Higher Categorical Symmetries and Defect Algebras

Let $\mathcal{C}$ be a fusion category over $k$. A finite semisimple right $\mathcal{C}$-module category consists of a $k$-linear abelian category $\mathcal{M}$, semisimple in the sense of Semisimple objects and semisimple abelian categories, with finitely many isomorphism classes of simple objects and finite-dimensional $k$-vector spaces ${\mathrm{Hom}}_{\mathcal{M}}(m,n)$ for all $m,n\in \mathcal{M}$, together with a bilinear action functor $\odot :\mathcal{M}\times \mathcal{C}\to \mathcal{M}$ and natural isomorphisms ${\alpha }_{m,X,Y}:(m\odot X)\odot Y⟶m\odot (X\otimes Y),{\rho }_m:m\odot \mathbf{1}⟶m,$ which satisfy the module pentagon and triangle equations obtained from the associator and unitors of $\mathcal{C}$. The left-action convention is defined by reversing the order of the action factors.

A module functor $F:\mathcal{M}\to \mathcal{N}$ is a linear functor with natural action comparisons $F(m\odot X)\cong F(m)\odot X$ satisfying the module associativity and unit equations. A module-natural transformation is a natural transformation commuting with those comparisons. These objects, functors, and transformations form a 2-category denoted $\mathrm{Mod}(\mathcal{C})$ with ordinary composition of module functors and horizontal and vertical composition of module-natural transformations. The action comparison for a composite is the composite of the two action comparisons; identity functors have identity comparisons.

This definition alone does not assert that there are finitely many equivalence classes of simple module categories, nor that $\mathrm{Mod}(\mathcal{C})$ is a fusion 2-category.

Fix an algebraically closed field $k$ of characteristic zero and a finite group $G$ (Group and abelian group). A finite semisimple category is understood in the sense of Semisimple objects and semisimple abelian categories, with simple objects as in Simple object. A finite semisimple $G$-graded $k$-linear category is a finite semisimple $k$-linear category $\mathcal{A}$ together with a direct-sum decomposition $\mathcal{A}={⨁}_{g\in G}{\mathcal{A}}_g\mathrm{.}$ An object in ${\mathcal{A}}_g$ is homogeneous of degree $g$. A grading-preserving linear functor sends each ${\mathcal{A}}_g$ into the corresponding degree-$g$ summand, and its 2-morphisms are natural transformations. Such categories, functors, and natural transformations form the underlying 2-category denoted $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G$.

For finite semisimple categories $\mathcal{A},\mathcal{B}$ with simple-label sets $S,T$, their finite semisimple tensor product $\mathcal{A}⊠\mathcal{B}$ has simple labels $S\times T$ and $\mathrm{Hom}(s⊠t,s^{\mathrm{\prime }}⊠t^{\mathrm{\prime }})={\mathrm{Hom}}_{\mathcal{A}}(s,s^{\mathrm{\prime }}){\otimes }_k{\mathrm{Hom}}_{\mathcal{B}}(t,t^{\mathrm{\prime }}),$ extended to direct sums. For a concrete skeleton, choose finite sets $S_g$ of simple labels and replace ${\mathcal{A}}_g$ by the category of finite-dimensional $k$-vector-space families indexed by $S_g$. A linear functor is represented by a finite matrix of finite-dimensional vector spaces within each grade. This skeleton fixes the meaning of “finite” and the homogeneous components; it does not assert any physical realization.

Suppose a modular ribbon fusion category $\mathcal{C}$ satisfies the hypotheses of The weighted Reshetikhin–Turaev TQFT and modular-functor factorization, including its weighted-Lagrangian bordism convention and proved state-space, composition, and factorization maps. The weighted RT background interface for the present page means using those TQFT state spaces and bordism maps as a separate 2+1-dimensional mathematical background alongside the algebraic defect 2-category $\mathrm{Mod}(\mathcal{C})$. The interface asserts no canonical functor from a module category, module functor, or module-natural transformation to a physical defect, and no classification of 3+1-dimensional phases.

The actual supplier The weighted Reshetikhin–Turaev TQFT and modular-functor factorization now proves the full marked weighted-Lagrangian RT functor and geometric factorization, including presentation/coupon independence, exact cap normalization, signature–Maslov composition, cylinders, duality, disconnected and empty cases, and both sewing inverses. Its weight action is ${\kappa }^{+n}$ for the prescribed $+{\mu }_W$ composition, with the actual cap-reference correction for arbitrary real object Lagrangians. This establishes the background used in the Definition for every category satisfying the stated modular hypotheses. Full AC enters through the boundary duality/topology suppliers and countable choice through framed coupon evaluation. The algebraic $\mathrm{Mod}(\mathcal{C})$ construction remains independently defined; no canonical physical realization or phase classification is inferred.

In a monoidal category $\mathcal{C}$, an algebra object is an object $A$ with multiplication $m_A:A\otimes A\to A$ and unit ${\eta }_A:\mathbf{1}\to A$ satisfying associativity and unit equations with the associator and unitors of $\mathcal{C}$. It is separable if there is an $A$-bimodule morphism ${\sigma }_A:A\to A\otimes A$ with $m_A{\sigma }_A=1_A$. The composite $e_A={\sigma }_A{\eta }_A:\mathbf{1}\to A\otimes A$ is its separability element. It satisfies $m_A{e}_A={\eta }_A$ and the centrality equation $(m_A\otimes 1_A)\circ (1_A\otimes e_A)=(1_A\otimes m_A)\circ (e_A\otimes 1_A):A⟶A\otimes A,$ with associators inserted. In element notation these are ${\sum }_i{x}_i{y}_i=1_A$ and ${\sum }_{i}a{x}_i\otimes y_i={\sum }_i{x}_i\otimes y_{i}a$.

A left $A$-module is an object $M\in \mathcal{C}$ with an associative unital action $l_M:A\otimes M\to M$; a right $A$-module has an associative unital action $r_M:M\otimes A\to M$. Write ${\mathrm{Mod}}_{\mathcal{C}}^l(A)$ and ${\mathrm{Mod}}_{\mathcal{C}}^r(A)$ for their categories and module morphisms. The category of left $A$-modules is a right $\mathcal{C}$-module category by $M\odot X:=M\otimes X$, with left $A$-action obtained by associating $A\otimes (M\otimes X)$ to $(A\otimes M)\otimes X$ and applying $l_M\otimes 1_X$, and the inherited associator and unitor. An $A$-$B$ bimodule is an object $X\in \mathcal{C}$ with commuting left and right actions $A\otimes X\to X$ and $X\otimes B\to X$, each associative and unital. A bimodule morphism preserves both actions.

The regular $A$-$A$ bimodule is the object $A$ with both actions given by $m_A$.

Given an $A$-$B$ bimodule $X$ and a $B$-$D$ bimodule $Y$, a morphism $q:X\otimes Y\to Z$ is balanced over $B$ if $q\circ (r_X\otimes 1_Y)=q\circ (1_X\otimes l_Y):X\otimes B\otimes Y⟶Z\mathrm{.}$ A relative tensor product $X{\otimes }_{B}Y$, when it exists, is an object with a balanced map $q_B:X\otimes Y\to X{\otimes }_{B}Y$ universal among balanced maps. Existence is not part of this definition.

Fix a field $k$. A weak linear 2-category $\mathcal{B}$ consists of a class of objects and, for each ordered pair $(A,B)$, a $k$-linear category $\mathcal{B}(A,B)$. Its objects are called 1-morphisms and its morphisms are called 2-morphisms. Each triple $A,B,C$ has a $k$-bilinear composition functor $\mathcal{B}(B,C)\times \mathcal{B}(A,B)⟶\mathcal{B}(A,C),(g,f)⟼g\circ f,$ and each $A$ has an identity 1-morphism $1_A\in \mathcal{B}(A,A)$. A $k$-linear category here means that every hom set is a $k$-vector space and composition is bilinear.

For composable $f:A\to B$, $g:B\to C$, and $h:C\to D$, there is a natural invertible 2-morphism $a_{h,g,f}:(h\circ g)\circ f⟹h\circ (g\circ f)\mathrm{.}$ For each $f:A\to B$ there are natural invertible unit 2-morphisms ${\mathrm{\ell }}_f:1_B\circ f⟹f,r_f:f\circ 1_A⟹f\mathrm{.}$ The associators satisfy the pentagon equation on every fourfold composite; the two unitors and associator satisfy the triangle equation on every pair of composable 1-morphisms. These equations are equalities of 2-morphisms between the indicated parallel 1-morphisms. Vertical composition is composition in each hom category. Horizontal composition is the displayed functor, so functoriality gives the interchange equation between horizontal and vertical composition.

When the associators and unitors are identities, this is a strict 2-category in the sense of Strict 2-category. No strictness is imposed in the weak definition.

The source and target of every 2-morphism lie in one hom category. Associators change only the parentheses of a composable string; they do not change its endpoints. No monoidal product on the objects is part of this definition.

Let $\mathcal{C}$ be a fusion category over an algebraically closed field. Write $\mathcal{D}=\mathcal{C}⊠{\mathcal{C}}^{\mathrm{r}\mathrm{e}\mathrm{v}}$, acting on $\mathcal{C}$ by $(X⊠Y)\cdot V=X\otimes V\otimes Y$. Choose duals with ${\mathrm{ev}}_i:i\otimes i^{\ast }\to \mathbf{1}$ and ${\mathrm{coev}}_i:\mathbf{1}\to i^{\ast }\otimes i$. The canonical algebra $K={\underset{}{\mathrm{End}}}_{\mathcal{D}}(\mathbf{1})$ is ${⨁}_{i}i⊠i^{\ast }$, with multiplication induced by composition of evaluations and unit the inclusion of the unit summand. Projection $\lambda :K\to \mathbf{1}⊠\mathbf{1}$ to that summand makes $\beta =\lambda \mu$ a nondegenerate pairing. Its inverse $c:\mathbf{1}\to K\otimes K$ defines a Frobenius comultiplication $\mathrm{\Delta }=(\mu \otimes \mathrm{i}\mathrm{d})(\mathrm{i}\mathrm{d}\otimes c)=(\mathrm{i}\mathrm{d}\otimes \mu )(c\otimes \mathrm{i}\mathrm{d})\mathrm{.}$ This comultiplication is coassociative, is a $K$-bimodule morphism, and has counit $\lambda$, with $\lambda \eta =1$. No nonvanishing dimension or pivotal structure is assumed.

Assume the Axiom of Choice. Let $k$ be algebraically closed of characteristic zero, let $G$ be a finite product of general linear groups, and let $X$ be a reduced affine variety of finite type with an algebraic $G$-action. If the differential of $f_x:G\to X$, $g\mapsto gx$, is surjective onto $T_{x}X$ at the identity for every $x\in X$, then every orbit is open and there are finitely many orbits. Empty products and the empty variety are allowed.

Let $A$ be an algebra object in a $k$-linear monoidal category and $B$ an $A$-bimodule. Suppose $e:\mathbf{1}\to A\otimes A$ satisfies $\mu e=\eta ,(\mu \otimes \mathrm{i}\mathrm{d})(\mathrm{i}\mathrm{d}\otimes e)=(\mathrm{i}\mathrm{d}\otimes \mu )(e\otimes \mathrm{i}\mathrm{d})\mathrm{.}$ Set $C^n=\mathrm{Hom}(A^{\otimes n},B)$, with the Hochschild differential given by the alternating sum of the left action, adjacent multiplications, and the right action. Its normalized subcomplex consists of maps vanishing whenever any input is the unit $\eta$. For every $n\ge 1$, define $h_n(f)={\lambda }_B({\mathrm{i}\mathrm{d}}_A\otimes f)(e\otimes {\mathrm{i}\mathrm{d}}_{A^{\otimes (n-1)}}),$ where ${\lambda }_B$ denotes the left $A$-action. Then $h$ preserves the normalized subcomplex and $dh+hd=\mathrm{i}\mathrm{d}$ in every positive degree.

Let $\mathcal{B}$ be a weak linear $2$-category. A monad on $A$ in $\mathcal{B}$ is a $1$-morphism $T:A\to A$ with $2$-morphisms $\eta :1_A\Rightarrow T$ and $\mu :T\circ T\Rightarrow T$ satisfying $\mu \cdot (\mu \ast 1_T)=\mu \cdot (1_T\ast \mu ),\mu \cdot (\eta \ast 1_T)=1_T=\mu \cdot (1_T\ast \eta ),$ where $\cdot$ is vertical composition, $\ast$ is horizontal composition, and the associator and unitors of $\mathcal{B}$ are inserted. Equivalently it is an algebra object in the monoidal endomorphism category $\mathcal{B}(A,A)$. This extends the ordinary-category definition in Monad on a category. It is separable if there is a $2$-morphism $\sigma :T\Rightarrow T\circ T$ with $\mu \cdot \sigma =1_T,\sigma \cdot \mu =(\mu \ast 1_T)\cdot (1_T\ast \sigma )=(1_T\ast \mu )\cdot (\sigma \ast 1_T),$ again with coherent associators. The last two equalities say that $\sigma$ is a map of regular $T$-bimodules.

An adjunction $L⊣U$ in $\mathcal{B}$, for $L:A\to B$ and $U:B\to A$, consists of $2$-morphisms $\eta :1_A\Rightarrow UL$ and $\epsilon :LU\Rightarrow 1_B$ with $(\epsilon \ast 1_L)\cdot (1_L\ast \eta )=1_L,(1_U\ast \epsilon )\cdot (\eta \ast 1_U)=1_U,$ with associators and unitors inserted. These are the same triangle equations as in Adjunction by unit, counit, and the triangle identities, now for $1$- and $2$-morphisms of $\mathcal{B}$. Its induced monad is $(UL,\eta ,1_U\ast \epsilon \ast 1_L)$.

A finite semisimple linear 2-category is a weak linear 2-category in which:

• every hom category is a finite semisimple $k$-linear category (it has finitely many simple isomorphism classes, finite-dimensional hom spaces, and every object is a finite direct sum of simples);
• there is a zero object, and each pair $A,B$ has a biproduct $A\oplus B$ with inclusions and projections ${\iota }_A,{\iota }_B,{\pi }_A,{\pi }_B$ satisfying ${\pi }_i\circ {\iota }_j\cong 0$ for $i\mathrm{\ne }j$, ${\pi }_i\circ {\iota }_i\cong 1$, and ${\iota }_A\circ {\pi }_A\oplus {\iota }_B\circ {\pi }_B\cong 1_{A\oplus B}$; finite biproducts are preserved by composition;
• every 1-morphism has both a left and a right adjoint (Adjunction by unit, counit, and the triangle identities), with unit and counit 2-morphisms satisfying the two triangle identities;
• every separable monad has a separable splitting: a separable monad $T:A\to A$ is represented by an object $A_T$ and an adjunction $L⊣U$, where $L:A\to A_T$ and $U:A_T\to A$, in the sense of Adjunction by unit, counit, and the triangle identities. Its induced monad $UL$ is isomorphic to $T$ as a monad, and its counit $\epsilon :LU\Rightarrow 1_{A_T}$ has a section $s:1_{A_T}\Rightarrow LU$ as a 2-morphism in $\mathcal{B}(A_T,A_T)$, with $\epsilon \circ s=1_{1_{A_T}}$ (in a bimodule presentation this section is a bimodule map);
• there are finitely many equivalence classes of simple objects, where an object $A$ is simple when its identity 1-morphism is simple in $\mathcal{B}(A,A)$.

The splitting condition concerns monads in the 2-category; it is stronger than the existence of finite direct sums in each hom category. A choice of separable splitting is not part of the property.

Let $\mathcal{C}$ be a braided fusion category and let $\mathcal{M},\mathcal{N}$ be finite semisimple right $\mathcal{C}$-module categories. A relative Deligne product $\mathcal{M}{⊠}_{\mathcal{C}}\mathcal{N}$ in $\mathrm{Mod}(\mathcal{C})$ means a finite semisimple right $\mathcal{C}$-module category $\mathcal{P}$ equipped with an exact bilinear functor $b:\mathcal{M}\times \mathcal{N}\to \mathcal{P}$ that is balanced over the middle $\mathcal{C}$-action and compatible with the outer right $\mathcal{C}$-action, and is universal among such functors: for every finite semisimple right $\mathcal{C}$-module category $\mathcal{D}$, composition with $b$ gives an equivalence between right $\mathcal{C}$-module functors $\mathcal{P}\to \mathcal{D}$ and exact bilinear middle-balanced right $\mathcal{C}$-module functors $\mathcal{M}\times \mathcal{N}\to \mathcal{D}$.

For the balancing convention, a right module category $\mathcal{N}$ is regarded as a left module category by $X▹n:=n\odot X$. Its left module associator $(X\otimes Y)▹n\to X▹(Y▹n)$ is the composite of $n\odot c_{Y,X}^{-1}$ with the inverse right-module associator. “Balanced” means that $b(m\odot X,n)\cong b(m,X▹n)$ naturally in all variables, compatibly with the unit and associativity constraints. This definition fixes the universal property and the braiding convention; it does not assert that a product exists.

A weak monoidal 2-category is a weak 2-category $\mathcal{B}$ together with the following data.

1. A bilinear tensor pseudofunctor $\otimes :\mathcal{B}\times \mathcal{B}\to \mathcal{B}$. On 1- and 2-morphisms it has invertible comparison 2-morphisms for composition and identities, natural in each variable and satisfying the pseudofunctor composition and unit equations.
2. A unit object $I$.
3. Pseudonatural equivalences $a_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)$ and ${\mathrm{\ell }}_A:I\otimes A\to A$, $r_A:A\otimes I\to A$. A pseudonatural transformation has a 1-morphism component at every object and an invertible naturality 2-morphism for every 1-morphism; the naturality squares for composites and identities commute. A modification between pseudonatural transformations has a 2-morphism component at every object, and its naturality square for each 1-morphism commutes. “Equivalence” means that an inverse pseudonatural transformation and invertible unit and counit modifications are supplied.
4. An invertible pentagonator modification ${\mathrm{\Pi }}_{A,B,C,D}$ from the composite $a_{A,B,C\otimes D}\circ a_{A\otimes B,C,D}$ to $(1_A\otimes a_{B,C,D})\circ a_{A,B\otimes C,D}\circ (a_{A,B,C}\otimes 1_D)\mathrm{.}$ There are three invertible unit modifications, with sources and targets ${\lambda }_{A,B}:{\mathrm{\ell }}_{A\otimes B}\circ a_{I,A,B}⟹{\mathrm{\ell }}_A\otimes 1_B,$ ${\rho }_{A,B}:r_{A\otimes B}⟹(1_A\otimes r_B)\circ a_{A,B,I},$ ${\tau }_{A,B}:(1_A\otimes {\mathrm{\ell }}_B)\circ a_{A,I,B}⟹r_A\otimes 1_B\mathrm{.}$

All pseudofunctor comparison maps, transformations, and modifications are natural in their object, 1-morphism, and 2-morphism inputs. Their equations are part of the structure: the pentagonator obeys the five-object coherence equation (the two pastings between the endpoints of the 4-dimensional associahedron agree); the three unit modifications obey the triangle and double-unit equations; and inserting a unit in the five-object equation agrees with those unit equations. The tensor pseudofunctor comparisons satisfy their composition, unit, and naturality equations. These are equalities of parallel 2-morphisms; they are not consequences of object-level associativity.

The ordinary associator pentagon is represented schematically below. In a weak monoidal 2-category its two boundary composites are compared by $\mathrm{\Pi }$, rather than required to be literally equal.

diagram failed to render:
& (A\otimes B)\otimes(C\otimes D) \arrow[rr,"a_{A,B,C\otimes D}"] & & A\otimes(B\otimes(C\otimes D)) \\
((A\otimes B)\otimes C)\otimes D \arrow[ur,"a_{A\otimes B,C,D}"] \arrow[dr,"a_{A,B,C}\otimes 1_D"'] & & & \\
& (A\otimes(B\otimes C))\otimes D \arrow[r,"a_{A,B\otimes C,D}"'] & A\otimes((B\otimes C)\otimes D) \arrow[ur,"1_A\otimes a_{B,C,D}"'] &

This definition uses the weak presentation directly. It invokes no global strictification theorem.

A Gray-monoid presentation is an alternative set of specified data; no strictification assertion is needed here. Its underlying linear $2$-category is strict. It has an object $I$, strict linear $2$-functors $L_A=A\mathrm{\square }-$ and $R_A=-\mathrm{\square }A$ for every object $A$, and invertible $2$-cells ${\varphi }_{f,g}:(f\mathrm{\square }{B}^{\mathrm{\prime }})\circ (A\mathrm{\square }g)⟹(A^{\mathrm{\prime }}\mathrm{\square }g)\circ (f\mathrm{\square }B)$ for $f:A\to A^{\mathrm{\prime }}$ and $g:B\to B^{\mathrm{\prime }}$. Write a dot for vertical composition, with the rightmost factor applied first, and $\circ$ for horizontal composition and whiskering. The complete semistrict equations are as follows.

1. $L_{A}B=R_{B}A$, denoted $A\mathrm{\square }B$.
2. $L_I=R_I=\mathrm{i}\mathrm{d}$, $L_A{L}_B=L_{A\mathrm{\square }B}$, $R_B{R}_A=R_{A\mathrm{\square }B}$, and $L_A{R}_B=R_B{L}_A$, as equalities of $2$-functors.
3. ${\varphi }_{f,1_B}=1_{f\mathrm{\square }B}$ and ${\varphi }_{1_A,g}=1_{A\mathrm{\square }g}$.
4. For composable $f,f^{\mathrm{\prime }}$ and $g,g^{\mathrm{\prime }}$, ${\varphi }_{f^{\mathrm{\prime }}\circ f,g}=({\varphi }_{f^{\mathrm{\prime }},g}\circ (f\mathrm{\square }B))\cdot ((f^{\mathrm{\prime }}\mathrm{\square }{B}^{\mathrm{\prime }})\circ {\varphi }_{f,g}),$ ${\varphi }_{f,g^{\mathrm{\prime }}\circ g}=((A^{\mathrm{\prime }}\mathrm{\square }{g}^{\mathrm{\prime }})\circ {\varphi }_{f,g})\cdot ({\varphi }_{f,g^{\mathrm{\prime }}}\circ (A\mathrm{\square }g))\mathrm{.}$
5. For $\alpha :f\Rightarrow f^{\mathrm{\prime }}$ and $\beta :g\Rightarrow g^{\mathrm{\prime }}$, ${\varphi }_{f^{\mathrm{\prime }},g}\cdot ((\alpha \mathrm{\square }{B}^{\mathrm{\prime }})\circ (A\mathrm{\square }g))=((A^{\mathrm{\prime }}\mathrm{\square }g)\circ (\alpha \mathrm{\square }B))\cdot {\varphi }_{f,g},$ ${\varphi }_{f,g^{\mathrm{\prime }}}\cdot ((f\mathrm{\square }{B}^{\mathrm{\prime }})\circ (A\mathrm{\square }\beta ))=((A^{\mathrm{\prime }}\mathrm{\square }\beta )\circ (f\mathrm{\square }B))\cdot {\varphi }_{f,g}\mathrm{.}$
6. ${\varphi }_{A\mathrm{\square }g,h}=A\mathrm{\square }{\varphi }_{g,h}$, ${\varphi }_{f\mathrm{\square }B,h}={\varphi }_{f,B\mathrm{\square }h}$, and ${\varphi }_{f,g\mathrm{\square }C}={\varphi }_{f,g}\mathrm{\square }C$.

Thus even in this semistrict presentation the two tensors of $1$-morphisms are compared by the specified interchanger; it is not discarded. The weak presentation above is used directly for the module-category and graded-vector-space constructions on this page.

Let $(\mathcal{C},\otimes ,\mathbf{1},c)$ be a braided monoidal category and let $(A,m,\eta )$ be an algebra object (Separable algebra objects and bimodules in a fusion category). Define the monoidal opposite ${\mathcal{C}}^{\mathrm{m}\mathrm{p}}$ to have the same objects and morphisms, tensor product $X{\otimes }_{\mathrm{m}\mathrm{p}}Y:=Y\otimes X$, associator ${\alpha }_{X,Y,Z}^{\mathrm{m}\mathrm{p}}:={\alpha }_{Z,Y,X}^{-1}$, and unitors ${\lambda }_X^{\mathrm{m}\mathrm{p}}:={\rho }_X$, ${\rho }_X^{\mathrm{m}\mathrm{p}}:={\lambda }_X$. The braiding defines a strong monoidal equivalence $J:{\mathcal{C}}^{\mathrm{m}\mathrm{p}}\to \mathcal{C}$ that is the identity on the underlying category and has tensorator $c_{Y,X}^{-1}:X\otimes Y\to Y\otimes X$.

Regard $A$ as an algebra in ${\mathcal{C}}^{\mathrm{m}\mathrm{p}}$ with the same unit and multiplication $m:A{\otimes }_{\mathrm{m}\mathrm{p}}A=A\otimes A\to A$. Its transport along $J$ is the opposite algebra $A^{\mathrm{o}\mathrm{p}}$ in $\mathcal{C}$; its multiplication is $m_{A^{\mathrm{o}\mathrm{p}}}=m\circ c_{A,A}^{-1}:A\otimes A\to A,$ and its unit is $\eta$. This convention fixes the braiding direction. The hexagon identities are the tensorator coherence of $J$, and its unit coherence follows from the braiding unit equations.

Use the canonical algebra $K$ in $\mathcal{D}=\mathcal{C}⊠{\mathcal{C}}^{\mathrm{r}\mathrm{e}\mathrm{v}}$ from The canonical algebra of a fusion category is Frobenius. Without a separability or characteristic-zero assumption, the functor $J:\mathcal{C}\to {\mathrm{Mod}}_{\mathcal{D}}(K)$ given by $J(V)=(V⊠\mathbf{1})\otimes K$ is an equivalence. There are natural right $K$-module isomorphisms ${\rho }_V:(V⊠\mathbf{1})K⟶(\mathbf{1}⊠V)K\mathrm{.}$ They obey ${\rho }_{\mathbf{1}}=\mathrm{i}\mathrm{d}$ and the sliding equation: ${\rho }_{V\otimes W}$ is the composite of ${\mathrm{i}\mathrm{d}}_{V⊠\mathbf{1}}\otimes {\rho }_W$, the interchange of the two separate Deligne factors, and ${\mathrm{i}\mathrm{d}}_{\mathbf{1}⊠W}\otimes {\rho }_V$. In particular any right $K$-linear map $J(V)\to J(W)$ uniquely equals $f\otimes {\mathrm{i}\mathrm{d}}_K$ for $f:V\to W$.

Let $\mathcal{C}$ be a fusion category over an algebraically closed field $k$ and $\mathcal{M}$ a finite semisimple right $\mathcal{C}$-module category. There is an algebra $A$ and a right module equivalence $\mathcal{M}\simeq {\mathrm{Mod}}_{\mathcal{C}}^l(A)$. This assertion does not yet assert separability. Fix the convention ${\mathrm{Hom}}_{\mathcal{C}}(X,[m,n])={\mathrm{Hom}}_{\mathcal{M}}(m\odot X,n)\mathrm{.}$ One can take $A=[m,m]$ whenever $m$ is a module generator, meaning every object of $\mathcal{M}$ is a quotient of $m\odot X$ for some $X\in \mathcal{C}$. In particular the direct sum of the simple representatives is such a generator. The multiplication is specified below. For any such algebras $A,B$, a right exact right module functor from left $A$-modules to left $B$-modules is uniquely, up to canonical isomorphism, $P\mapsto Q{\otimes }_{A}P$ for a $B$-$A$ bimodule $Q$. The equivalence identifies module-natural transformations with bimodule maps. No braiding is assumed.

For a finite semisimple left $\mathcal{C}$-module category and a generator $m$ with every $n$ a quotient of $X\odot m$, the left-action version is ${\mathrm{Hom}}_{\mathcal{C}}(X,[m,n])={\mathrm{Hom}}_{\mathcal{M}}(X\odot m,n)$ and $n\mapsto [m,n]$ is an equivalence to right modules over $A=[m,m]$, with $[m,X\odot n]\cong X\otimes [m,n]$. This version likewise does not require or assert separability.