ScalingStacks

2.1. Semistrict Monoidal 2-Categories

Since 2​𝖒𝖺𝗍2\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:

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

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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 β€˜weak’ 3-categories, or β€˜tricategories’, seemingly more general than that of semistrict 3-categories, and indeed intended to be β€˜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 (β€˜triequivalence’) to a semistrict one, so for many purposes semistrict 3-categories are β€˜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 π’ž~\tilde{\mathcal{C}} is a semistrict 3-category with one object βˆ—\ast, let π’ž=hom⁑(βˆ—,βˆ—)\mathcal{C}={\rm hom}(\ast,\ast). This is a 2-category equipped with a 2-functor

βŠ—:π’žβŠ—Gπ’žβ†’π’ž\otimes:\mathcal{C}\otimes_{\rm G}\mathcal{C}\to\mathcal{C}

coming from composition in π’ž~\tilde{\mathcal{C}}, as well as a functor i:β„β†’π’ži\colon\mathcal{I}\to\mathcal{C} coming from the identity of βˆ—\ast in π’ž~\tilde{\mathcal{C}}.

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Lemma 3. Suppose π’ž~\tilde{\mathcal{C}} is a semistrict 3-category with one object, and let (π’ž,βŠ—,i)(\mathcal{C},\otimes,i) be defined as above. Then π’ž\mathcal{C} is a 2-category, βŠ—:π’žβŠ—Gπ’žβ†’π’ž\otimes\colon\mathcal{C}\otimes_{\rm G}\mathcal{C}\to\mathcal{C} and i:β„β†’π’ži\colon\mathcal{I}\to\mathcal{C} are 2-functors, and the following diagrams commute:

  1. (1)

    Associativity:

    π’žβŠ—Gπ’žβŠ—Gπ’ž{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}\otimes_{\rm G}\mathcal{C}}π’žβŠ—Gπ’ž{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}}π’žβŠ—Gπ’ž{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}}π’ž{\lx@inpgf@ignorespaces\mathcal{C}}βŠ—βŠ—Gπ’ž\scriptstyle{\lx@inpgf@ignorespaces\otimes\otimes_{\rm G}\mathcal{C}}π’žβŠ—GβŠ—\scriptstyle{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\otimes}βŠ—\scriptstyle{\lx@inpgf@ignorespaces\otimes}βŠ—\scriptstyle{\lx@inpgf@ignorespaces\otimes}
  2. (2)

    Unit law:

    β„βŠ—Gπ’ž{\lx@inpgf@ignorespaces\mathcal{I}\otimes_{\rm G}\mathcal{C}}π’žβŠ—Gπ’ž{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}}π’ž{\lx@inpgf@ignorespaces\mathcal{C}}iβŠ—Gπ’ž\scriptstyle{\lx@inpgf@ignorespaces i\otimes_{\rm G}\mathcal{C}}β‰…\scriptstyle{\lx@inpgf@ignorespaces\cong}βŠ—\scriptstyle{\lx@inpgf@ignorespaces\otimes}β€ƒβ€ƒβ€ƒπ’žβŠ—Gℐ{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{I}}π’žβŠ—Gπ’ž{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}}π’ž{\lx@inpgf@ignorespaces\mathcal{C}}π’žβŠ—Gi\scriptstyle{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}i}β‰…\scriptstyle{\lx@inpgf@ignorespaces\cong}βŠ—\scriptstyle{\lx@inpgf@ignorespaces\otimes}

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

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Proof. This is a straightforward consequence of the definition of semistrict 3-categories as categories enriched over 2​𝖒𝖺𝗍2\mathsf{Cat} with its Gray tensor product. ∎

There is thus no harm in thinking of a semistrict monoidal 2-category as a triple (π’ž,βŠ—,i)(\mathcal{C},\otimes,i) satisfying the associativity and unit law conditions of LemmaΒ 3. Since the 2-functor ii is determined by the object of π’ž\mathcal{C} obtained by applying i:β„β†’π’ži\colon\mathcal{I}\to\mathcal{C} to the one object in ℐ\mathcal{I}, we can also think of a semistrict monoidal 2-category as a triple (π’ž,βŠ—,I)(\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 βŠ—f,g\otimes_{f,g} for the 2-morphism βŠ—(Ξ³f,g)\otimes(\gamma_{f,g}) in π’ž\mathcal{C}.

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

  1. (1)

    An object Iβˆˆπ’žI\in\mathcal{C}.

  2. (2)

    For any two objects A,BA,B in π’ž\mathcal{C}, an object AβŠ—BA\otimes B in π’ž\mathcal{C}.

  3. (3)

    For any 1-morphism f:Aβ†’Aβ€²f\colon A\to A^{\prime} and any object Bβˆˆπ’žB\in\mathcal{C} a 1-morphism fβŠ—B:AβŠ—Bβ†’Aβ€²βŠ—Bf\otimes B\colon A\otimes B\to A^{\prime}\otimes B.

  4. (4)

    For any 1-morphism g:Bβ†’Bβ€²g\colon B\to B^{\prime} and any object Aβˆˆπ’žA\in\mathcal{C} a 1-morphism AβŠ—g:AβŠ—Bβ†’AβŠ—Bβ€²A\otimes g\colon A\otimes B\to A\otimes B^{\prime}.

  5. (5)

    For any object Bβˆˆπ’žB\in\mathcal{C} and any 2-morphism Ξ±:fβ‡’fβ€²\alpha\colon f\Rightarrow f^{\prime} a 2-morphism Ξ±βŠ—B:fβŠ—Bβ‡’fβ€²βŠ—B\alpha\otimes B\colon f\otimes B\Rightarrow f^{\prime}\otimes B.

  6. (6)

    For any object Aβˆˆπ’žA\in\mathcal{C} and any 2-morphism Ξ²:gβ‡’gβ€²\beta\colon g\Rightarrow g^{\prime} a 2-morphism AβŠ—Ξ²:AβŠ—gβ‡’AβŠ—gβ€²A\otimes\beta\colon A\otimes g\Rightarrow A\otimes g^{\prime}.

  7. (7)

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

    AβŠ—B{\lx@inpgf@ignorespaces A\otimes B}AβŠ—Bβ€²{\lx@inpgf@ignorespaces A\otimes B^{\prime}}Aβ€²βŠ—B{\lx@inpgf@ignorespaces A^{\prime}\otimes B}Aβ€²βŠ—Bβ€²{\lx@inpgf@ignorespaces A^{\prime}\otimes B^{\prime}}AβŠ—g\scriptstyle{\lx@inpgf@ignorespaces A\otimes g}β‡“βŠ—f,g{\lx@inpgf@ignorespaces\Downarrow\otimes_{f,g}}fβŠ—B\scriptstyle{\lx@inpgf@ignorespaces f\otimes B}fβŠ—Bβ€²\scriptstyle{\lx@inpgf@ignorespaces f\otimes B^{\prime}}Aβ€²βŠ—g\scriptstyle{\lx@inpgf@ignorespaces A^{\prime}\otimes g}

Moreover, these data must satisfy the following conditions.

  • (i)

    For any object Aβˆˆπ’žA\in\mathcal{C} we have AβŠ—βˆ’:π’žβ†’π’žA\otimes-\;\colon\mathcal{C}\to\mathcal{C} and βˆ’βŠ—A:π’žβ†’π’ž-\otimes A\colon\mathcal{C}\to\mathcal{C} are 2-functors.

  • (ii)

    For xx any object, morphism or 2-morphism of π’ž\mathcal{C} we have xβŠ—I=IβŠ—x=xx\otimes I=I\otimes x=x.

  • (iii)

    For xx any object, morphism or 2-morphism of π’ž\mathcal{C}, and for all objects A,Bβˆˆπ’žA,B\in\mathcal{C} we have AβŠ—(BβŠ—x)=(AβŠ—B)βŠ—xA\otimes(B\otimes x)=(A\otimes B)\otimes x, AβŠ—(xβŠ—B)=(AβŠ—x)βŠ—BA\otimes(x\otimes B)=(A\otimes x)\otimes B and xβŠ—(AβŠ—B)=(xβŠ—A)βŠ—Bx\otimes(A\otimes B)=(x\otimes A)\otimes B.

  • (iv)

    For any 1-morphisms f:Aβ†’Aβ€²f\colon A\to A^{\prime}, g:Bβ†’Bβ€²g\colon B\to B^{\prime} and h:Cβ†’Cβ€²h\colon C\to C^{\prime} in π’ž\mathcal{C} we have ⨂AβŠ—g,h=Aβ¨‚βŠ—g,h\bigotimes_{A\otimes g,h}=A\bigotimes\otimes_{g,h}, ⨂f​yβŠ—B,h=⨂f,BβŠ—h\bigotimes_{fy\otimes B,h}=\bigotimes_{f,B\otimes h} and ⨂f,gβŠ—C=⨂f,gβŠ—C\bigotimes_{f,g\otimes C}=\bigotimes_{f,g}\otimes C.

  • (v)

    For any objects A,Bβˆˆπ’žA,B\in\mathcal{C} we have 1AβŠ—B=AβŠ—1B=1AβŠ—B1_{A}\otimes B=A\otimes 1_{B}=1_{A\otimes B}, and for any 1-morphisms f:Aβ†’Aβ€²f\colon A\to A^{\prime}, g:Bβ†’Bβ€²g\colon B\to B^{\prime} in π’ž\mathcal{C} we have ⨂1A,g=1AβŠ—g\bigotimes_{1_{A},g}=1_{A\otimes g} and ⨂f,1B=1fβŠ—B\bigotimes_{f,1_{B}}=1_{f\otimes B}.

  • (vi)

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

    AβŠ—B{\lx@inpgf@ignorespaces A\otimes B}⇓AβŠ—Ξ²{\lx@inpgf@ignorespaces\Downarrow A\otimes\beta}AβŠ—Bβ€²{\lx@inpgf@ignorespaces A\otimes B^{\prime}}Aβ€²βŠ—B{\lx@inpgf@ignorespaces A^{\prime}\otimes B}⇓Aβ€²βŠ—Ξ²{\lx@inpgf@ignorespaces\Downarrow A^{\prime}\otimes\beta}Aβ€²βŠ—Bβ€²{\lx@inpgf@ignorespaces A^{\prime}\otimes B^{\prime}}β‡“βŠ—f,g{\lx@inpgf@ignorespaces\Downarrow\otimes_{f,g}}β‡“βŠ—f,gβ€²{\lx@inpgf@ignorespaces\Downarrow\otimes_{f,g^{\prime}}}
  • (vii)

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

    AβŠ—B{\lx@inpgf@ignorespaces A\otimes B}β‡“Ξ±βŠ—B{\lx@inpgf@ignorespaces\Downarrow\alpha\otimes B}Aβ€²βŠ—B{\lx@inpgf@ignorespaces A^{\prime}\otimes B}AβŠ—Bβ€²{\lx@inpgf@ignorespaces A\otimes B^{\prime}}β‡“Ξ±βŠ—Bβ€²{\lx@inpgf@ignorespaces\Downarrow\alpha\otimes B^{\prime}}Aβ€²βŠ—Bβ€²{\lx@inpgf@ignorespaces A^{\prime}\otimes B^{\prime}}β‡‘βŠ—f,g{\lx@inpgf@ignorespaces\Uparrow\otimes_{f,g}}β‡‘βŠ—fβ€²,g{\lx@inpgf@ignorespaces\Uparrow\otimes_{f^{\prime},g}}
  • (viii)

    For any 1-morphisms f:Aβ†’Aβ€²f\colon A\to A^{\prime}, g:Bβ†’Bβ€²g\colon B\to B^{\prime} and gβ€²:Bβ€²β†’Bβ€²β€²g^{\prime}\colon B^{\prime}\to B^{\prime\prime} the 2-isomorphism ⨂f,g​gβ€²\bigotimes_{f,gg^{\prime}} coincides with the pasting of ⨂f,g\bigotimes_{f,g} and ⨂f,gβ€²\bigotimes_{f,g^{\prime}} as in the following diagram.

    AβŠ—B{\lx@inpgf@ignorespaces A\otimes B}AβŠ—Bβ€²{\lx@inpgf@ignorespaces A\otimes B^{\prime}}AβŠ—Bβ€²β€²{\lx@inpgf@ignorespaces A\otimes B^{\prime\prime}}Aβ€²βŠ—B{\lx@inpgf@ignorespaces A^{\prime}\otimes B}Aβ€²βŠ—Bβ€²{\lx@inpgf@ignorespaces A^{\prime}\otimes B^{\prime}}Aβ€²βŠ—Bβ€²β€²{\lx@inpgf@ignorespaces A^{\prime}\otimes B^{\prime\prime}}β‡“βŠ—f,g{\lx@inpgf@ignorespaces\Downarrow\otimes_{f,g}}β‡“βŠ—f,gβ€²{\lx@inpgf@ignorespaces\Downarrow\otimes_{f,g^{\prime}}}

    For any 1-morphisms f:Aβ†’Aβ€²f\colon A\to A^{\prime}, fβ€²:Aβ€²β†’Aβ€²β€²f^{\prime}\colon A^{\prime}\to A^{\prime\prime} and g:Bβ†’Bβ€²g\colon B\to B^{\prime} the 2-isomorphism ⨂f​fβ€²,g\bigotimes_{ff^{\prime},g} coincides with the pasting of ⨂f,g\bigotimes_{f,g} and ⨂f,gβ€²\bigotimes_{f,g^{\prime}} in a similar way.

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

Note that condition (v​i​i​i)(viii) and the invertibility of the 22-morphism βŠ—f,g\otimes_{f,g} imply that βŠ—1A,g=1g\otimes_{1_{A},g}=1_{g} and βŠ—f,1B=1f\otimes_{f,1_{B}}=1_{f}, for any f:Aβ†’Aβ€²f:A\to A^{\prime} and any g:Bβ†’Bβ€²g:B\to B^{\prime}.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

John C. Baez, Martin Neuchl

Original source: arXiv:q-alg/9511013v2