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.

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.

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.

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.

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.

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.