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.

We have to show that $\mathcal{Z}(\mathcal{C})$ bears a monoidal structure $(\mathcal{Z}(\mathcal{C}),\otimes_{\mathcal{Z}(\mathcal{C})},I)$, such that all the requirements for a monoidal category given in Definition 4 are satisfied.

(Ad 4.1): The object $I\in\mathcal{Z}(\mathcal{C})$ is $(I,1_{-},1_{1_{(-\otimes-)}})$.

The tensor product of objects: (Ad 4.2): The tensor product of two objects $(A,R_{A,-},\tilde{R}_{(A|-,-)})\otimes_{\mathcal{Z}(\mathcal{C})}(B,R_{B,-},\tilde{R}_{(B|-,-)})$ is defined to be the triple $(A\otimes B,(R_{A}\otimes R_{B})_{-},(\tilde{R}_{A}\otimes\tilde{R}_{B})_{(-,-)})$, where:

1. (1)

   The underlying $\mathcal{C}$-object is the tensor product $A\otimes B$ in $\mathcal{C}$.

2. (2)

   By Remark (9), the underlying pseudonatural equivalence $(R_{A}\otimes R_{B})_{-}:(A\otimes B)\otimes-\Rightarrow-\otimes(A\otimes B)$ assigns a 1-morphism $(R_{A}\otimes R_{B})_{X}$ to any object $X\in\mathcal{C}$ and a 2-morphism $(R_{A}\otimes R_{B})_{f}$ to any 1-morphism $f\colon X\to Y$. These are given as follows:

   $$(R_{A}\otimes R_{B})_{X}=(A\otimes R_{B,X})(R_{A,X}\otimes B),$$

   $$(R_{A}\otimes R_{B})_{f}=((A\otimes R_{B,f})\circ(R_{A,Y}\otimes B))\cdot(A\otimes R_{B,X})\circ(R_{A,f}\otimes B)),$$

   or in terms of a diagram:

   0N8A

   Remark 12. This will imply that the braiding in $\mathcal{Z}(\mathcal{C})$ satisfies $((\bullet\otimes\bullet)\otimes{\to})$.

   To show that these data constitute a pseudonatural equivalence, we have to show that $(\bullet\otimes{\to}\to)$ and $(\bullet\otimes{\Downarrow})$ hold. This can be done easily by pasting together the corresponding diagrams for $R_{A,-}$ and $R_{B,-}$.

3. (3)

   The underlying modification $(\tilde{R}_{A}\otimes\tilde{R}_{B})_{(X,Y)}:(A\otimes R_{B,X}\otimes Y)(R_{A,X}\otimes B\otimes Y)(X\otimes A\otimes R_{B,Y})(X\otimes R_{A,Y}\otimes B)\Rightarrow(A\otimes R_{B,X\otimes Y})(R_{A,X\otimes Y}\otimes B)$ is defined to be the pasting:

   Again it is easy to verify that this satisfies $(\bullet\otimes({\to}\otimes\bullet))$ and $(\bullet\otimes(\bullet\otimes{\to}))$ and hence is a modification.

To show that this definition gives in fact an object in $\mathcal{Z}(\mathcal{C})$, we have to verify that $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$ is satisfied. The following picture shows the tetrahedron. Those vertices in the picture that are vertices of the tetrahedron are written in big capitals. The remaining vertices occur since they are needed for the decomposition.

The following picture gives a decomposition of the tetrahedron into four smaller commutative diagrams.

Two of them are tetrahedra of the form $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$, tensored by an object from the left and the right, respectively. The upper of the two triangular prisms commutes by the axioms $4.(vii)$ with $\alpha=\tilde{R}_{(A|X,Y)}$ and $g=R_{Z,B}$, together with $4.(viii)$. The lower commutes by $4.(vi)$, applied to $\beta=\tilde{R}_{(B|Y,Z)}$ and $f=R_{A,X}$ together with $4.(viii)$. One can verify that this tensor product is in fact associative.

We shall often write $(A\otimes B,R_{A}\otimes R_{B},\tilde{R}_{A}\otimes\tilde{R}_{B})$ as a shorthand symbol for the tensor product of objects in $\mathcal{Z}(\mathcal{C})$.

The tensor product of an object and a morphism:

(Ad 4.3): Let $(f,R_{f,-})\colon(A,R_{A,-},\tilde{R}_{(A|-,-)})\to(A^{\prime},R_{A^{\prime},-},\tilde{R}_{(A^{\prime}|-,-)})$ be a morphism in $\mathcal{Z}(\mathcal{C})$ and let $(B,R_{B,-},\tilde{R}_{(B|-,-)})$ be an object. Their tensor product is the morphism given by the pair

or in terms of a diagram:

(Ad 4.4) Let $(f,R_{f,-}):(B,R_{B,-},\tilde{R}_{(B|-,-)})\to(B^{\prime},R_{B^{\prime},-},\tilde{R}_{(B^{\prime}|-,-)})$ be a morphism in $\mathcal{Z}(\mathcal{C})$ and let $(A,R_{A,-},\tilde{R}_{(A|-,-)})$ be an object. Their tensor product is the pair

or in terms of a diagram:

To verify that these formulas really define morphisms in $\mathcal{Z}(\mathcal{C})$, one must check that $({\to}\otimes{\to})$ and $({\to}\otimes(\bullet\otimes\bullet))$ hold. We only do this for $4.4$; the other case being similar. To show $({\to}\otimes{\to})$ one pastes together two cubes, one being the $({\to}\otimes{\to})$ cube for $f\colon B\to B^{\prime}$ and $g\colon X\to Y$ tensored on the left by $A$, the other being a special case of $5.(vii)$. For $({\to}\otimes(\bullet\otimes\bullet))$ we must show the following diagram commutes:

We cut it into one rectangular and two triangular prisms. To see that the left triangular prism commutes, we apply $({\to}\otimes(\bullet\otimes\bullet))$ to $(f,R_{f,-})$, tensored on the left by $A$. The rectangular prism commmutes by $5.(vi)$ and $5.(viii)$, applied to the 2-morphism $R_{f,Y}$. The right triangular prism commutes by $5.(iv)$ and $5.(vii)$, applied to the 2-morphism $\tilde{R}_{(A|X,Y)}$.

The tensor product of an object and a 2-morphism:

(Ad 4.5): For any object $(A,R_{A,-},\tilde{R}_{(A|-,-)})$ and any 2-morphism $\alpha\colon(f,R_{f,-})\Rightarrow(f^{\prime},R_{f^{\prime},-})$ we have a 2-morphism

(Ad 4.6): For any object $(B,R_{B,-},\tilde{R}_{(B|-,-)})$ and any 2-morphism $\alpha:(g,R_{g,-})\Rightarrow(g^{\prime},R_{g^{\prime},-})$ we have a 2-morphism

We must verify that these are 2-morphisms in $\mathcal{Z}(\mathcal{C})$, so we must check $({\Downarrow}\otimes\bullet)$. We do this only for $4.5$.

The upper prism commutes by $({\Downarrow}\otimes\bullet)$ tensored from the left by $A$. The lower prism commutes by an application of the axiom $4.(vi)$ for monoidal $2$-categories to the 2-morphism $\alpha$.

The tensor product of morphisms:

(Ad 4.7): For any morphisms $(f,R_{f,-}):(A,R_{A},\tilde{R}_{A})\to(A^{\prime},R_{A^{\prime}},\tilde{R}_{A^{\prime}})$ and $(g,R_{g,-}):(B,R_{B},\tilde{R}_{B})\to(B^{\prime},R_{B^{\prime}},\tilde{R}_{B^{\prime}})$ we have a 2-isomorphism:

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

The top cube commutes by $4.(iv),(vi),(viii)$, applied to the 2-morphism $A\otimes R_{g,X}$. The bottom cube commutes by $4.(iv),(vii),(viii)$, applied to the 2-morphism $R_{f,X}\otimes B$.

We have to verify that these data satisfy the conditions $4.(i)-(viii)$. These follow from the corresponding conditions holding in $\mathcal{C}$.

$(\bullet\otimes\bullet)$: 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: