ScalingStacks

Lemma 4. A semistrict monoidal 2-category consists of a 2-category $\mathcal{C}$ together with:

1. (1)

   An object $I\in\mathcal{C}$.

2. (2)

   For any two objects $A,B$ in $\mathcal{C}$, an object $A\otimes B$ in $\mathcal{C}$.

3. (3)

   For any 1-morphism $f\colon A\to A^{\prime}$ and any object $B\in\mathcal{C}$ a 1-morphism $f\otimes B\colon A\otimes B\to A^{\prime}\otimes B$.

4. (4)

   For any 1-morphism $g\colon B\to B^{\prime}$ and any object $A\in\mathcal{C}$ a 1-morphism $A\otimes g\colon A\otimes B\to A\otimes B^{\prime}$.

5. (5)

   For any object $B\in\mathcal{C}$ and any 2-morphism $\alpha\colon f\Rightarrow f^{\prime}$ a 2-morphism $\alpha\otimes B\colon f\otimes B\Rightarrow f^{\prime}\otimes B$.

6. (6)

   For any object $A\in\mathcal{C}$ and any 2-morphism $\beta\colon g\Rightarrow g^{\prime}$ a 2-morphism $A\otimes\beta\colon A\otimes g\Rightarrow A\otimes g^{\prime}$.

7. (7)

   For any two 1-morphisms $f\colon A\to A^{\prime}$ and $g\colon B\to B^{\prime}$ a 2-isomorphism

Moreover, these data must satisfy the following conditions.

• (i)

  For any object $A\in\mathcal{C}$ we have $A\otimes-\;\colon\mathcal{C}\to\mathcal{C}$ and $-\otimes A\colon\mathcal{C}\to\mathcal{C}$ are 2-functors.

• (ii)

  For $x$ any object, morphism or 2-morphism of $\mathcal{C}$ we have $x\otimes I=I\otimes x=x$.

• (iii)

  For $x$ any object, morphism or 2-morphism of $\mathcal{C}$, and for all objects $A,B\in\mathcal{C}$ we have $A\otimes(B\otimes x)=(A\otimes B)\otimes x$, $A\otimes(x\otimes B)=(A\otimes x)\otimes B$ and $x\otimes(A\otimes B)=(x\otimes A)\otimes B$.

• (iv)

  For any 1-morphisms $f\colon A\to A^{\prime}$, $g\colon B\to B^{\prime}$ and $h\colon C\to C^{\prime}$ in $\mathcal{C}$ we have $\bigotimes_{A\otimes g,h}=A\bigotimes\otimes_{g,h}$, $\bigotimes_{fy\otimes B,h}=\bigotimes_{f,B\otimes h}$ and $\bigotimes_{f,g\otimes C}=\bigotimes_{f,g}\otimes C$.

• (v)

  For any objects $A,B\in\mathcal{C}$ we have $1_{A}\otimes B=A\otimes 1_{B}=1_{A\otimes B}$, and for any 1-morphisms $f\colon A\to A^{\prime}$, $g\colon B\to B^{\prime}$ in $\mathcal{C}$ we have $\bigotimes_{1_{A},g}=1_{A\otimes g}$ and $\bigotimes_{f,1_{B}}=1_{f\otimes B}$.

• (vi)

  For any 1-morphism $f:A\to A^{\prime}$, any 1-morphisms $g,g^{\prime}\colon B\to B^{\prime}$, and any 2-morphism $\beta\colon g\Rightarrow g^{\prime}$ the following diagram commutes:

• (vii)

  For any 1-morphism $g:B\to B^{\prime}$, any 1-morphisms $f,f^{\prime}\colon A\to A^{\prime}$, and any 2-morphism $\alpha\colon f\Rightarrow f^{\prime}$, the following diagram commutes:

• (viii)

  For any 1-morphisms $f\colon A\to A^{\prime}$, $g\colon B\to B^{\prime}$ and $g^{\prime}\colon B^{\prime}\to B^{\prime\prime}$ the 2-isomorphism $\bigotimes_{f,gg^{\prime}}$ coincides with the pasting of $\bigotimes_{f,g}$ and $\bigotimes_{f,g^{\prime}}$ as in the following diagram.

  For any 1-morphisms $f\colon A\to A^{\prime}$, $f^{\prime}\colon A^{\prime}\to A^{\prime\prime}$ and $g\colon B\to B^{\prime}$ the 2-isomorphism $\bigotimes_{ff^{\prime},g}$ coincides with the pasting of $\bigotimes_{f,g}$ and $\bigotimes_{f,g^{\prime}}$ in a similar way.