Finite semisimple fusion data for non-invertible defects

A strict $2$-category has objects and, for each ordered pair $(A,B)$, a category ${\mathrm{Hom}}_{\mathcal{C}}(A,B)$ whose objects are $1$-morphisms and whose morphisms are $2$-morphisms. It has identity $1$-morphisms and composition functors ${\mathrm{Hom}}_{\mathcal{C}}(B,C)\times {\mathrm{Hom}}_{\mathcal{C}}(A,B)\to {\mathrm{Hom}}_{\mathcal{C}}(A,C)$ that are strictly associative and unital. Morphisms in a Hom category compose vertically, denoted $\cdot$; the composition functors give horizontal composition, denoted ${\circ }_h$. Their functoriality is the interchange law: for vertically composable $2$-morphisms ${\alpha }_1,{\alpha }_2$ and ${\beta }_1,{\beta }_2$ with the indicated horizontal composites defined,

and horizontal composition preserves identity $2$-morphisms. A strict $k$-linear $2$-category has $k$-vector spaces of $2$-morphisms, bilinear vertical and horizontal composition, and the same strict identity and composition laws. A strict $k$-linear $2$-functor assigns objects and functors on Hom categories, preserving identities and horizontal and vertical compositions as equalities; it is $k$-linear when each Hom functor is $k$-linear.

Fix a field $k$. In the ordinary categorical case, a finite semisimple fusion category over $k$ is a finite semisimple tensor category in the sense of the earlier fusion-and-multifusion definition. Concretely, it is a locally small category (each Hom collection is a set) whose Hom spaces are $k$-vector spaces and whose composition maps are $k$-bilinear. A monomorphism $u:Y\to X$ is a morphism for which precomposition $\mathrm{Hom}(Z,Y)\to \mathrm{Hom}(Z,X)$ is injective for every object $Z$; a subobject of $X$ is an isomorphism class of monomorphisms into $X$. An epimorphism is defined by the dual cancellation condition. The category is abelian: it has finite biproducts, kernels, and cokernels, every monomorphism is a kernel, and every epimorphism is a cokernel. Each Hom space is finite-dimensional, and every object has finite length, meaning a finite composition series with simple quotients. A simple object is nonzero with only the zero subobject and itself as subobjects; every object is a finite direct sum of simple objects, and only finitely many simple isomorphism classes occur. It has a $k$-bilinear tensor bifunctor, a tensor unit $\mathbf{1}$ with $\mathrm{End}(\mathbf{1})\cong k$, and a natural associator ${\alpha }_{X,Y,Z}:(X\otimes Y)\otimes Z\to X\otimes (Y\otimes Z)$ and left and right unitors satisfying the pentagon and triangle axioms.

For an object $X$, a left dual $X^L$ consists of morphisms ${\mathrm{ev}}_L:X^L\otimes X\to \mathbf{1}$ and ${\mathrm{coev}}_L:\mathbf{1}\to X\otimes X^L$ for which $({\mathrm{id}}_X\otimes {\mathrm{ev}}_L)\circ {\alpha }_{X,X^L,X}\circ ({\mathrm{coev}}_L\otimes {\mathrm{id}}_X)={\mathrm{id}}_X,({\mathrm{ev}}_L\otimes {\mathrm{id}}_{X^L})\circ {\alpha }_{X^L,X,X^L}^{-1}\circ ({\mathrm{id}}_{X^L}\otimes {\mathrm{coev}}_L)={\mathrm{id}}_{X^L}\mathrm{.}$ A right dual $X^R$ consists of ${\mathrm{ev}}_R:X\otimes X^R\to \mathbf{1}$ and ${\mathrm{coev}}_R:\mathbf{1}\to X^R\otimes X$ for which $({\mathrm{ev}}_R\otimes {\mathrm{id}}_X)\circ {\alpha }_{X,X^R,X}^{-1}\circ ({\mathrm{id}}_X\otimes {\mathrm{coev}}_R)={\mathrm{id}}_X,({\mathrm{id}}_{X^R}\otimes {\mathrm{ev}}_R)\circ {\alpha }_{X^R,X,X^R}\circ ({\mathrm{coev}}_R\otimes {\mathrm{id}}_{X^R})={\mathrm{id}}_{X^R}\mathrm{.}$ The unitors are suppressed in these four composites. Rigidity means that each object has both kinds of dual.

For the ordinary fusion-category clauses below, write $\mathcal{C}$ for this category.

Choose representatives $i\in I$ of the finitely many simple-object isomorphism classes, with $0\in I$ representing $\mathbf{1}$. For each $i,j,\mathrm{\ell }\in I$, the fusion multiplicity $N_{ij}^{\mathrm{\ell }}\in {\mathbb{Z}}_{\ge 0}$ is the number of copies of $\mathrm{\ell }$ in a semisimple decomposition of $i\otimes j$, so $i\otimes j\cong {⨁}_{\mathrm{\ell }\in I}{N}_{ij}^{\mathrm{\ell }}\mathrm{\ell }$. The associated fusion space is $V_{ij}^{\mathrm{\ell }}:={\mathrm{Hom}}_{\mathcal{C}}(i\otimes j,\mathrm{\ell })$. For $D_{\mathrm{\ell }}:={\mathrm{End}}_{\mathcal{C}}(\mathrm{\ell })$, simplicity makes $D_{\mathrm{\ell }}$ a finite-dimensional division $k$-algebra and $V_{ij}^{\mathrm{\ell }}$ a left $D_{\mathrm{\ell }}$-module by postcomposition; its rank as a left $D_{\mathrm{\ell }}$-module is $N_{ij}^{\mathrm{\ell }}$. When $D_{\mathrm{\ell }}=k$ (in particular for absolutely simple objects), ${\dim }_k{V}_{ij}^{\mathrm{\ell }}=N_{ij}^{\mathrm{\ell }}$.

For simple $i,j,k,\mathrm{\ell }$, the associator induces an isomorphism between the two fusion-channel spaces ${⨁}_{m\in I}{V}_{mk}^{\mathrm{\ell }}{\otimes }_{D_m}{V}_{ij}^m\stackrel{\cong }{\to }{⨁}_{n\in I}{V}_{in}^{\mathrm{\ell }}{\otimes }_{D_n}{V}_{jk}^n\mathrm{.}$ Here $V_{mk}^{\mathrm{\ell }}$ is a right $D_m$-module by $g\cdot d=g\circ (d\otimes {\mathrm{id}}_k)$, $V_{ij}^m$ is a left $D_m$-module by $d\cdot f=d\circ f$, $V_{in}^{\mathrm{\ell }}$ is a right $D_n$-module by $f\cdot d=f\circ ({\mathrm{id}}_i\otimes d)$, and $V_{jk}^n$ is a left $D_n$-module by postcomposition. The isomorphism is defined by the semisimple decomposition identifications with ${\mathrm{Hom}}_{\mathcal{C}}((i\otimes j)\otimes k,\mathrm{\ell })$ and ${\mathrm{Hom}}_{\mathcal{C}}(i\otimes (j\otimes k),\mathrm{\ell })$, and the middle map sends $h$ to $h\circ {\alpha }_{i,j,k}^{-1}$. Thus its domain and codomain are the stated fusion-channel spaces, not unspecified multiplicity symbols. In chosen decompositions and bases these are the associator matrices; their coherence is the pentagon. The fusion datum includes the actual associator, unitors, and both duality structures, not just the integers $N_{ij}^{\mathrm{\ell }}$; the unitors satisfy the triangle and the dual maps satisfy the four displayed snake identities.

A tensor-invertible object $D$ is an object for which some object $E$ satisfies $D\otimes E\cong \mathbf{1}$ and $E\otimes D\cong \mathbf{1}$. An object with no such two-sided inverse is non-invertible. Simplicity and tensor-invertibility are separate properties: an object can be simple and non-invertible, or a direct sum of invertible simple objects and non-invertible as a whole.

For the finite semisimple fusion 2-category convention, assume that $k$ is algebraically closed of characteristic zero. A strict $k$-linear 2-category $\mathcal{C}$ has objects $A,B,\dots$, a $k$-linear category ${\mathrm{Hom}}_{\mathcal{C}}(A,B)$ for each ordered pair, 1-morphisms as its objects, and 2-morphisms as its morphisms; horizontal composition is $k$-bilinear and strictly associative and unital, and satisfies interchange. A $k$-linear category is one whose morphism spaces are $k$-vector spaces and whose composition is $k$-bilinear. A finite semisimple $k$-linear category is an abelian $k$-linear category with finite-dimensional Hom spaces, finitely many isomorphism classes of simple objects, and every object a finite direct sum of simple objects; in each such category a simple object is nonzero with no nonzero proper subobject, using the ordinary categorical subobject convention above.

A functor between categories is fully faithful when it induces a bijection on morphism sets for each pair of objects. For 1-morphisms $p:A\to B$ and $q:B\to A$, an adjunction $p⊣q$ consists of 2-morphisms (unit) $\eta :{\mathrm{id}}_A\Rightarrow q\circ p$ and (counit) $ϵ:p\circ q\Rightarrow {\mathrm{id}}_B$ satisfying $(ϵ\star p)\cdot (p\star \eta )={\mathrm{id}}_p$ and $(q\star ϵ)\cdot (\eta \star q)={\mathrm{id}}_q$, where $\star$ denotes whiskering by 1-morphisms and $\cdot$ denotes vertical composition of 2-morphisms. A 1-morphism $f:A\to B$ is adjointable when there are 1-morphisms $l,r:B\to A$ with $l⊣f⊣r$, with each adjunction carrying the unit, counit, and triangle identities just specified.

A zero object $0$ is an object for which each Hom category ${\mathrm{Hom}}_{\mathcal{C}}(X,0)$ and ${\mathrm{Hom}}_{\mathcal{C}}(0,X)$ is equivalent to the zero category, the category with one object and one morphism. A zero 1-morphism is a zero object in its Hom category. A 1-morphism $f:A\to B$ is fully faithful when for every object $X$, postcomposition with $f$ gives a fully faithful functor ${\mathrm{Hom}}_{\mathcal{C}}(X,A)\to {\mathrm{Hom}}_{\mathcal{C}}(X,B)$. A subobject of $B$ is an equivalence class of fully faithful 1-morphisms $f:A\to B$, where $f:A\to B$ and $g:A^{\mathrm{\prime }}\to B$ represent the same class exactly when there is an equivalence $u:A\to A^{\mathrm{\prime }}$ and an invertible 2-morphism $g\circ u\Rightarrow f$. A zero subobject is represented by a fully faithful 1-morphism whose domain is equivalent to a zero object. A 1-morphism $f:A\to B$ is an equivalence if there is a 1-morphism $g:B\to A$ and invertible 2-morphisms $g\circ f\Rightarrow {\mathrm{id}}_A$ and $f\circ g\Rightarrow {\mathrm{id}}_B$; two objects are equivalent when there is such an equivalence between them. A nonzero object (one not equivalent to a zero object) is simple when all its subobjects are represented by either a zero-domain 1-morphism or an equivalence.

An additive category is a category with a zero object and finite biproducts. A 2-category is additive when each Hom category is additive in this sense, it has a zero object, and finite direct sums of objects exist. For $A_1,A_2$, a direct sum $A_1\oplus A_2$ has inclusions ${\iota }_i:A_i\to A_1\oplus A_2$ and projections ${\rho }_i:A_1\oplus A_2\to A_i$ with ${\rho }_i{\iota }_i\cong {\mathrm{id}}_{A_i}$, ${\rho }_i{\iota }_j$ a zero 1-morphism for $i\mathrm{\ne }j$, and ${\mathrm{id}}_{A_1\oplus A_2}\cong {\iota }_1{\rho }_1\oplus {\iota }_2{\rho }_2$ in its endomorphism category.

A monad on $B$ is a 1-morphism $E:B\to B$ with unit $\eta :{\mathrm{id}}_B\Rightarrow E$ and multiplication $m:E\circ E\Rightarrow E$ satisfying

It is separable when there is an $E$-bimodule 2-morphism $c:E\Rightarrow E\circ E$ such that

The first equation says that $c$ respects the left and right $E$-actions, and the second says that it is a section of multiplication. An isomorphism of monads $\tau :E\Rightarrow F$ is an invertible 2-morphism respecting the units and multiplications: $\tau \cdot {\eta }_E={\eta }_F$ and $\tau \cdot m_E=m_F\cdot (F\star \tau )\cdot (\tau \star E)$.

A separable splitting of $E:B\to B$ consists of an object $A$, an adjunction $\rho ⊣\iota$ with $\rho :B\to A$, $\iota :A\to B$, unit $\theta :{\mathrm{id}}_B\Rightarrow \iota \circ \rho$, and counit $ϵ:\rho \circ \iota \Rightarrow {\mathrm{id}}_A$, a 2-morphism section $s:{\mathrm{id}}_A\Rightarrow \rho \circ \iota$ with $ϵ\cdot s={\mathrm{id}}_{{\mathrm{id}}_A}$, and an isomorphism of monads $E\cong \iota \circ \rho$, where $\iota \circ \rho$ has unit $\theta$ and multiplication $\iota \star ϵ\star \rho$. A 2-category is finite semisimple when its Hom categories are finite semisimple, every 1-morphism is adjointable, it is additive, every separable monad has a separable splitting, and it has finitely many equivalence classes of simple objects.

Use the following semistrict monoidal 2-category convention (a Gray monoid). It adds a unit object $I$, strict $k$-linear 2-functors $L_A=A\otimes -$ and $R_A=-\otimes A$ for every object $A$, and for every pair of 1-morphisms $f:A\to A^{\mathrm{\prime }}$ and $g:B\to B^{\mathrm{\prime }}$ an invertible interchanger ${\varphi }_{f,g}:(f\otimes B^{\mathrm{\prime }})\circ (A\otimes g)\Rightarrow (A^{\mathrm{\prime }}\otimes g)\circ (f\otimes B)\mathrm{.}$ Write $A\otimes B=L_A(B)=R_B(A)$. The tensor is strictly unital and associative: $L_I=R_I=\mathrm{id}$, $L_A{L}_B=L_{A\otimes B}$, $R_B{R}_A=R_{A\otimes B}$, and $L_A{R}_B=R_B{L}_A$. The interchanger is the identity when either 1-morphism is an identity, is natural in both 2-morphisms, is compatible with composition in each 1-morphism variable, and is compatible with tensoring by an object in either variable. These are coherence axioms on the data, not consequences of a fusion table.

An object $A^R$ is a right dual of $A$ when there are evaluation and coevaluation 1-morphisms $e_R:A\otimes A^R\to I$ and $i_R:I\to A^R\otimes A$ and invertible 2-morphisms exhibiting $(e_R\otimes A)\circ (A\otimes i_R)\cong {\mathrm{id}}_A$ and $(A^R\otimes e_R)\circ (i_R\otimes A^R)\cong {\mathrm{id}}_{A^R}$. A left dual $A^L$ has evaluation $e_L:A^L\otimes A\to I$ and coevaluation $i_L:I\to A\otimes A^L$, with invertible 2-morphisms exhibiting $(A\otimes e_L)\circ (i_L\otimes A)\cong {\mathrm{id}}_A$ and $(e_L\otimes A^L)\circ (A^L\otimes i_L)\cong {\mathrm{id}}_{A^L}$. A fusion 2-category in this convention is a finite semisimple monoidal 2-category in which every object has left and right duals and the monoidal unit is simple.

Neither finite semisimplicity nor either categorical definition asserts unitarity: a dagger structure, positivity, and any unitary coherence are additional hypotheses and data. These are abstract mathematical structures. They do not construct defect operators or junctions in a quantum field theory, specify spacetime support, or prove topological invariance; a physical realization requires a separately specified model and its own construction and proof.

The ordinary associator pentagon below displays the two composites that the axiom requires to agree. This diagram records that axiom; no proof or example verification relies on the drawing.

Fusion and multifusion categories

Monoidal category

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§2.1–2.2, 2.10–2.11, 4.1, and 4.16

Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §§1.2.2–1.4.2 and 2.1.1–2.1.6

For the rank assertion in the Definition, choose a semisimple decomposition $i\otimes j\cong {⨁}_{r\in I}{r}^{\oplus N_{ij}^r}$. A nonzero morphism $f:r\to \mathrm{\ell }$ between simple objects has zero kernel, since a nonzero kernel would have to be all of $r$, and has image all of $\mathrm{\ell }$, since the image is a nonzero subobject of the simple object $\mathrm{\ell }$. Thus $f$ is both monic and epic, hence an isomorphism in the abelian category. Therefore $\mathrm{Hom}(r,\mathrm{\ell })=0$ for $r\mathrm{\ne }\mathrm{\ell }$, while $\mathrm{Hom}(\mathrm{\ell },\mathrm{\ell })=D_{\mathrm{\ell }}$ and every nonzero element of $D_{\mathrm{\ell }}$ is invertible. Applying ${\mathrm{Hom}}_{\mathcal{C}}(-,\mathrm{\ell })$ to the finite direct sum gives $V_{ij}^{\mathrm{\ell }}\cong D_{\mathrm{\ell }}^{\oplus N_{ij}^{\mathrm{\ell }}}$ as a left $D_{\mathrm{\ell }}$-module by postcomposition. Its rank is therefore $N_{ij}^{\mathrm{\ell }}$.