In this post I will follow mostly section 3 of Liang Kong and Zhi-Hao Zhang's review paper “An invitation to topological orders and category theory ”. In particular, I will introduce another type of enriched category called \(\mathbb{C}-\)linear categories, as well as additional necessary concepts, such as semisimplic ity, simplicity and the main idea behind fusion categories, modular tensor categories and finally unitary modular tensor categories, which are the key concept behind topological order.
\(\mathbb{C-}\)linear categories
Definition ( \(\mathbb{C-}\)linear category)
A \(\mathbb{C-}\)linear category \(\boldsymbol{C}\) is a category enriched over the category of finite dimensional vector spaces over \(\mathbb{C}\). In other words, it is a category in which each Hom-set is equipped with the structure of a finite-dimensional vector space over \(\mathbb{C}\), such that the composition of morphisms is \(\mathbb{C-}\)bilinear.
Definition (Direct sum in a \(\mathbb{C-}\)linear category)
A direct sum in a \(\mathbb{C-}\)linear category \(\boldsymbol{C}\) is a biproduct in \(\boldsymbol{C}\). In other words, given objects \(A_{1},\dots,A_{n}\in\text{ob}(\boldsymbol{C})\), a direct sum is an object \(x\in\text{ob}(\text{\ensuremath{\mathcal{C}}})\) equipped with morphisms (injections) \(\iota_{i}:A_{i}\to A\) and (projections) \(\pi_{i}:A\to A_{i}\) , \(\forall i\) such that
\[\begin{align} \pi_{i}\circ\iota_{j} & =\delta_{ij}\text{id}_{x},\\ \sum_{j=1}^{n}\iota_{j}\circ\pi_{j} & =\text{id}_{x}. \end{align}\]
Observation
Since a direct sum is a biproduct, it is a universal construction, i.e. it is unique up to a unique isomorphism. Thus, we can talk about THE direct sum of objects.
Lemma
Let \(\mathcal{C}\) be a \(\mathbb{C}-\)linear category, and let \(0\) be its zero object. Then \(x\oplus0\simeq x\).
Lemma (Matrix representation)
A morphism \(f:A_{1}\oplus\cdots\oplus A_{n}\to B_{1}\oplus\cdots B_{m}\) can be represented as an \(m\times n\) matrix of morphisms \(f_{\mu}:A_{i}\to B_{j}\) as described in Post 2 on the reconstruction of linear algebra from categories.
Definition (Direct sum of \(\mathbb{C-}\)linear categories)
Let \(\boldsymbol{C},\boldsymbol{D}\) be \(\mathbb{C}-\)linear categories. Their direct sum, denoted by \(\boldsymbol{C}\oplus\boldsymbol{D}\) is the \(\mathbb{C}-\)linear category corresponding to the cartesian product \(\boldsymbol{C}\times\boldsymbol{D}\) with the \(\mathbb{C}-\)linear structure of the Hom-set \(\text{Hom}_{\boldsymbol{C}\oplus\boldsymbol{D}}((A,B),(A',B'))=\text{Hom}_{\boldsymbol{C}}((A,B),(A',B'))\oplus\text{Hom}_{\boldsymbol{D}}((A,B),(A',B'))\), where on the right \(\oplus\) stands for the usual direct sum of vector spaces.
Definition ( \(\mathbb{C-}\)linear functor)
A \(\mathbb{C-}\)linear functor between \(\mathbb{C}-\)linear categories \(\boldsymbol{C},\boldsymbol{D}\) is a functor \(F:\boldsymbol{C}\to\boldsymbol{D}\) such that \(F_{A,B}:\text{Hom}_{\boldsymbol{C}}(A,B)\to\text{Hom}_{\boldsymbol{D}}(F(A),F(B))\) is a \(\mathbb{C}-\)linear map for every \(x\) and \(y\).
Definition (Simple object)
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear category. An object \(A\in\text{ob}(\boldsymbol{C})\) is called simple if \(\text{Hom}_{\boldsymbol{C}}(A,A)\simeq\mathbb{C}\).
Definition (Disjoint objects)
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear category. Two objects \(A,B\in\text{ob}(\boldsymbol{C})\) are called disjoint if their Hom-set is the zero vecto space \(\text{Hom}_{\boldsymbol{C}}(A,B)=0=\text{Hom}_{\boldsymbol{C}}(B,A)\).
Definition (Semisimple category)
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear category. It is semisimple if
- The direct sum of finitely many objects exists in \(\boldsymbol{C}\)
- There exists a collection of mutually disjoint simple objects \(\left\{ A_{i}\right\} _{i\in I}\) such that every object in \(\text{ob}(\boldsymbol{C})\) is a finite direct sum of objects in \(\left\{ A_{i}\right\} _{i\in I}\)
Definition (Finite semisimple category)
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear category. It is finite semisimple if it is semisimple and the set \(I\) of “basis object ” indices is finite.
Definition (Indecomposable object)
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear category. An object \(A\in\text{ob}(\boldsymbol{C})\) is called indecomposable if any decomposition \(A=A_{1}\oplus A_{2}\) is trival, i.e. \(A_{1}=0\) or \(A_{2}=0\).
Observation
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear category. All simple objects are indecomposable, but not all indecomposable objects are simple.
Observation
When \(\boldsymbol{C}\) is a semisimple category, an object \(x\in\text{ob}(\text{\ensuremath{\boldsymbol{C}}})\) is indecomposable if and only if it is simple.
Theorem
A \(\mathbb{C}-\)linear category \(\boldsymbol{C}\) is finite semisimple if and only if there are finitely many simple objects in \(\boldsymbol{C}\).
Proof
Let \(\left\{ x_{i}\right\} _{i\in I}\) be a collection of mutually disjoint simple objects such that every object in \(\boldsymbol{C}\) is a direct sum of objects in \(\left\{ x_{i}\right\} _{i\in I}\). By the matrix representation lemma, every morphism in \(\boldsymbol{C}\) can be written as a block-diagonal matrix with coefficients in \(\mathbb{C}\). More precisely
\[\begin{equation} \text{Hom}_{\mathcal{C}}\left(\bigoplus_{i\in I}A_{i}^{\oplus n_{i}},\bigoplus_{j\in I}A_{j}^{\oplus m_{j}}\right)\simeq M_{m_{i}\times n_{i}}\left(\mathbb{C}\right), \end{equation}\] where \(M_{m\times n}\left(\mathbb{C}\right)\) represents the space of \(m\times n\) matrices in \(\mathbb{C}\). Thus, every simple object in \(\text{ob}(\boldsymbol{C})\) is isomorphic to exactly one object in \(\left\{ x_{i}\right\} _{i\in I}\). In other words, each of the isomorphism classes of simple objects in \(\boldsymbol{C}\) contains exactly one object in \(\left\{ A_{i}\right\} _{i\in I}\), which proves the theorem.
Definition (Set of irreps of a category)
The set of irreps of a semisimple \(\boldsymbol{C}\) is defined as the set of isomorphism classes of simple objects in \(\boldsymbol{C}\), and is denoted by \(\text{Irr}(\boldsymbol{C})\).
Theorem
Any finite semisimple category \(\boldsymbol{C}\) is equivalent to the direct sum \(\text{\textbf{Vec}}^{\oplus n}\) of \(n\) copies of \(\text{\textbf{Vec}}\), where \(n\) is the number of isomorphism classes of simple objects of \(\boldsymbol{C}\), or rather \(n=|\text{Irr}(\boldsymbol{C})|\).
Observation
The notation of irreps follows from Maschke's theorem, which states that if \(G\) is a finite group, then \(\text{\textbf{Rep}}(G)\) is a finite semisimple category. A simple object in \(\textbf{Rep}(G)\) is also called an irreducible \(G-\)representation.
Definition (Interaction with monoidal structure)
A \(\mathbb{C}-\)linear monoidal category \(\boldsymbol{C}\) is both a \(\mathbb{C}-\)linear category and a monoidal category such that the tensor product functor \(\otimes:\boldsymbol{C}\times\boldsymbol{C}\to\boldsymbol{C}\) is \(\mathbb{C}-\)bilinear, i.e., \(\forall y\) both \(f"=A\otimes\text{\ensuremath{\sim}}":\boldsymbol{C}\to\boldsymbol{C}\) and \(g"=\sim\otimes A":\boldsymbol{C}\to\boldsymbol{C}\) are \(\mathbb{C}-\)bilinear functors for each \(A\in\text{ob}(\boldsymbol{C})\).
Definition (Interaction with unitary structure)
A dagger structure on a \(\mathbb{C}-\)linear monoidal category \(\boldsymbol{C}\) is compatible with the monoidal structure if
- \(f^{\dagger}\otimes g^{\dagger}=\left(f\otimes g\right)^{\dagger}\) for all morphisms \(f,g\) in \(\boldsymbol{C}\)
- The morphisms \(\alpha_{A,B,C},\lambda_{A},\rho_{A}\) are all unitary for all \(A,B,C\in\text{ob}(\boldsymbol{C})\)
A unitary monoidal ( \(\mathbb{C}-\)linear) category is a \(\mathbb{C}-\)linear category equipped with a compatible unitary structure.
Theorem (Interaction with duals)
Let \(\boldsymbol{C}\) be a \(\mathbb{C}-\)linear rigid monoidal category. Taking duals defines a \(\mathbb{C}-\)linear functor \(\delta^{L}:\boldsymbol{C}^{\text{op}}\to\boldsymbol{C}\). In particular, taking left duals preserves direct sums, i.e. there exists a canonical isomorphism \(\left(A_{1}\oplus\cdots\oplus A_{n}\right)^{*}\simeq A_{1}^{*}\oplus\cdots\oplus A_{n}^{*}\) for all \(A_{1},\dots,A_{n}\in\text{ob}(\boldsymbol{C})\).
Proof
Fusion categories
Definition (Multi-fusion category)
A multi-fusion category is a \(\mathbb{C}-\)linear rigid monoidal finite semisimple category.
Definition (Fusion category)
A fusion category is a multi-fusion category such that the monoidal unit \(1\) is a simple object.
Definition (Fusion rules)
Let \(\boldsymbol{C}\) be a multi-fusion category. Since \(\boldsymbol{C}\) is semi-simple, the tensor product of two simple objects \(x,y\in\text{ob}(\boldsymbol{C})\) is the direct sum of simple objects. Thus, we have
\[\begin{equation} x\otimes y\simeq\bigoplus_{z\in\text{Irr}\left(\boldsymbol{C}\right)}N_{xy}^{z}z, \end{equation}\] for non-negative integers \(N_{xy}^{z}\in\mathbb{N}^{0}\), where \(nz\) denotes the direct sum of \(n\) copies of \(z\). The set \(\left\{ N_{xy}^{z}\right\} _{xy\in\text{Irr}\left(\boldsymbol{C}\right)}\) is called the set of fusion rules of \(\boldsymbol{C}\).
Definition (Grothendick ring)
From the existence of fusion rules, its clear that the isomorphism classes of simple objects of a multi-fusion category \(\boldsymbol{C}\) generate a ring, where the multiplication is given by the tensor product of \(\boldsymbol{C}\). The structure constants of the ring are the fusion rules of \(\boldsymbol{C}\). This ring is called the fusion ring, or Grothendieck ring of \(\boldsymbol{C}\), denoted by \(\text{Gr}\left(\boldsymbol{C}\right)\).
Definition (Unitary fusion category)
A unitary fusion category is a fusion category equipped with a compatible unitary structure.
Definition (Quantum dimension of objects in a unitary fusion category)
Let \(\boldsymbol{C}\) be a unitary fusion category and let \(A\in\text{ob}(\boldsymbol{C})\) be an object in the category. The quantum dimension of \(A\) is
\[\begin{equation} \dim\left(A\right)=\sqrt{\left(\eta_{A}^{\dagger}\circ\eta_{A}\right)\left(\varepsilon_{A}\circ\varepsilon_{A}^{\dagger}\right)}. \end{equation}\]
Definition (Quantum dimension of a unitary fusion category)
Let \(\boldsymbol{C}\) be a unitary fusion category. The quantum dimension of \(\boldsymbol{C}\) is defined as
\[\begin{equation} \dim\left(\boldsymbol{C}\right)=\sum_{x\in\text{Irr}\left(\boldsymbol{C}\right)}\dim\left(x\right)^{2}. \end{equation}\]
Observation
The morphisms \(\varepsilon_{A}\) and \(\eta_{A}\) can always be rescalled such that
\[\begin{equation} \eta_{A}^{\dagger}\circ\eta_{A}=\varepsilon_{A}\circ\varepsilon_{A}^{\dagger}=\dim(A). \end{equation}\]
Definition (Normalized duals)
A choice of duals in a unitary fusion category \(\boldsymbol{C}\) is normalized if the above equation holds for every object in \(\boldsymbol{C}\).
Observation
With a normalized choice of duals, the left and right dual functors are equal.
Definition (Pivotal fusion category)
A pivotal fusion category is a fusion category equipped with a pivotal structure.
Lemma
Let \(\boldsymbol{C}\) be a unitary fusion category. For every \(A,B\in\text{ob}\left(\boldsymbol{C}\right)\), we have
\[\begin{align} \dim\left(A\oplus B\right) & =\dim\left(A\right)+\dim\left(B\right),\\ \dim\left(A\otimes B\right) & =\dim\left(A\right)\dim\left(B\right), \end{align}\] i.e. \(\dim\) defines a ring homomorphism \(\text{Gr}\left(\boldsymbol{C}\right)\to\mathbb{R}\).
Theorem
From the previous lemma, follows
\[\begin{equation} \dim\left(x\right)\dim\left(y\right)=\sum_{z\in\text{Irr}\left(\boldsymbol{C}\right)}N_{xy}^{z}\dim\left(z\right), \end{equation}\] for every simple object \(x,y\in\text{ob}\left(\boldsymbol{C}\right)\).
Observation
We have previously seen that a pivotal structure on a rigid monoidal category \(\boldsymbol{C}\) is a monoidal natural isomorphism \(\pi_{A}:A\to A^{**}\). In a pivotal fusion category we can similarly close the world-line of an object \(A\), however there are two different ways to close the world line which may not be equal.
Definition (Left and Right quantum dimensions)
The left and right quantum dimension of an object in a pivotal fusion category \(\boldsymbol{C}\) are defined as
\[\begin{equation} \dim^{L}(A)=\varepsilon_{A^{*}}\circ\left(\text{id}_{A}\otimes\pi_{A}\right)\circ\eta_{A}, \end{equation}\]
\[\begin{equation} \dim^{R}(A)=\varepsilon_{A}\circ\left(\pi_{A}^{-1}\otimes\text{id}_{A}\right)\circ\eta_{A^{*}}. \end{equation}\]
Observation
In post 5 we showed that in a dagger pivotal category, the pivotal structure is determined by the dagger duals. This is the same for a unitary fusion category, which admits a unique pivotal structure determined by the normalized choice of duals. This pivotal structure is automatically spherical and compatible with the unitary structure.
Lemma
The quantum dimension of a fusion category is independent of the choice of pivotal or spherical structures, but the quantum dimensions of objects depend on the choice of pivotal or spherical structures.
Definition (Frobenius-Perron dimension)
Let \(\boldsymbol{C}\) be a fusion category with fusion rules \(N_{xy}^{z}\). We can interpret these as a matrix with elements \(\left[N_{x}\right]_{zy}\) of non-negative elements. By the Frobenius-Perron theorem, it admits a largest non-negative eigenvalue, which we call the Frobenius-Perron dimension of \(x\). The Frobenius-Perron dimension of \(\boldsymbol{C}\) is
\[\begin{equation} \text{FPdim}(\boldsymbol{C})=\sum_{x\in\text{Irr}\left(\boldsymbol{C}\right)}\text{FPdim}(x)^{2}. \end{equation}\]
Definition (Pseudo-unitary category)
A fusion category \(\boldsymbol{C}\) is called pseudo-unitary if
\[\begin{equation} \dim\left(\boldsymbol{C}\right)=\text{FPdim}(\boldsymbol{C}). \end{equation}\]
Definition (Unitary braided monoidal category)
A unitary braided monoidal category is \(\mathbb{C-}\)linear braided monoidal category equipped with a compatible unitary structure.