Weak monoidal 2-category

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.