z} = (\id_y\otimes\phi^{(x)}_z) \circ (\phi^{(x)}_y\otimes\id_x), \quad \forall y,z\in\cat{C}\]
  • Morphism $f\colon(x,\phi^{(x)})\to(y,\phi^{(y)})$ is a morphism $f\colon x\to y$ in $\cat{C}$ such that

    \[(\id_z\otimes f)\circ\phi^{(x)}_z = \phi^{(y)}_z\circ(f\otimes\id_z), \quad \forall z\in\cat{C}\]
  • Tensor product is

    \[(x,\phi^{(x)})\otimes(y,\phi^{(y)}) = \left( x\otimes y, \, (\phi^{(x)}\otimes\id_y)\otimes(\id_x\otimes\phi^{(y)}) \right)\]

The Drinfeld center $\cat{Z}(\cat{C})$ is naturally a braided tensor category, with braiding isomorphism given by

\[\sigma_{(x,\phi^{(x)}),\,(y,\phi^{(y)})} \colon (x,\phi^{(x)})\otimes(y,\phi^{(y)}) \to (y,\phi^{(y)})\otimes(x,\phi^{(x)})\]

Ribbon category

Dual

Let’s introduce the concept of dual in a tensor category first. This is a generalization of the dual vector space: for a vector space $V$ over field $F$, its dual space $V^\vee$ is defined as the set of all linear maps $\phi\colon V\to F$.

The right dual of an object $x\in\cat{C}$ is an object $x^\vee$ with two morphisms:

\[e_x\colon x^\vee\otimes x\to\1, \quad i_x\colon\1\to x\otimes x^\vee\]

such that the composition $(\id_x\otimes e_x)\circ(i_x\otimes\id_x)=\id_x$:

\[x \xrightarrow{i_x\otimes\id_x} (x\otimes x^\vee)\otimes x = x\otimes(x^\vee\otimes x) \xrightarrow{\id_x\otimes e_x} x\otimes\1 = x\]

and $(e_x\otimes\id_{x^\vee})\circ(\id_{x^\vee}\otimes i_x)=\id_{x^\vee}$:

\[x^\vee \xrightarrow{\id_{x^\vee}\otimes i_x} x^\vee\otimes(x\otimes x^\vee) = (x^\vee\otimes x)\otimes x^\vee \xrightarrow{e_x\otimes\id_{x^\vee}} \1\otimes x^\vee = x^\vee\]

The above conditions are called rigidity axioms. In some sense, $e_x$ and $i_x$ are called annihilation and creation morphisms, as they can annihilate/create objects to/from “vacuum”, just as the $\hat{a}$ and $\hat{a}^\dagger$ operators in quantum mechanics.

Similarly, we can define the left dual with the following morphisms:

\[e'_x \colon x\otimes \ldual{x}\to\1, \quad i'_x\colon \1\to\ldual{x}\otimes x\]

and similar rigidity axioms. By definition, if $x$ is the right dual of $y$, then $y$ is the left dual of $x$, and vice versa. For a tensor category $\cat{C}$, if every object $x\in\cat{C}$ has left/right dual, then $\cat{C}$ is called left/right rigid. If both types of duals exist, then it’s called rigid (or autonomous).

In the graphical notation, the dual of an object is represented by simply reversing the arrow. In addition, the unit object $\1$, or “vacuum”, can be neglected. So $e_x$ and $i_x$ correspond to the following diagrams:

dual

The rigidity axioms (for right dual) then become

rigidity-axioms

It can be shown that $\Hom_\cat{C}(x,y)$ is isomorphic to $\Hom_\cat{C}(y^\vee,x^\vee)$, therefore for every morphism $f\in\Hom_\cat{C}(x,y)$, we can define the dual as its image $f^\vee\in\Hom_\cat{C}(y^\vee,x^\vee)$:

\[f^\vee \coloneqq (e_y\otimes\id_x) \circ (\id_{y^\vee}\otimes f\otimes\id_{x^\vee}) \circ (\id_{y^\vee}\otimes i_x)\]

or in the graphical notation:

f-dual

If combining the dual with tensor and braiding structures, we can then find some important identities:

  • $(x\otimes y)^\vee = y^\vee\otimes x^\vee$

    dual-and-monoidal

  • $(e_y\otimes\id_x)\circ\sigma_{x,y^\vee\otimes y} = \sigma_{x,\1}\circ(\id_x\otimes e_y)$

    lemma-e-sigma

  • $e_{x\otimes y} = (e_x\otimes e_y)\circ(\sigma_{y^\vee,x^\vee\otimes x}\otimes\id_y)$

    e-and-braiding

  • $i_{x\otimes y} = (\id_x\otimes\sigma_{x^\vee,y^\vee\otimes y})\circ(i_x\otimes i_y)$

    i-and-braiding

  • $e_y\circ(\id_{y^\vee}\otimes f) = e_x\circ(f^\vee\otimes\id_x), \quad (f\otimes\id_{x^\vee})\circ i_x = (\id_y\otimes f^\vee)\circ i_y$

    dual-morphism-identities

  • $\sigma_{x,y}^\vee = \sigma_{x^\vee,y^\vee}$

    dual-and-braiding

where $\sigma$ can be replaced by $\sigma^{-1}$ in these equations.

Pivotal and spherical structure

For a finite-dimensional vector space $V$, we know that its double dual $V^{\vee\vee}$ is isomorphic to $V$. Its categorical generalization (over a right rigid category $\cat{C}$) is the pivotal structure, which is given by the following natural isomorphism:

\[\delta_x \colon x \overset\sim\to x^{\vee\vee}, \quad \forall x\in\cat{C}\]

such that

\[\delta_{x\otimes y} = \delta_x\otimes\delta_y, \quad \delta_\1 = \id_\1, \quad \delta_{x^\vee} = (\delta_x^\vee)^{-1}\]

The definition of right dual gives

\[e_x\colon x^\vee\otimes x^{\vee\vee}\to\1, \quad i_x\colon\1\to x^{\vee\vee}\otimes x^\vee\]

and therefore $x^{\vee\vee}$ is the left dual of $x^\vee$. By renaming $x^\vee$ to $y$, we can see that $\ldual{y}=y^\vee$.

In a pivotal category $\cat{C}$, a morphism $f\in\Hom_\cat{C}(x,y)$ will be identified with its double dual $f^{\vee\vee}\in\Hom_\cat{C}(x^{\vee\vee},y^{\vee\vee})\simeq\Hom_\cat{C}(x,y)$:

double-dual

For an endomorphism $f\in\End_{\cat{C}}(x)$, where $\cat{C}$ is a pivotal category, we can define the left and right (pivotal) traces as

\[\begin{aligned} \tr_{\text{L}} f &\colon \1 \xrightarrow{i_{x^\vee}} x^\vee\otimes x^{\vee\vee} \xrightarrow{\id_{x^\vee}\otimes\delta_x^{-1}} x^\vee\otimes x \xrightarrow{\id_{x^\vee}\otimes f} x^\vee\otimes x \xrightarrow{e_x} \1 \\ \tr_{\text{R}} f &\colon \1 \xrightarrow{i_x} x\otimes x^\vee \xrightarrow{f\otimes\id_{x^\vee}} x\otimes x^\vee \xrightarrow{\delta_x\otimes\id_{x^\vee}} x^{\vee\vee}\otimes x^\vee \xrightarrow{e_{x^\vee}} \1 \end{aligned}\]

In particular, when $f=\id_x$, we can define the left and right dimension of $x\in\cat{C}$ as

\[\dim_{\text{L}}x\coloneqq\tr_{\text{L}}\id_x, \quad \dim_{\text{R}}x\coloneqq\tr_{\text{R}}\id_x\]

Note that the left/right trace is not the same thing as the left/right dual (we only use left dual in the definition, for example), as $\cat{C}$ is already pivotal.

The graphical representation of trace and dimension are the following:

trace-dimension

If the left and right traces of every endomorphism $f\in\End_{\cat{C}}(x)$ coincide, then $\cat{C}$ is called spherical.

Ribbon

A ribbon category (or tortile category) is a braided pivotal category equipped with a twist (or balancing isomorphism):

\[\theta_x \colon x \overset\sim\to x, \quad \forall x\in\cat{C}\]

such that the balancing axioms are satisfied:

\[\theta_{x\otimes y} = \sigma_{y,x} \circ \sigma_{x,y} \circ (\theta_x\otimes\theta_y), \quad \theta_\1 = \id_\1, \quad \theta_{x^\vee} = \theta_x^\vee\]

We can explicitly construct $\theta$ via the following natural isomorphism:

\[\psi_x \colon x^{\vee\vee} \overset\sim\to x, \quad \psi_x \coloneqq (\id_x\otimes e_{x^\vee}) \circ (\id_x\otimes\sigma_{x^{\vee\vee},x^\vee}^{-1}) \circ (i_x\otimes\id_{x^{\vee\vee}}), \quad \forall x\in\cat{C}\]

then

\[\theta_x = \psi_x \circ \delta_x, \quad \forall x\in\cat{C}\]

The graphical notation of $\theta$ can be deduced from the definition of $\psi$, once we identify $x$ and $x^{\vee\vee}$ (i.e. simply ignore $\delta$):

twist

However, if imagining such thing as a 1-dimensional line or string in $\R^3$, then we may think that $\theta=\id$, which is not true in general. Therefore, we may turn to use a 2D object, or a “ribbon”, to correctly represent $\theta$:

TODO:

When trying to straighten it, we will get a ribbon with a “twist”, and clearly it’s not the same as a flat one. With such “ribbon” representation, the identity $\theta_{x\otimes y}=\sigma_{y,x}\circ\sigma_{x,y}\circ(\theta_x\otimes\theta_y)$ can be visualized as:

TODO:

Every ribbon category admits a spherical structure. It can be seen from the following procedure:

TODO:

Summary

At the end of this part, we list of all the “ingredients” that can be added in a tensor category:

Ingredients Meanings
Tensor (monoidal) With tensor product
Braided With braiding
Symmetric Tensor product is symmetric
Rigid (autonomous) With left/right duals
Pivotal (sovereign) Left/right duals coincide
Spherical Left/right traces coincide
Ribbon (tortile) With twist

References