Monoidal category ($(\mathcal{C},\otimes ,I)$)

A category equipped with a monoidal tensor product, a monoidal unit object, and natural associators and unitors satisfying the pentagon and triangle axioms.

Cartesian product with singleton unit and disjoint union with empty unit each make the Category of sets monoidal. Associators and unitors simply retag or rebracket the same elements, so the pentagon and triangle identities hold elementwise. These structures differ even up to monoidal equivalence: any category equivalence preserves initial and terminal objects, so it cannot identify the empty unit with the singleton unit.

The associator and unitors of a monoidal category satisfyThus the two ways to remove a unit between two tensor factors agree. Together with the pentagon identity for a monoidal category this implies all unit identities derived from the monoidal pentagon and triangle and the monoidal coherence theorem.

For an associator $a$ in a monoidal category, the two paths from $((W\otimes X)\otimes Y)\otimes Z$ to $W\otimes (X\otimes (Y\otimes Z))$ agree:This five-vertex diagram is the basic compatibility for reassociating four tensor factors. The monoidal coherence theorem extends it to arbitrary finite tensor expressions.

The associator $a$ and unitors of a monoidal category satisfyFor the first identity, postcompose the pentagon on $X,Y,I,Z$ by $1_X\otimes (1_Y\otimes {\lambda }_Z)$, then use the triangle twice and cancel $a_{X,Y,Z}$. This gives the first equality after tensoring with $1_Z$; set $Z=I$ and cancel using the natural invertible right unitor. Reversing tensor order proves the second identity. Naturality of the left unitor at ${\lambda }_X$ gives ${\lambda }_{I\otimes X}=1_I\otimes {\lambda }_X$; comparing the second identity with the triangle on $I,I,X$ gives $({\lambda }_I\otimes 1_X)=({\rho }_I\otimes 1_X)$ and hence the last identity. These are consequences of the axioms, not extra axioms imported from the monoidal coherence theorem.

An invertible morphism $y:X\otimes X\to X\otimes X$ satisfying $(y\otimes 1)(1\otimes y)(y\otimes 1)=(1\otimes y)(y\otimes 1)(1\otimes y)$ with coherent parentheses. It produces representations of the braid groups; a self-braiding is an example.

Objects $X,Y$ with an evaluation morphism $e:X\otimes Y\to I$ and a coevaluation morphism $n:I\to Y\otimes X$ satisfying both snake identities. These are categorical duality data, distinct from the locally convex dual pair.

Closing an identity strand using compatible evaluation morphisms and coevaluation morphisms gives the categorical dimension in a pivotal or ribbon setting. For the standard two-dimensional quantum enveloping algebra of sl2 module, the weighted contraction has diagonal coefficients $q^{-1},q$, giving quantum dimension $q+q^{-1}=[2{]}_q$. The chosen duality and framing conventions matter.

The positive square root of the sum of the squared quantum dimensions of all simple anyons. The vacuum anyon has $d_1=1$ and antianyons have equal quantum dimensions.

The two triangular identities $(e\otimes 1_X)(1_X\otimes n)=1_X$ and $(1_Y\otimes e)(n\otimes 1_Y)=1_Y$, with canonical associators and unitors inserted, for a dual pair in a monoidal category.

The unit-valued duality map $n:I\to Y\otimes X$ of a dual pair in a monoidal category, constrained together with evaluation by both snake identities.

For a dual pair in a monoidal category, the contraction $e:X\otimes Y\to I$. In a left closed monoidal category, evaluation also denotes the adjunction map $[M,N]\otimes M\to N$.

Using the convention $-\otimes M⊣[M,-]$, every tensoring-on-the-right functor has a right adjoint functor. Evaluation is $[M,N]\otimes M\to N$. Naming left and right closure varies in the literature, so the adjunction fixes the convention.

The categorical dual of a monoid object: an object with coassociative comultiplication and a counit, with the ambient constraints included.

The structure map $\epsilon :C\to I$ of a comonoid, with $(\epsilon \otimes 1)\Delta$ and $(1\otimes \epsilon )\Delta$ equal to the inverse unitors. This is distinct from the counit of an adjunction.

The structure map $\Delta :C\to C\otimes C$ of a comonoid; it is coassociative up to the ambient associator.

A morphism $f:C\to D$ satisfying ${\Delta }_{D}f=(f\otimes f){\Delta }_C$ and ${\epsilon }_{D}f={\epsilon }_C$.

An object with multiplication $m:A\otimes A\to A$ and unit $j:I\to A$ satisfying associativity and unit diagrams using the ambient associators and unitors.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

In a symmetric monoidal category, a convolution-invertible scalar pairing $\gamma :H\otimes H\to I$ on a bimonoid inducing the comodule braiding $u\otimes v\mapsto \sum v_{(0)}\otimes u_{(0)}\gamma (u_{(1)},v_{(1)})$. Its multiplicativity, normalization and commutation axioms express the hexagon, unit and colinearity conditions. The notation abbreviates morphisms and ambient symmetries, not a requirement for elements.

A morphism $f:A\to B$ satisfying $f{m}_A=m_B(f\otimes f)$ and $f{j}_A=j_B$.

A monoidal category with natural invertible braidings $c_{X,Y}:X\otimes Y\to Y\otimes X$ satisfying the two hexagon axioms. The equation $c_{Y,X}{c}_{X,Y}=1$ is an additional symmetric condition, not a braided axiom.

A finite semisimple ribbon braided monoidal category with nondegenerate modular S-matrix. For a unitary anyon theory, this means that the only transparent anyon is the vacuum anyon. Such data define a three-dimensional topological quantum field theory.

The net chiral central charge of an edge theory. For a bosonic unitary modular tensor category, its residue modulo eight is determined by the topological spins and quantum dimensions through $e^{2\pi ic/8}={\mathcal{D}}^{-1}{\sum }_a{d}_a^2{\theta }_a$. This phase corrects the signature dependence in the Witten-Reshetikhin-Turaev invariant.

Cutting one nonseparating circle on each handle of a closed orientable surface gives a sphere with $g$ pairs of oppositely oriented boundary circles. The topological quantum field theory gluing rule sums over the paired anyon labels $a,\bar{a}$. The resulting anyon fusion matrix formula becomes $\dim V({\Sigma }_g)={\mathcal{D}}^{2g-2}{\sum }_a{d}_a^{2-2g}$ by the Verlinde formula.

The matrix of Hopf link evaluations divided by the total quantum dimension $\mathcal{D}$. In a unitary modular tensor category it is a symmetric unitary matrix, with $S_{1a}=d_a/\mathcal{D}$ and $S_{\bar{a}b}=S_{ab}^{\ast }$. Its columns diagonalize the anyon fusion matrices through the Verlinde formula.

The simultaneous diagonalization formula for the anyon fusion matrices of a modular tensor category. The eigenvalue of $N_b$ in column $x$ of the modular S-matrix is $S_{bx}/S_{1x}$.

The braided monoidal category generated by one object: objects are parenthesized tensor expressions in that object and the unit, and morphisms are structural isomorphisms and crossings subject to precisely the monoidal category and braiding axioms.

The braided monoidal category with objects $n\ge 0$, endomorphism groups $B_n$, no arrows between unequal objects, tensor given by juxtaposition, and block-crossing braiding. Its tensor is strict.

A monoidal functor between braided monoidal categories compatible with their braidings: $F(c_{X,Y})F_2(X,Y)=F_2(Y,X)c_{FX,FY}$.

The natural tensor interchange isomorphism in a braided monoidal category, constrained by the two hexagon axioms.

The compatibility equations between braiding and the associator in a braided monoidal category. Omitting the associators from the notation, they read $c_{a,b\otimes c}=(1_b\otimes c_{a,c})(c_{a,b}\otimes 1_c)$ and $c_{a\otimes b,c}=(c_{a,c}\otimes 1_b)(1_a\otimes c_{b,c})$. Thus exchanging a composite with another object is the sequence of exchanges of its factors.

A functor with natural maps $F_2:F(X\otimes Y)\to FX\otimes FY$ and $F_0:FI\to I$ satisfying the reversed monoidal functor associativity and unit diagrams. No invertibility is required.

A natural transformation $\tau :F\Rightarrow G$ satisfying $G_2{\tau }_{X\otimes Y}=({\tau }_X\otimes {\tau }_Y)F_2$ and $G_0{\tau }_I=F_0$.

Here the unqualified term allows lax comparison maps $F_2:FX\otimes FY\to F(X\otimes Y)$ and $F_0:I\to FI$, natural and compatible with associators and unitors. Invertible comparison maps give a strong monoidal functor. The opposite direction gives an opmonoidal functor.

A lax monoidal functor $\phi$ and opmonoidal functor $\psi$ on the same underlying functor, satisfying ${\psi }_{A\otimes B,C}{\phi }_{A,B\otimes C}=({\phi }_{A,B}\otimes 1)(1\otimes {\psi }_{B,C})$ and ${\psi }_{A,B\otimes C}{\phi }_{A\otimes B,C}=(1\otimes {\phi }_{B,C})({\psi }_{A,B}\otimes 1)$ in coherent notation. It transports dual pairs in a monoidal category and Frobenius monoids.

A monoidal functor whose tensor and unit comparison maps are isomorphisms.

A strong monoidal functor whose comparison maps are identities. It preserves the tensor, unit and structural constraints exactly; its source need not be a strict monoidal category.

A monoidal category whose associators and unitors are identities; the tensor and unit laws are then equalities of objects and morphisms.

Every diagram formed solely from the canonical associators, unitors and their inverses commutes. Consequently structural reparenthesizations can be suppressed in calculations; this statement does not make distinct braidings equal.

Orient associativity towards right association and unit maps towards deletion of unit symbols. A lexicographic measure consisting of the number of leaves and the sum of the sizes of left subtrees decreases at every step. All overlapping reductions have commuting completions: the essential cases are the pentagon identity for a monoidal category, the triangle identity for a monoidal category, and the derived left and right unit identities. Induction on the decreasing measure makes every normalization arrow equal, proving that every structural arrow is determined by its endpoints.

Associate to a finite ordered word $w$ the right-associated tensor $N(w)$, retaining a terminal unit: $N(\varnothing )=I$ and $N(Aw)=A\otimes N(w)$. Define concatenation maps recursively byThe unit identities derived from the monoidal pentagon and triangle give $c_{u,\varnothing }={\rho }_{N(u)}$. Induction on $u$, using the pentagon in the induction step and left-unitor compatibility in the base case, givesNormalize a letter by ${\rho }_A^{-1}$, a formal unit by $1_I$, and a tensor expression recursively using $c$. The displayed identities show that normalization commutes with each generating associator or unitor, and therefore also with inverses, tensor products and composites of them. Every structural arrow from expression $S$ to expression $T$ is consequently ${\nu }_T^{-1}{\nu }_S$, proving monoidal coherence without presupposing it.

The natural isomorphisms ${\lambda }_X:I\otimes X\to X$ and ${\rho }_X:X\otimes I\to X$ in a monoidal category. Together with the associator they satisfy the triangle axiom.

The natural isomorphism ${\alpha }_{X,Y,Z}:(X\otimes Y)\otimes Z\to X\otimes (Y\otimes Z)$ in a monoidal category. It satisfies the pentagon axiom.

The distinguished object $I$ in a monoidal category, with the natural unitors $I\otimes X\cong X\cong X\otimes I$.

The bifunctor $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ in a monoidal category. Its associativity is expressed by the associator, rather than equality in general.