ScalingStacks

As noted in Section 1.1, the center construction applied to a monoid yields its usual center, because a certain square must commute. However, as one would expect from the weakening principle, when $\mathcal{C}$ is a monoidal category the corresponding square need only commute up to a specified natural isomorphism. An object of $\mathcal{Z}(\mathcal{C})$ thus turns out to be an object $A\in\mathcal{C}$ equipped with a natural isomorphism $R_{A,-}\colon A\otimes-\Rightarrow-\otimes A$ satisfying various coherence laws, such as the commutativity of following diagram:

for any objects $X,Y\in\mathcal{C}$. Of course, this diagram is part of the definition of a braided monoidal category. Similarly, the morphisms in $\mathcal{Z}(\mathcal{C})$ also work out to have properties that form part of the definition of a braided monoidal category.

Heuristic 4-categorical computations suggest how these patterns should continue when $\mathcal{C}$ is a monoidal 2-category. We thus define $\mathcal{Z}(\mathcal{C})$ as follows.

Objects in $\mathcal{Z}(\mathcal{C})$:

An object of $\mathcal{Z}(\mathcal{C})$ is a triple $(A,R_{A,-},\tilde{R}_{(A|-,-)})$ consisting of:

1. (1)

   an object $A\in\mathcal{C}$

2. (2)

   a pseudonatural equivalence $R_{A,-}\colon A\otimes-\Rightarrow-\otimes A$

3. (3)

   an invertible modification $\tilde{R}_{(A|-,-)}$, giving for any objects $X,Y\in\mathcal{C}$ a 2-isomorphism

such that for any objects $X,Y,Z\in\mathcal{C}$, the tetrahedron $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$ commutes.

Here we mean that the diagram $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$ commutes with objects $A,X,Y,Z$, and with the modification $\tilde{R}_{(-|-,-)}$ in the definition of a braided monoidal 2-category replaced by the above $\tilde{R}_{(A|-,-)}$. Throughout the following we use the hieroglyphical notation in this way. Also, we use letters near the beginning of the alphabet to denote objects of $\mathcal{C}$ underlying objects in $\mathcal{Z}(\mathcal{C})$, and letters near the end to denote objects of $\mathcal{C}$ being used as such.

Morphisms in $\mathcal{Z}(\mathcal{C})$:

A morphism in $\mathcal{Z}(\mathcal{C})$ from $(A,R_{A,-},\tilde{R}_{(A|-,-)})$ to $(B,R_{B,-},\tilde{R}_{(B|-,-)})$ is a pair $(f,R_{f,-})$ consisting of:

1. (1)

   a morphism $f\colon A\to B$

2. (2)

   an invertible modification $R_{f,-}$, giving for any object $X\in\mathcal{C}$ a 2-isomorphism

such that the prism $({\to}\otimes(\bullet\otimes\bullet))$ commutes.

2-Morphisms in $\mathcal{Z}(\mathcal{C})$:

A 2-morphism $\alpha$ in $\mathcal{Z}(\mathcal{C})$ from $(f,R_{f,-})$ to $(g,R_{g,-})$ is

1. (1)

   a 2-morphism $\alpha\colon f\Rightarrow g$ in $\mathcal{C}$

such that $({\Downarrow}\otimes\bullet)$ commutes.

We define the composition operations in $\mathcal{Z}(\mathcal{C})$ as follows. Composition of morphisms is defined by:

where $f\colon A\to B$ and $g\colon B\to C$ are the underlying 1-morphisms in $\mathcal{C}$. Note that for any object $X\in\mathcal{C}$, the 2-morphism $((f\otimes X)\circ R_{g,X})\cdot(R_{f,X}\circ(X\otimes g))$ equals the back of the following diagram:

To show that the composite of morphisms in $\mathcal{Z}(\mathcal{C})$ is again a morphism, we have to check that $({\to}\otimes{\to})$ and $({\to}\otimes(\bullet\otimes\bullet))$ commute. These can be seen by pasting together two diagrams of the form $({\to}\otimes{\to})$ and $({\to}\otimes(\bullet\otimes\bullet))$, respectively.

Vertical and horizontal composition of 2-morphisms is defined the same as in $\mathcal{C}$; one can check that these composites again satisfy $({\Downarrow}\otimes\bullet)$ by pasting together two diagrams of this form.