ScalingStacks

$(\bullet\otimes\bullet)$: For any two objects we have the morphism

in $\mathcal{Z}(\mathcal{C})$, where the 2-morphism $R_{R_{A,B},X}$ is defined to be the pasting:

First we have to show that $R_{R_{A,B},-}$ satisfies $({\to}\otimes{\to})$ and hence is a modification. This is shown in the following diagram (or follows from the fact that it is a pasting of modifications).

The front and the back side of the cube are the 2-morphisms $R_{R_{A,B},X}$ and $R_{R_{A,B},X^{\prime}}$, respectively. The top and the bottom are $\otimes_{R_{A,B},f}$ and $\otimes_{f,R_{A,B}}$, respectively. The left and the right side are the 2-morphisms corresponding to the pseudonatural transformations in the tensor product of the objects $A$ and $B$, $(R_{A}\otimes R_{B})_{f}$ and $(R_{B}\otimes R_{A})_{f}$, respectively.

The top triangular prism commutes by $(\bullet\otimes(\bullet\otimes{\to}))$. The bottom triangular prism commutes by $(\bullet\otimes({\to}\otimes\bullet))$. The cube in the middle commutes by $(\bullet\otimes(\bullet\otimes{\to}))^{\prime}$, which is a consequence of $(\bullet\otimes{\Downarrow})$ and $(\bullet\otimes{\to}\to)$ as indicated in Lemma 8.

Next, to show that we have really defined a morphism in $\mathcal{Z}(\mathcal{C})$, we have to verify $({\to}\otimes(\bullet\otimes\bullet))$. This means we have to check the commutativity of the following diagram.

As shown in the diagram below, we decompose this diagram in the following way: 1) Three tetrahedra of the form $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$. 2) One prism of the form $(\bullet\otimes(\bullet\otimes{\to}))$, namely $(A\otimes(X\otimes(BY\to YB)))$ (second row, right). 3) One prism of the form $(\bullet\otimes({\to}\otimes\bullet))$, namely $(A\otimes((BX\to XB)\otimes Y))$ (second row, left). 4) One prism of the form $(\bullet\otimes{\Downarrow})$, namely $(A\otimes\tilde{R}_{(B|X,Y)})$ (in the middle of the first row). All of these diagrams commute by our assumptions.

$({\to}\otimes\bullet)$: For any 1-morphism $(f,R_{f,-})\colon(A,R_{A},\tilde{R}_{A})\to(A^{\prime},R_{A^{\prime}},\tilde{R}_{A^{\prime}})$ and any object $(B,R_{B},\tilde{R}_{B})\in\mathcal{Z}(\mathcal{C})$ we have a 2-isomorphism

The following diagram shows that $R_{f,B}$ satisfies $({\Downarrow}\otimes\bullet)$ and is therefore a 2-morphism in $\mathcal{Z}(\mathcal{C})$.

The left and right sides are the 2-morphisms $R_{R_{A,B},X}$ and $R_{R_{A^{\prime},B},X}$, respectively. The front and the back sides are pastings as in our treatment in Section 3.2 of the tensor product of an object and a morphism in $\mathcal{Z}(\mathcal{C})$.

We decompose this cube into two commutative triangular prisms of the form $({\to}\otimes(\bullet\otimes\bullet))$, correspoding to $(f\otimes(B\otimes X))$ and $(f\otimes(X\otimes B))$, and one cube of the form $({\to}\otimes{\to})$, namely $(A\to A^{\prime}\otimes BX\to XB)$.

$(\bullet\otimes{\to})$: For any 1-morphism $(g,R_{g,-}):(B,R_{B},\tilde{R}_{B})\to(B^{\prime},R_{B^{\prime}},\tilde{R}_{B^{\prime}})$ and any object $(A,R_{A},\tilde{R}_{A})\in\mathcal{Z}(\mathcal{C})$, we have a 2-iso

The following diagram shows that $R_{A,g}$ satisfies $({\Downarrow}\otimes\bullet)$ and is thus a 2-morphism in $\mathcal{Z}(\mathcal{C})$.

The decomposition is similar to the one before.

$((\bullet\otimes\bullet)\otimes\bullet)$: For any objects $(A,R_{A},\tilde{R}_{A}),(B,R_{B},\tilde{R}_{B}),(C,R_{C},\tilde{R}_{C})\in\mathcal{Z}(\mathcal{C})$ we have the 2-isomorphism $\tilde{R}_{(A,B|C)}:=1_{(R_{A}\otimes R_{B})_{C}}$:

$((\bullet\otimes(\bullet\otimes\bullet))$: For any objects $(A,R_{A},\tilde{R}_{A}),(B,R_{B},\tilde{R}_{B}),(C,R_{C},\tilde{R}_{C})\in\mathcal{Z}(\mathcal{C})$ we have the 2-isomorphism $\tilde{R}_{(A|B,C)}$:

To verify that $\tilde{R}_{(A|-,-)}$ is a 2-morphism in $\mathcal{Z}(\mathcal{C})$, we have to check $({\Downarrow}\otimes\bullet)$. The next diagram gives the proof.

The top triangle corresponds to the 2-morphism $\tilde{R}_{(A|B,C)}\otimes X$. The bottom triangle corresponds to the 2-morphism $X\otimes\tilde{R}_{(A|B,C)}$. The back side is $R_{R_{A,B\otimes C},X}$, the left front side is $R_{R_{A,B}\otimes C,X}$ and the right front side is $R_{B\otimes R_{A,C},X}$. The decomposition is indicated in the diagram.

Now we have to verify that these data satisfy all the axioms of a braided monoidal 2-category. The tetrahedron $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$ commutes by the definition of the objects of $\mathcal{Z}(\mathcal{C})$. The diagram $((\bullet\otimes\bullet)\otimes(\bullet\otimes\bullet))$ commutes by the definition of the tensor product of two objects in $\mathcal{Z}(\mathcal{C})$. By the same definition can be shown that $((\bullet\otimes\bullet\otimes\bullet)\otimes\bullet)$ commutes in $\mathcal{Z}(\mathcal{C})$. Note that because of our special choice of the 2-morphism $R_{R_{A,B},-}$ that completes the morphism $R_{A,B}$ to a morphism in $\mathcal{Z}(\mathcal{C})$, the two 2-morphisms $S^{+}$ and $S^{-}$ are equal in $\mathcal{Z}(\mathcal{C})$.

The other axioms of a braided monoidal 2-category are either part of our definitions, or else we have indicated within our Remarks which definitions imply them. We may summarize by stating: