Section 2. (0N8L) · ScalingStacks

2. Definitions

We begin by defining semistrict monoidal 2-categories and semistrict braided monoidal 2-categories. Following traditional practice among category theorists [18, 23], we use ‘2-category’ to mean what Kapranov and Voevodsky [21] call a strict 2-category, and ‘2-functor’ to mean what Kapranov and Voevodsky call a strict 2-functor. Composition of 1-morphisms, the horizontal composition of a 1-morphism and a 2-morphism (in either order) and the horizontal composition of 2-morphisms is denoted by $\circ$ or simply juxtaposition. Vertical composition of 2-morphisms is denoted by $\cdot$ . We use the ordering in which, for example, the composite of $f\colon A\to B$ and $g\colon B\to C$ is denoted $f\circ g$ .

We use $\mathcal{C}\otimes_{\rm G}\mathcal{D}$ to denote Gordon, Power, and Street’s [17] ‘Gray’ tensor product of the 2-categories $\mathcal{C}$ and $\mathcal{D}$ . This differs from Gray’s original version [18] in being the ‘pseudo’ rather than the ‘lax’ weakening of the Cartesian product. For readers unfamiliar with these distinctions, let us simply recall that given a 1-morphism $f\colon A\to A^{\prime}$ in $\mathcal{C}$ and a 1-morphism $g\colon B\to B^{\prime}$ in $\mathcal{D}$ , the Cartesian product $\mathcal{C}\times\mathcal{D}$ contains a commuting square

Following the ‘lax’ approach to weakening, which consists of replacing equations by morphisms, Gray’s original product of $\mathcal{C}$ and $\mathcal{D}$ instead contains a square commuting only up to a specified 2-morphism:

Following the ‘pseudo’ approach, which consists of replacing equations by isomorphisms (or, more generally, equivalences), Gordon, Power, and Street additionally require $\gamma_{f,g}$ to be an isomorphism. We use their version of the Gray tensor product as part of a systematic adherence to the ‘pseudo’ approach.

The category $2\mathsf{Cat}$ with 2-categories as objects and 2-functors as morphisms becomes a monoidal category $(2\mathsf{Cat},\otimes_{\rm G},\mathcal{I})$ when equipped with the Gray tensor product and the unit object $\mathcal{I}$ , the 2-category with one object, one morphism and one 2-morphism. This monoidal category is symmetric, with the symmetry

S_{\mathcal{C},\mathcal{D}}\colon\mathcal{C}\otimes_{\rm G}\mathcal{D}\to\mathcal{D}\otimes_{\rm G}\mathcal{C}

given by:

\begin{array}[]{ccc}(B,A)\mapsto(B,A)&(f,1)\mapsto(1,f)&(1,g)\mapsto(g,1)\\ (\alpha,1)\mapsto(1,\alpha)&(1,\beta)\mapsto(\beta,1)&\gamma_{f,g}\mapsto\gamma_{g,f}^{-1}\end{array}

2.1. Semistrict Monoidal 2-Categories

Since $2\mathsf{Cat}$ is monoidal when equipped with the Gray tensor product, we may use enriched category theory [22] to efficiently define semistrict 3-categories and monoidal 2-categories:

0N7U

Definition 1. A semistrict 3-category is a category enriched over $(2\mathsf{Cat},\otimes_{\rm G},\mathcal{I})$ .

0N7V

Definition 2. A semistrict monoidal 2-category is a semistrict 3-category with one object.

Gordon, Power and Street [17] have given a definition of ‘weak’ 3-categories, or ‘tricategories’, seemingly more general than that of semistrict 3-categories, and indeed intended to be ‘maximally general’ in some sense. For example, associativity and identity laws hold as equations in a semistrict 3-category, but only hold up to specified equivalence in a weak one. However, these authors have shown that every weak 3-category is equivalent in a precise sense (‘triequivalence’) to a semistrict one, so for many purposes semistrict 3-categories are ‘sufficiently general’. Defining a weak monoidal 2-category to be a weak 3-category with one object, it follows from their proof that any one of these is triequivalent to a semistrict monoidal 2-category. So again, while not maximally general, semistrict monoidal 2-categories are sufficiently general for many purposes.

Often we shall think of a semistrict monoidal 2-category as a 2-category with extra structure. More precisely, if $\tilde{\mathcal{C}}$ is a semistrict 3-category with one object $\ast$ , let $\mathcal{C}={\rm hom}(\ast,\ast)$ . This is a 2-category equipped with a 2-functor

\otimes:\mathcal{C}\otimes_{\rm G}\mathcal{C}\to\mathcal{C}

coming from composition in $\tilde{\mathcal{C}}$ , as well as a functor $i\colon\mathcal{I}\to\mathcal{C}$ coming from the identity of $\ast$ in $\tilde{\mathcal{C}}$ .

0N7W

Lemma 3. Suppose $\tilde{\mathcal{C}}$ is a semistrict 3-category with one object, and let $(\mathcal{C},\otimes,i)$ be defined as above. Then $\mathcal{C}$ is a 2-category, $\otimes\colon\mathcal{C}\otimes_{\rm G}\mathcal{C}\to\mathcal{C}$ and $i\colon\mathcal{I}\to\mathcal{C}$ are 2-functors, and the following diagrams commute:

(1)

Associativity:

(2)

Unit law:

Conversely, for any $(\mathcal{C},\otimes,i)$ with these properties, there is a unique semistrict 3-category $\tilde{\mathcal{C}}$ with one object from which $(\mathcal{C},\otimes,i)$ arises as above.

0N7X

Proof. This is a straightforward consequence of the definition of semistrict 3-categories as categories enriched over $2\mathsf{Cat}$ with its Gray tensor product. ∎

There is thus no harm in thinking of a semistrict monoidal 2-category as a triple $(\mathcal{C},\otimes,i)$ satisfying the associativity and unit law conditions of Lemma 3. Since the 2-functor $i$ is determined by the object of $\mathcal{C}$ obtained by applying $i\colon\mathcal{I}\to\mathcal{C}$ to the one object in $\mathcal{I}$ , we can also think of a semistrict monoidal 2-category as a triple $(\mathcal{C},\otimes,I)$ .

One may further unpack our definition of a semistrict monoidal 2-category and obtain the same explicit list of operations and laws that Kapranov and Voevodsky take as their definition [21]. Here the standard machinery of 2-categorical commutative diagrams becomes very handy [23]. In what follows we write $\otimes_{f,g}$ for the 2-morphism $\otimes(\gamma_{f,g})$ in $\mathcal{C}$ .

0N7Y

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

(1)

An object $I\in\mathcal{C}$ .
(2)

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

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)

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)

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)

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.

0N7Z

Proof. This is a straightforward verification. In particular, conditions (v), (vi) and (vii) come from the coherence laws satisfied by $\gamma_{f,g}$ in the Gray tensor product. ∎

Note that condition $(viii)$ and the invertibility of the $2$ -morphism $\otimes_{f,g}$ imply that $\otimes_{1_{A},g}=1_{g}$ and $\otimes_{f,1_{B}}=1_{f}$ , for any $f:A\to A^{\prime}$ and any $g:B\to B^{\prime}$ .

2.2. Semistrict Braided Monoidal 2-Categories

To efficiently define braided monoidal 2-categories it is useful to exploit the fact that $(2\mathsf{Cat},\otimes_{\rm G},\mathcal{I})$ is closed, i.e., enriched over itself [17]. Put more explicitly, what this means is that $2\mathsf{Cat}$ can be regarded as a semistrict 3-category having small 2-categories as objects, 2-functors as morphisms, ‘pseudonatural transformations’ as 2-morphisms, and ‘modifications’ as 3-morphisms [3, 23]. A pseudonatural transformation $T$ between 2-functors $\mathcal{F},\mathcal{G}\colon\mathcal{C}\to\mathcal{D}$ assigns to each object $A\in\mathcal{C}$ a morphism $T_{A}\colon\mathcal{F}(A)\to\mathcal{G}(A)$ which satisfies the definition of a natural transformation only up to a specified isomorphism. Thus, $T$ also assigns to each morphism $f\colon A\to B$ in $\mathcal{C}$ a 2-isomorphism $T_{f}$ as follows:

These 2-morphisms $T_{f}$ must in turn satisfy some equational laws of their own. First, for any identity morphism $1_{A}\colon A\to A$ , we require $T_{1_{A}}=1_{T_{A}}$ . Second, given a composable pair of morphisms $f\colon A\to B$ , $g\colon B\to C$ , the 2-morphism $T_{fg}$ is given by the following pasting:

Third, given morphisms $f,f^{\prime}\colon A\to B$ and a 2-morphism $\alpha\colon f\Rightarrow f^{\prime}$ , the following diagram commutes:

Given two pseudonatural transformations $S,T\colon\mathcal{F}\Rightarrow\mathcal{G}$ , a modification $\alpha$ from $S$ to $T$ assigns to each object $A\in\mathcal{C}$ a 2-morphism $\alpha_{A}\colon S_{A}\Rightarrow T_{A}$ . Moreover, for any morphism $F\colon A\to B$ , the following diagram is required to commute:

As explained in the introduction, in an $n$ -category the notion of ‘isomorphism’ can be weakened to a recursively defined notion of ‘equivalence’. In the case of $2\mathsf{Cat}$ this gives the following concepts. A modification $\alpha$ from the pseudonatural transformation $S$ to the pseudonatural transformation $T$ is ‘invertible’ if there is a modification $\alpha^{-1}$ from $T$ to $S$ such that $\alpha\alpha^{-1}=1_{S}$ and $\alpha^{-1}\alpha=1_{T}$ . A pseudonatural transformation $T$ from $\mathcal{F}$ to $\mathcal{G}$ is a ‘pseudonatural equivalence’ if there is a pseudonatural transformation $\overline{T}\colon\mathcal{G}\to F$ and invertible modifications

\alpha_{1}\colon T\overline{T}\to 1_{\mathcal{F}},\qquad\alpha_{2}\colon\overline{T}T\to 1_{\mathcal{G}}.

There is a similar notion at the level of 2-functors, but we will not need it.

Every semistrict monoidal 2-category has a second, ‘opposite’ tensor product:

0N80

Lemma 5. Suppose $(\mathcal{C},\otimes,I)$ is a semistrict monoidal 2-category. Then $(\mathcal{C},\otimes^{\rm op},I)$ is also a semistrict monoidal 2-category, where $\otimes^{\rm op}=S_{\mathcal{C},\mathcal{C}}\circ\otimes$ .

0N81

Proof. Straightforward. ∎

There is an analogous opposite tensor product for strict monoidal categories, and a strict braided monoidal category is just a strict monoidal category equipped with a natural isomorphism $R\colon\otimes\Rightarrow\otimes^{\rm op}$ , the ‘braiding’, such that the following triangles commute:

The definition of a semistrict braided monoidal 2-category is very similar. However, instead of a strict monoidal category, one starts with a semistrict monoidal 2-category. Instead of the braiding being a natural transformation, it is a pseudonatural equivalence. Instead of the equations above holding ‘on the nose’, they hold up to specified invertible modifications. Finally, these modifications must satisfy 3 new coherence laws discovered by Kapranov and Voevodsky, together with the equation $S^{+}=S^{-}$ discussed in Section 1.3.

In all that follows, in diagrams we sometimes denote the tensor product of objects simply by juxtaposition. We also label some clauses in the definition using the ‘hieroglyphic’ notation invented by Kapranov and Voevodsky.

0N82

Definition 6. A braided monoidal 2-category $(\mathcal{C},\otimes,I,R,\tilde{R}_{(-|-,-)},\tilde{R}_{(-,-|-)})$ consists of:

(1)

A semistrict monoidal 2-category $(\mathcal{C},\otimes,1)$
(2)

A pseudonatural equivalence $R\colon\otimes\Rightarrow\otimes^{\rm op}$

(3)

Two invertible modifications $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$ , giving for any objects $A,B,C\in\mathcal{C}$ the 2-isomorphisms

These data must satisfy the following conditions. First, for all objects $A,B,C,D\in\mathcal{C}$ the following diagrams commute:

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

\begin{array}[]{ll}1.\>\tilde{R}_{(A\otimes B,C|D)}&2.\>=\>\tilde{R}_{(A,B|D)}\otimes C\\ 3.\>=\>A\otimes\tilde{R}_{(B,C|D)}&4.\>=\>\tilde{R}_{(A,B\otimes C|D)}\end{array}

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

\begin{array}[]{ll}1.\>=\>\tilde{R}_{(A|B,C\otimes D)}&2.\>=\>B\otimes\tilde{R}_{(A|C,D)}\\ 3.\>=\>\tilde{R}_{(A|B,C)}\otimes D&4.\>=\>\tilde{R}_{(A|B\otimes C,D)}\end{array}

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

\begin{array}[]{lll}1.\>=\>\tilde{R}_{(A,B|C)}\otimes D&2.\>=\>C\otimes\tilde{R}_{(A,B|D)}&3.\>=\>A\otimes\tilde{R}_{(B|C,D)}\\ 4.\>=\>\tilde{R}_{(A|C,D)}\otimes B&5.\>=\>\tilde{R}_{(A,B|C\otimes D)}&6.\>=\>\otimes_{(R_{A,C},R_{B,D})}\\ 7.\>=\>\tilde{R}_{(A\otimes B|C,D)}&\end{array}

Second, for any objects $A,B,C\in\mathcal{C}$ , we define two 2-isomorphisms corresponding to two proofs of the Yang–Baxter hexagon in a braided monoidal category:

We refer to these 2-morphisms as $S^{+}_{A,B,C}$ and $S^{-}_{A,B,C}$ , respectively. We require them to be equal:

$(S^{+}=S^{-})$ :

We can unpack this definition to obtain an explicit list of operations and laws. In this form the definition is essentially due to Kapranov and Voevodsky, though with certain differences, which we list at the end of this section.

0N83

Lemma 7. A braided monoidal 2-category $(\mathcal{C},\otimes,1,R,\tilde{R}_{(-|-,-)},\tilde{R}_{(-,-|-)})$ consists of the following data:

(1)

A semistrict monoidal 2-category $(\mathcal{C},\otimes,1)$
(2)

$(\bullet\otimes\bullet)$ For any two objects $A,B\in\mathcal{C}$ an equivalence $R_{A,B}\colon A\otimes B\to B\otimes A$

(3)

$({\to}\otimes\bullet)$ For any 1-morphism $f:A\to A^{\prime}$ and any object $B\in\mathcal{C}$ a 2-isomorphism

(4)

$(\bullet\otimes{\to})$ For any object $A\in\mathcal{C}$ and any 1-morphism $g\colon B\to B^{\prime}$ a 2-isomorphism

(5)

$((\bullet\otimes\bullet)\otimes\bullet)$ For any objects $A,B,C\in\mathcal{C}$ a 2-iso

(6)

$(\bullet\otimes(\bullet\otimes\bullet))$ For any objects $A,B,C\in\mathcal{C}$ a 2-isomorphism

Moreover, these data must satisfy the following conditions:

$({\to}\otimes{\to})$ For any 1-morphisms $f\colon A\to A^{\prime}$ and $g\colon B\to B^{\prime}$ the following cube commutes:

1.\>=\>\otimes_{f,g}\qquad 2.\>=\>\otimes_{g,f}\qquad 3.\>=\>R_{A,g}\qquad 4.\>=\>R_{A^{\prime},g}\qquad 5.\>=\>R_{f,B^{\prime}}\qquad 6.\>=\>R_{f,B}

$(\bullet\otimes{\Downarrow})$ For any object $A\in\mathcal{C}$ , any 1-morphisms $f,f^{\prime}\colon B\to B^{\prime}$ , and any 2-morphism $\beta\colon f\Rightarrow f^{\prime}$ , the following prism commutes:

$({\Downarrow}\otimes\bullet)$ A similar prism, left to the reader.

$(\to{\to}\otimes\bullet)$ For any pair of 1-morphisms $A\stackrel{{\scriptstyle f}}{{\to}}A^{\prime}\stackrel{{\scriptstyle f^{\prime}}}{{\to}}A^{\prime\prime}$ and any object $B\in\mathcal{C}$ , the 2-isomorphism $R_{ff^{\prime},B}$ coincides with the pasting

$(\bullet\otimes{\to}\to)$ A similar pasting law, left to the reader.

$((\bullet\otimes\bullet)\otimes{\to})$ For any objects $A,B,C\in\mathcal{C}$ and any 1-morphism $f\colon C\to C^{\prime}$ , the following triangular prism commutes:

1.\>=\>A\otimes R_{B,f}\qquad 2.\>=\>R_{A,f}\otimes B\qquad 3.\>=\>\tilde{R}_{(A,B|C)}\qquad 4.\>=\>\tilde{R}_{(A,B|C^{\prime})}\qquad 5.\>=\>R_{AB,f}

$({\to}\otimes(\bullet\otimes\bullet))$ A similar prism, left to the reader.

$(({\to}\otimes\bullet)\otimes\bullet)$ For any objects $A,B,C\in\mathcal{C}$ and any 1-morphism $f\colon A\to A^{\prime}$ , the following triangular prism commutes:

1.\>=\>\otimes_{(f,R_{B,C})}\qquad 2.\>=\>R_{f,C}\otimes B\qquad 3.\>=\>\tilde{R}_{(A,B|C)}\qquad 4.\>=\>\tilde{R}_{(A^{\prime},B|C)}\qquad 5.\>=\>R_{f\otimes B,C}

$((\bullet\otimes{\to})\otimes\bullet)$ , $(\bullet\otimes({\to}\otimes\bullet))$ and $(\bullet\otimes(\bullet\otimes{\to}))$ Similar prisms, left to the reader.

$((\bullet\otimes\bullet\otimes\bullet)\otimes\bullet)$ , $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$ , $((\bullet\otimes\bullet)\otimes(\bullet\otimes\bullet))$ As in Definition 6.

$S^{+}=S^{-}$ As in Definition 6.

0N84

Proof. The 1-equivalences $R_{A,B}$ and 2-isomorphisms $R_{f,B}$ and $R_{A,g}$ comprise the pseudonatural equivalence $R\colon\otimes\to\otimes^{\rm op}$ , and conditions $({\to}\otimes{\to})$ , $({\bullet}\otimes{\Downarrow})$ , $({\Downarrow}\otimes{\bullet})$ , $(\to{\to}\otimes{\bullet})$ and $({\bullet}\otimes{\to}\to)$ state that it is indeed a pseudonatural transformation. The 2-morphisms $\tilde{R}_{(A|B,C)}$ and $\tilde{R}_{(A,B|C)}$ comprise the invertible modifications $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$ , and the commuting triangular prisms state that these are indeed modifications, expressing naturality in each argument. The remaining 4 conditions come from Definition 6. ∎

Note that by $(\to{\to}\otimes\bullet)$ resp. $(\bullet\otimes{\to}\to)$ and by the invertibility of the respective 2-morphisms, for any objects $A,B\in\mathcal{C}$ we have $R_{A,1_{B}}=1_{R_{A,B}}$ and $R_{1_{A},B}=1_{R_{A,B}}$ .

The above lemma makes it clear that our definition of braided monoidal 2-category differs from that of Kapranov and Voevodsky in precisely the following points:

(1)

Invertibility of the braiding. Our definition implies that the 1-morphisms $R_{A,B}$ are equivalences. Kapranov and Voevodsky make no invertibility assumptions on these 1-morphisms. Our definition would agree with theirs on this point, and otherwise stay the same, if we required $R\colon\otimes\to\otimes^{\rm op}$ to be merely a pseudonatural transformation, rather than a pseudonatural equivalence.
(2)

$S^{+}=S^{-}$ . As already noted, Kapranov and Voevodsky omit this condition.
(3)

Naturality of $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$ . Our definition implies the commutativity of 6 triangular prisms expressing the naturality in each argument of these modifications. Kapranov and Voevodsky substitute cubes for 4 of these prisms, namely $(\bullet\otimes({\to}\otimes\bullet))$ , $(\bullet\otimes(\bullet\otimes{\to}))$ , $((\bullet\otimes{\to})\otimes\bullet)$ and $(({\to}\otimes\bullet)\otimes\bullet)$ . By the following lemma one can deduce these cubes from the remaining data — but not, it appears, vice versa. In personal communication, Kapranov agreed that all these prisms should hold.

0N85

Lemma 8. For any three objects $A,B,C\in\mathcal{C}$ and any morphism $f\colon B\to B^{\prime}$ , the following cube commutes.

\begin{array}[]{lll}1.\>=\>A\otimes R_{f,C}&2.\>=\>R_{f,C}\otimes A&3.\>=\>R_{A,R_{B^{\prime},C}}\\ 4.\>=\>R_{A,R_{B,C}}&5.\>=\>R_{A,C\otimes f}&6.\>=\>R_{A,f\otimes C}\end{array}

0N86

Proof. This is an special case of the axiom $(\bullet\otimes{\Downarrow})$ together with $(\bullet\otimes{\to}\to)$ . ∎

We refer to this cube with the hieroglyph $(\bullet\otimes({\to}\otimes\bullet))^{\prime}$ . One can similarly prove the analogous cube corresponding to the hieroglyph $(\bullet\otimes(\bullet\otimes{\to}))^{\prime}$ commutes. Moreover, we can prove the commutativity of cubes corresponding to the hieroglyphs $((\bullet\otimes{\to})\otimes\bullet)^{\prime}$ and $(({\to}\otimes\bullet)\otimes\bullet)^{\prime}$ using $({\Downarrow}\otimes\bullet)$ and $(\to{\to}\otimes\bullet)$ .

John C. Baez, Martin Neuchl

Original source: arXiv:q-alg/9511013v2