A 3-categorical perspective on G-crossed braided categories
AA 3-categorical perspective on G -crossed braided categories Corey Jones, David Penneys, and David ReutterSeptember 2, 2020
Abstract
A braided monoidal category may be considered a 3-category with one object and one 1-morphism. Inthis paper, we show that, more generally, 3-categories with one object and 1-morphisms given by elementsof a group G correspond to G -crossed braided categories, certain mathematical structures which haveemerged as important invariants of low-dimensional quantum field theories. More precisely, we show thatthe 4-category of 3-categories C equipped with a 3-functor B G → C which is essentially surjective onobjects and 1-morphisms is equivalent to the 2-category of G -crossed braided categories. This providesa uniform approach to various constructions of G -crossed braided categories. G - crossed braided categories [EGNO15, § § G -crossed braided categories arise from global symmetries in (1+1)D chiral conformal field theory ([Kir02,Kir01, M¨ug05]) and (2+1)D topological phases of matter [BBCW19], and as invariants of three-dimensionalhomotopy quantum field theories [Tur10, SW18]. They are a central object of study in the theory of G -extensions of fusion categories [ENO10, GNN09, CGPW16]. In this article we describe a higher categoricalapproach to G -crossed braided categories, which unifies these perspectives.When G is trivial, a G -crossed braided category is exactly a braided monoidal category. It is well-known that braided monoidal categories are ‘the same as’ 3-categories with exactly one object and one1-morphism [BD95, Table 21] and [CG11]. This is an instance of the Delooping Hypothesis [BS10, § k -fold degenerate ( n + k )-categories with k -fold monoidal n -categories. However, twicedegenerate 3-categories, 3-functors, transformations, modifications, and perturbations form a 4-category,whereas braided monoidal categories, braided monoidal functors, and monoidal natural transformationsonly form a 2-category. This discrepancy can be resolved by viewing ‘2-fold degeneracy’ as a structure on a3-category rather than a property , namely the structure of a 1-surjective pointing [BS10, Sec 5.6]. Explicitly,the Delooping Hypothesis may then be understood as asserting that the 4-category of 3-categories equippedwith 1-surjective pointings and pointing-preserving higher morphisms between them is in fact a 2-category(all hom 2-categories between 2-morphisms are contractible) and is equivalent to the 2-category of braidedmonoidal categories.Rather than pointing by something contractible (i.e., a point), we can also study ‘pointings’ by othercategories. In this article, we show that 1-surjective G - pointed G viewed as a 1-category B G with one object, are ‘the same as’ G -crossedbraided categories. Theorem A.
The 4-category Cat G of 1-surjective G -pointed -categories and pointing-preserving highermorphisms (see Definition 3.2) is equivalent to the 2-category G CrsBrd of G -crossed braided categories. Inparticular, every hom -category between parallel -morphisms in Cat G is contractible. In this article, by a 3 -category we mean an algebraic tricategory in the sense of [Gur13, Def 4.1], and by functor , transforma-tion , modification , and perturbation , we mean the corresponding notions of trihomomorphism, tritransformation, trimodification,and perturbation of [Gur13, Def 4.10, 4.16, 4.18, 4.21]. A functor between n -categories G → C is k - surjective if it is essentially surjective on objects and on j -morphisms for all1 ≤ j ≤ k . A k - surjective pointing on an n -category C is a k -surjective functor ∗ → C . We never actually work with a 4-category, as all our results can be stated and proven at the level of 2-categories. SeeRemark 3.3 for more details. a r X i v : . [ m a t h . C T ] S e p e prove Theorem A as follows. First, we show in Theorem 3.4 and Corollary 3.5 that 3 Cat G is 2-truncated by showing it is equivalent to the strict sub-2-category 3 Cat st G of strict G -pointed 3-categories,whose objects are Gray -categories with precisely one object, whose sets of 1-morphisms is exactly G , andcomposition of 1-morphisms is the group multiplication. Then in Theorem 4.1, we construct a strict 2-equivalence between 3 Cat st G and the strict 2-category G CrsBrd st of strict G -crossed braided categories. Finally,by [Gal17], every G -crossed braided category is equivalent to a strict one (see Definition 4.5 for more details),so that G CrsBrd is equivalent to its full 2-subcategory G CrsBrd st . In summary, we construct the followingzig-zag of strict equivalences, where the hooked arrows denote inclusions of full subcategories.3 Cat G Cat st G G CrsBrd st G CrsBrd ∼ Thm. 3.4 ∼ Thm. 4.1 ∼ [Gal17] (1)For the trivial group G = { e } , Theorem A specializes to the Delooping Hypothesis for twice degenerate3-categories (also see [CG11], which uses so called ‘iconic natural transformations’ rather than pointings). Corollary B.
The 4-category Cat { e } of 1-surjective pointed 3-categories is equivalent to the 2-category ofbraided monoidal categories. Our main theorem was inspired by, and is closely related to, the following two results: Passing from a G -pointed 3-category to the associated G -crossed braided category generalizes a result of [BGM19] whichconstructs G -crossed braided categories from group actions on 2-categories; see Example 1.11 for moredetails. A version of the construction of a G -pointed 3-category from a G -crossed braided category isdiscussed in [Cui19], and we use this construction in Section 4.3 to prove essential surjectivity of the 2-functor 3 Cat st G → G CrsBrd st . G -crossed braided categories from G -pointed -categories In the proof of Theorem A, we construct the equivalence 3
Cat G ∼ G CrsBrd by passing through appropriatestrictifications, resulting in the zig-zag (1) of strict equivalences. For the reader’s convenience, we now sketcha direct construction of a G -crossed braided category (defined in § G -pointed 3-category,without passing through strictifications.For a group G , we denote by B G the delooping of G , i.e., G considered as a 1-category with one object.Let C be a 3-category equipped with a 3-functor π : B G → C .To construct the G -crossed braided category, we will make use of the graphical calculus of Gray -categories(outlined in Section 2.2 below) and hence assume that C has been strictified to a Gray -category. Unpackingthe (weak) 3-functor into the data ( π, µ π , ι π , ω π , λ π , ρ π ) as described in Appendix A, the G -crossed braidedcategory C may be constructed as follows. Strictifying the situation slightly, we may assume that C has onlyone object, i.e. is a Gray -monoid, a monoid object in
Gray viewed as a monoidal 2-category, and that theunderlying 2-functor of π is strict (the unitor and compositor data π , π of π is trivial).We write g C := π ( g ) ∈ C , and we define C g := C (1 C → g C ) for each g ∈ G . We denote the tensorator µ πg,h ∈ C ( g C ⊗ h C → gh C ) and unitor ι π ∗ ∈ C (1 C → e C ) of π by trivialent and univalent vertices respectively µ πg,h = gh C g C h C ι π ∗ = e C . We denote 1-morphisms a g ∈ C (1 C → g C ) by shaded disks as follows: a g = g C b h = h C c k = k C . Since a k -category may be viewed as an n -category for n ≥ k with only identity r -morphism for n ≥ r > k , it makes senseto talk about an n -functor from a k -category to an n -category. In fact, [Gut19] justifies working with this graphical calculus even in the context of weak 3-categories. g, h ∈ G , we define a tensor product ( a g , b h ) (cid:55)→ a g ⊗ b h by g C h C × (cid:55)−→ gh C , (2)and we define the associator ⊗ gh,k ◦ ( ⊗ g,h × C k ) ⇒ ⊗ g,hk ◦ ( C g × ⊗ h,k ) by g C h C gh C k C ghk C ω πg,h,k = ⇒ ghk C φ − = ⇒ h C k C hk C g C ghk C , (3)where φ denotes the interchanger in C (see § C := ι π ∗ ∈ C e . Unitors ⊗ e,g ◦ ( i × − ) ⇒ id C g and ⊗ g,e ◦ ( − × i ) ⇒ id C g are given respectively by e C g C g C λ πg = ⇒ g C and e C g C g C φ = ⇒ e C g C g C ρ πg = ⇒ g C . We define a G -action F g : C h → C ghg − by F g (cid:32) h C (cid:33) := h C g C g − C gh C ghg − C e C . (4)The functors F g come equipped with natural isomorphisms ψ g : ⊗ ghg − ,gkg − ◦ ( F g × F g ) ⇒ F g ◦ ⊗ h,k builtfrom the coherence isomorphisms ω π , λ π , ρ π and interchangers between two black nodes and between a blacknode and a shaded disk. For example, ψ gb h ,c k is given by φ ⇒ ω,λ,ρ ⇒ φ ⇒ ω,λ,ρ ⇒ ω,λ,ρ ⇒ . (5)The tensorator µ g,h : F g ◦ F h ⇒ F gh and the unit map ι h : id C h → F e | C h are defined similarly. The G -crossedbraiding natural isomorphisms β g,h : a g ⊗ b h → F g ( b h ) ⊗ a g are also defined similarly using the interchangerisomorphism φ of C : g C h C φ ⇒ h C g C gh C ∼ = ⇒ h C g C g − C gh C ghg − C g C gh C e C . (6)3 .2 The delooping hypothesis Recall ( n + k )-categories form an ( n + k + 1)-category, whereas k -fold monoidal n -categories only form an( n + 1)-category. Thus one should not think of ‘ k -fold degeneracy’ as a property of an ( n + k )-category C but rather as additional structure , namely the structure of a ( k − -surjective pointing , and require allmorphisms and higher morphisms between these categories to preserve pointings [BS10, Sec 5.6]. Explicitly,the Delooping Hypothesis may then be understood as asserting that ( k − n + k )-categories and pointing-preserving higher morphisms form an ( n + 1)-category which is equivalent to the( n + 1)-category of k -fold monoidal n -categories. This is an instance of a more general higher categoricalprinciple. Definition 1.1.
We call a functor F : C → D of n -categories k - surjective if it is essentially surjective onobjects and parallel r -morphisms for r ≤ k . By convention, any functor is ( − Hypothesis 1.2.
Let G be a n -category. The full ( n + 1) -subcategory of the under- ( n + 1) -category n Cat G / on the k -surjective functors out of G is an ( n − k ) -category, i.e., all hom ( k + 1) -categories between parallel ( n − k ) -morphisms are contractible. Remark 1.3.
We expect hypothesis 1.2 is a direct consequence of more common assumptions on the ( n +1)-category of n -categories: Namely, following [BS10, § F : C → D between n -categories is j -monic if it is essentially surjective on k -morphisms for all k > j (including k = n + 1,where we interpret surjectivity to mean faithfulness on n -morphisms). By [BS10, Hypothesis 17], the (weak)fibers of such a j -monic functor are expected to be (possibly poset-enriched) ( j − Dually,a functor G : C → D between n -categories is j -epic if for every n -category E , the pre-composition functor n Cat ( p, E ) : n Cat ( D → E ) → n Cat ( C → E ) is ( n − − j )-monic. In particular, any j -surjective functor in thesense of Definition 1.1 is j -epic. Combining these observations, given a k -surjective functor p : G → C andan n -category E , the pre-composition functor n Cat ( p, E ) : n Cat ( C → E ) → n Cat ( G → E ) is ( n − − k )-monic.Hence, its fiber at a g ∈ n Cat ( G → E ) is a (possibly poset-enriched) ( n − − k )-category. But the fiber ofthe pre-composition functor n Cat ( p, E ) at g : G → E is the hom-category n Cat G / ( g, p ) of the under-categoryof n -categories under G . Therefore, the full subcategory of n Cat G / on the k -surjective functors is a (possiblyposet-enriched) ( n − k )-category. Moreover, essential (0-)surjectivity of p : G → C (also cf. Footnote 9) shouldimply that the pre-composition functor n Cat ( p, E ) : n Cat ( C → E ) → n Cat ( G → E ) is n -conservative , andhence that the enriching posets of the (weak) fibers of n Cat ( p, E ) are honest sets. Example 1.4 ( k -fold monoidal n -categories) . In the case where G = ∗ is the terminal category, Hypothe-sis 1.2 asserts that ( k − k -fold degenerate) pointed ( n + k )-categories form an ( n + 1)-category.The Delooping Hypothesis [BS10, § n + 1)-category with the ( n + 1)-category of k -fold monoidal n -categories.An important consequence of Hypothesis 1.2 is that it allows us to study certain higher-categoricalobjects, namely k -surjective functors and their higher transformations, using lower-categorical machinery.In many instances, there exist concrete descriptions of the resulting low-dimensional categories which havebeen developed and appear in mathematics and physics independently.As a concrete example, it is easier to describe and work with the 1-category of monoids and monoid ho-momorphisms than its unpointed variant, the 2-category of categories, functors, and natural transformations. This notion of k -surjectivity does not coincide with the one used in [BS10], where a functor is said to be k -surjective if itis essentially surjective on k -morphisms. Many of the definitions and statements in this remark are extensively developed in the setting of ( ∞ , n + 1 , n -categories. However, we are not able to use these ( ∞ , n +1 , n +1)-category of n -categories. For example, our j -monomorphismsdo not coincide with the ( ∞ , j -monomorphisms (in this context also known as ( j − -truncated morphisms ) asthe latter only fulfill essential surjectivity conditions with respect to invertible cells. A functor between n -groupoids is j -monic if and only if its fibers are ( j − n -categories, j -monomorphisms have truncated fibers but the converse is not necessarily true. More generally, j -surjective functors are expected to correspond to ‘strong j -epimorphisms’ [BS10, Hypothesis 21], thatis, functors that have the left lifting property with respect to j -monomorphisms. Since the ( n + 1)-category of n -categories hasfinite limits, any such ‘strong j -epimorphism’ is in particular a j -epimorphism; see [BS10, Sec 5.5]. An n -functor F : C → D is n -conservative if it reflects n -isomorphisms, i.e., for every n -morphism α : f ⇒ g in D for which F ( α ) is an isomorphism, it follows that α is an isomorphism. G n + k = 0 n + k = 1 n + k = 2 n + k = 3 k = − ∅ k = 0 ∗ point monoid monoidal category monoidal 2-category k = 1 B G normal subgroup of G G -crossed monoid G -crossed braided categoryFigure 1: ( n + k )-categories equipped with k -surjective functors from G form an n -categoryIn this article, we focus on 1-surjective functors from the delooping B G of G , i.e., the 1-category withone object and endomorphisms G . Hypothesis 1.5 ( G -crossed delooping) . For n ≥ − , the ( n + 3) -category of 1-surjective functors from B G into ( n + 2) -categories is equivalent to the ( n + 1) -category of G -crossed braided n -categories. While we do not present a general definition of G -crossed braided n -category here, this hypothesisis a desideratum for any such definition (such as for example via M¨uller and Woike’s ‘little bundles’ op-erad [MW19]). Observe that the k = 1 version of the delooping hypothesis follows as a consequence for thetrivial group G = { e } .In the following, as a warm-up to our main theorem, we discuss the low-dimensional versions ( n = 0 and n = −
1) of Hypothesis 1.5 appearing in the last row of Figure 1.
Example 1.6 ( G -crossed monoids as G -pointed 2-categories) . The 3-category G of 2-categories C equipped with 1-surjective 2-functors B G → C is equivalent to the 1-category of G -crossed monoids , or‘ G -crossed braided 0-categories’, defined below. Explicitly, the 2-category G has • objects ( C , π C ) where C is a 2-category and π C : B G → C is a 1-surjective 2-functor, • A, α ) : ( C , π C ) → ( D , π D ) where A : C → D is a 2-functor and α : π D ⇒ A ◦ π C is aninvertible 2-transformation, • η, m ) : ( A, α ) ⇒ ( B, β ) where η : A ⇒ B is a 2-transformationB G CD . π C π D Bβ m (cid:86) B G CD π C π D Aα Bη • p : ( η, m ) (cid:86) ( ζ, n ) where p : η (cid:86) ζ is a 2-modification such that π D A ◦ π C B ◦ π C . αβ ζ ◦ π C n = π D A ◦ π C B ◦ π C αβ η ◦ π C ζ ◦ π C m p ◦ π C On the other hand, a natural decategorification of a G -crossed braided monoidal category is a G -gradedmonoid M = (cid:113) g ∈ G M g together with a group homomorphism π M : G → Aut( M ) such that the followingaxioms are satisfied: • π Mg ( m h ) ∈ M ghg − for all g ∈ G and m h ∈ M h , and5 m g · n h = π Mg ( n h ) · m g for all m ∈ M g and n h ∈ M h .We call such a pair ( M, π M ) a G -crossed monoid , or a ‘ G -crossed braided 0-category’. Morphisms ( M, π M ) → ( N, π N ) are G -graded monoid homomorphisms that intertwine the G -actions.To see that G is equivalent to the category of G -crossed monoids, we mirror our proof of Theorem A.One first shows that G is equivalent to the 1-category st G with • objects strict monoidal categories C whose set of objects is { g C } g ∈ G with 1 C = e C and tensor productgiven by the group multiplication, and • morphisms A : C → D are strict monoidal functors such that A ( g C ) = g D for all g ∈ G .The equivalence from st G to G -crossed monoids is given by taking hom from 1 C . We set M g := C (1 C → g C ), and the multiplication on M := (cid:113) g ∈ G M g is ⊗ in C . The G -action π M : G → Aut( M ) is givenby conjugation: π Mg ( m h ) := id g C ⊗ m h ⊗ id g − C ∈ M ghg − = C (1 C → ghg − C ) . One then verifies the G -crossed braiding axiom by a G -graded version of Eckmann-Hilton. A 1-morphism A ∈ st G ( C → D ) yields a G -graded monoid homomorphism by restricting to M g = C (1 C → g C ). This monoidhomomorphism is compatible with the G -actions by strictness of A . Finally, one verifies this construction isan equivalence of categories. Example 1.7 (Normal subgroups as G -pointed 1-categories) . The 2-category
Cat G of 1-categories C equippedwith 1-surjective functors B G → C is equivalent to the set of normal subgroups of G (which we may thinkof as the ‘0-category of G -crossed braided ( − Cat G has • objects ( C , π C ) where C is a category and π C : B G → C is a 1-surjective functor, • A, α ) : ( C , π C ) → ( D , π D ) where A : C → D is a functor and α : π D ⇒ A ◦ π C is a naturalisomorphism, and • η : ( A, α ) ⇒ ( B, β ) are natural transformations η : A ⇒ B such thatB G CD . π C π D Bβ = B G CD π C π D Aα Bη
It is straightforward to verify that this 2-category is equivalent to a set. Moreover, up to equivalence, thedata of a 1-surjective functor π C : B G → C is equivalent to the data of a normal subgroup of G , obtainedas the kernel of the surjective group homomorphism G → Aut C ( π C ( ∗ )). Hence, the 2-category Cat G isequivalent to the set of normal subgroups of G .Employing ‘categorical negative thinking’ as in [BS10, § G as a ‘ G -crossed braided ( − G -crossed braided ( − − § G -graded ( − G → Bool = ( { T, F } , ∧ ), where Bool denotes the Booleans which one may think of as the commutative monoid(symmetric monoidal 0-category) of ( − G -graded ( − G . This correspondence may be seen a further decategorifiedanalogue of our construction. Indeed, given ( C , π C ), the corresponding monoid homomorphism G → Bool isexactly given by g (cid:55)→ C (id π C ( ∗ ) → π C ( g )), where the latter is the Boolean which is true if id π C ( ∗ ) = π ( g ) andfalse otherwise. Example 1.8 (Shaded monoidal algebras) . In [GMP +
18, Defn. 3.18 and 3.26], the authors define the notionof a shaded monoidal algebra, which is an operadic approach to 2-categories with a chosen set of objects anda set of generating 1-morphisms. The statements of [GMP +
18, Thm. 3.21 and Cor. 3.23] can be understood6s examples of Hypothesis 1.2. Indeed, equipping a 2-category with a set of objects and a generating set of1-morphisms is equivalent to pointing by the free category on a graph Γ. Hence the 3-category of 1-surjectiveΓ-pointed 2-categories is equivalent to the 1-category of Γ-shaded monoidal algebras.
Remark 1.9 (Planar algebras) . Expanding on Example 1.8, Jones’ planar algebras [Jon99] reflect thephilosophy of Hypothesis 1.2. A 2-shaded planar algebra may be understood as a pivotal 2-category C withprecisely two objects ‘unshaded’ and ‘shaded’ together with a generating dualizable 1-morphism betweenthem with loop modulus δ . This choice of generating 1-morphism may be understood as equipping C with a1-surjective pivotal functor π C : T LJ ( δ ) → C , where T LJ ( δ ) is the free spherical 2-category on a dualizble1-morphism with quantum dimension δ . By (a pivotal version of) Hypothesis 1.2, such pivotal 2-categoriesand functors preserving this ‘TLJ-pointing’ actually form a 1-category, which is equivalent to the 1-categoryof 2-shaded planar algebras and planar algebra homomorphisms.Another instance of this philosophy appears in [HPT16] which shows the 2-category ModTens ∗ of pointedmodule tensor categories over a braided pivotal category V (defined in [HPT16, § ModTens ∗ is equivalent to the 1-category of anchored planar algebras in V . Our main theorem asserts an equivalence between 1-surjective functors B G → C and G -crossed braidedcategories. Starting with an arbitrary 3-functor π : B G → C we may factor it through a 1-surjective functor π (cid:48) : B G → C (cid:48) (where C (cid:48) is the subcategory of C with objects and 1-morphisms in the essential image of π , and all 2- and 3-morphisms between them) and apply our construction from § G -crossedbraided category. Most examples discussed below arise in this way. Example 1.10 (Delooped braided monoidal categories) . Let B be a braided monoidal category, and denotethe corresponding 3-category with one object and one 1-morphism by B B . Observe that every weak 3-functorB G → B B is automatically 1-surjective. Such 3-functors B G → B B factor through the maximal sub-3-groupoid B B × of B B , delooping the braided monoidal groupoid B × of invertible objects and morphisms in B . Assuming the homotopy hypothesis for algebraic trigroupoids, such functors correspond to homotopyclasses of maps from the classifying space B G to the 1-connected homotopy 3-type B B × .Such 1-connected 3-types are completely determined by the abelian group π (B B × ) = Inv( B ) of iso-morphism classes of invertible objects of B , the abelian group π (B B × ) = Aut(1 B ) of automorphisms ofthe tensor unit 1 B of B , and the k –invariant q ∈ H ( K (Inv( B ) , , Aut(1 B )) ∼ = Quad(Inv( B ) , Aut(1 B )), thegroup of quadratic functions on Inv( B ) valued in Aut(1 B ) [EM54], which is explicitly given by the quadraticfunction q : Inv( B ) → Aut(1 B ) given by q ( b ) := ev b ◦ β B b,b − ◦ coev b . Here, ev b : b − ⊗ b → I and coev b : I → b ⊗ b − denote a choice of pairing between b and b − and β b,b − : b ⊗ b − → b − ⊗ b denotes the braiding.By [Mac52, EM54], the group Quad(Inv( B ) , Aut(1 B )) is further isomorphic to the group H ab (Inv( B ) , Aut(1 B ))of abelian -cocycles ( α, β ), consisting of pairs of a group 3-cocycle α : Inv( B ) → Aut(1 B ) and a certain‘ α -twisted-bilinear’ form β : Inv( B ) → Aut(1 B ). We refer the reader to [Bra20, (1.2) and §
11] for moredetails.By the obstruction theory for homotopy classes of maps into such Postnikov towers (cf. [ENO10, Theorem1.3]), it follows that, up to natural isomorphism, 3-functors B G → B B correspond to the following data: • a 2-cocycle µ ∈ Z ( G, Inv( B )), up to coboundary; • a 3-cochain ω ∈ C ( G, Aut(1 B )) such that dω = ( α, β ) ∗ µ , where ( α, β ) ∗ µ ∈ Z ( G, Aut(1 B )) isthe 4-cocycle in the image of the Pontryagin-Whitehead morphism ( α, β ) ∗ : H ( G, Inv( B )) → The article [Lac11] constructs a model category structure on the category of
Gray -categories and
Gray -functors which restrictto a model structure on
Gray -groupoids. Even though it is shown that the corresponding homotopy category of
Gray -groupoidslocalized at the
Gray -equivalences is equivalent to the category of homotopy 3-types and homotopy classes of continuous maps,to the best of our knowledge, it has not yet been shown that this category is also equivalent to the 1-category whose objectsare
Gray -groupoids (or algebraic trigroupoids) and whose morphisms are natural equivalence classes of weak 3-functors. Under the isomorphism H ( K (Inv( B ) , , Aut(1 B )) ∼ = H ab (Inv( B ) , Aut(1 B )), the abelian 3-cocycle ( α, β ) correspondsto a map ( α, β ) ∗ : K (Inv( B ) , → K (Aut(1 B ) , α, β ) ∗ : H ( G, Inv( B )) → H ( G, Aut B (1 B )) is simply given by postcomposing a class ω : B G → K (Inv( B ) ,
2) with ( α, β ) ∗ . ( G, Aut B (1 B )) for the k -invariant ( α, β ) ∈ H ( K (Inv( B ) , , Aut(1 B )). An explicit expression forthe 4-cocycle ( α, β ) ∗ µ ∈ Z ( G, Aut(1 B )) is given by( α, β ) ∗ µ ( g, h, k, (cid:96) ) = β µ k,(cid:96) ,µ g,h α − µ ghk,(cid:96) ,µ gh,k ,µ g,h α µ ghk,(cid:96) ,µ g,hk ,µ h,k α − µ g,hk(cid:96) ,µ hk,(cid:96) ,µ h,k α µ g,hk(cid:96) ,µ h,k(cid:96) ,µ k,(cid:96) α − µ gh,k(cid:96) ,µ g,h ,µ k,(cid:96) α µ gh,k(cid:96) ,µ k,(cid:96) ,µ g,h (7)This explicit expression can also be obtained, up to conventions, by taking the trivial G -action in[CGPW16, Eq. (5.6)].In fact, after strictifying B to a strict braided monoidal category, so that B B is a Gray -category, thiscohomological data may be directly read off from the components of the weak 3-functor π : B G → B B , usingnotation from Appendix A, as follows: We may assume the underlying 2-functor of π is strictly unital, i.e., π g = id B for all g ∈ G . By (F-I).ii, this implies π g,h = id B for all g, h ∈ G . We write µ g,h := µ πg,h ∈ Inv( B ).By (F-II).iii, µ π id g , id h = id ∈ End( µ g,h ). Using the isomorphism ω πg,h,k : µ gh,k ⊗ µ g,h → µ g,hk ⊗ µ h,k , µ descends to a 2-cocycle in Z ( G, Inv( B )). To translate ω π into a 3-cochain in C ( G, Aut B (1 B )), we let C be a skeletalization of B × . In C , we may identify all automorphism spaces of C with Aut(1 B ), andhence recover the associator α in C as an element of Z (Inv( C ) , Aut(1 B )), and descend the isomorphisms ω πg,h,k : µ gh,k ⊗ µ g,h → µ g,hk ⊗ µ h,k to a 3-cochain ω in C ( G, Aut B (1 B )). Unpacking (F-1) leads to dω = ( α, β ) ∗ µ .We can now explicitly describe the G -crossed braided category resulting from our construction from thiscohomological data by interpreting the diagrams (2), (3), (4), (5), (6). • All g -graded components are B , • the monoidal structure is given by interpreting (2): a g ⊗ b h := µ g,h ⊗ a g ⊗ b h , with associator given byinterpreting (3): µ gh,k ⊗ µ g,h ⊗ a g ⊗ b h ⊗ c k ω πg,h,k ⊗ id −−−−−−→ µ g,hk ⊗ µ h,k ⊗ a g ⊗ b h ⊗ c k id ⊗ β − ag,µh,k ⊗ id −−−−−−−−−−→ µ g,hk ⊗ a g ⊗ µ h,k ⊗ b h ⊗ c k . • the G -action is given by interpreting (4): F g ( b h ) := µ gh,g − ⊗ µ g,h ⊗ b h ⊗ µ − g,g − , with tensorator ψ g given by interpreting (5). • the G -crossed braiding is given by interpreting (6).One can view the resulting G -crossed extension as a twisting of the trivial extension by a 2-cocycle[ENO10, Pf. of Thm. 1.3]. When B is fusion, this is a G -crossed zesting of the trivial G -crossed extension B (cid:2) Vec( G ) of B [DGP + Example 1.11 (Generalized relative center construction) . The article [BGM19] shows that every (weak) G -action on a 2-category may be strictified to a strict G -action on a strict 2-category, encoded by a grouphomomorphism π : G → Aut st ( B ), where Aut st ( B ) is the group of strict 2-equivalences of B which admitstrict inverses. From such a strict G -action, the authors then construct a G -crossed braided monoidal cate-gory Z G ( B ) whose g -graded component is the category of pseudonatural transformations and modifications PseudoNat (id B ⇒ π ( g )) . Since π ( e ) = id B , the trivial graded component is the Drinfeld center Z ( B ). Thisconstruction generalizes the construction of the relative center Z C ( D ) of a G -extension D of a fusion category C ; by [GNN09], Z C ( D ) is a G -crossed braided fusion category whose trivial graded component is Z ( C ).Our construction of a G -crossed braided monoidal category from a G -pointed 3-category may be un-derstood as a generalization of [BGM19] from G -actions on 2-categories, encoded by 3-functors B G → Cat from B G into the 3-category of 2-categories, to arbitrary 3-functors B G → C . In particular, we show inSection 3.2 that we may strictify a 1-surjective weak 3-functor B G → C to a Gray -functor B G → C (cid:48) into a Gray -category C (cid:48) equivalent to C , and construct a G -crossed braided category from this data. Unpacking (F-1) in C introduces six additional associator terms, one for every vertex of the hexagon commutative diagram.As these terms correspond to the two different ways to associate each of the vertex 1-cells in (F-1), the associators alternate α and α − around the diagram. The resulting 12 sided commutative diagram exactly reproduces, up to conventions, a simplificationof [CGPW16, Fig. 1] where the G -action is trivial. Five of these 12 terms give dω , while the other 7 terms give (7). Since thediagram commutes, we have dω = ( α, β ) ∗ µ as desired. xample 1.12 ( G -crossed extension theory for braided fusion categories) . Let C be a braided fusion category,and consider the monoidal 2-category Mod ( C ) of finite semisimple module categories [Gre10, DR18]. Givena monoidal 2-functor π : G → Mod ( C ), our construction produces the G -crossed braided fusion category (cid:77) g ∈ G Hom( C C → π ( g ) C ) ∼ = (cid:77) g ∈ G π ( g )which is a G -crossed braided extension of the e -graded piece End Mod( C ) ( C C ) ∼ = C . This G -crossed braidedcategory is equivalent to the G -crossed extension constructed in [ENO10] (which moreover gives an alternateproof that faithful G -crossed extensions of braided fusion categories are in fact classified by monoidal 2-functors G → Mod ( C )). Example 1.13 (Permutation crossed extensions) . Let C be a symmetric monoidal 3-category, and let A bean object of C . Then there exists a monoidal 2-functor π : S n → End( A (cid:2) n ), where (cid:2) denotes the symmetricmonoidal product in C . Our construction produces a S n -crossed braided category whose trivially graded pieceis End(id A (cid:2) n ). For example, if A is an object in the 3-category of fusion categories [DSPS13, Hau17, JFS17],there is an equivalence End(id A (cid:2) n ) ∼ = Z ( A (cid:2) n ) ∼ = Z ( A ) (cid:2) n , where Z ( A ) is the Drinfeld center of A , and the resulting S n -crossed braided category is what is known as a permutation crossed extension of Z ( A ) (cid:2) n . More generally, the article [GJ19] shows that such permutationcrossed extensions of C (cid:2) n exist for any modular tensor category. Example 1.14 (Conformal nets) . Consider the symmetric monoidal 3-category of coordinate free con-formal nets CN defined in [DH12, BDH15, BDH17, BDH19, BDH18]. A 3-functor B G → CN amounts toa conformal net A ∈ CN together with a generalized action of G on the net A by invertible topologicaldefects. Applied to such a 3-functor, our construction produces a G -crossed braided category whose trivialgraded component is the braided category End CN (1 A ) = Rep ( A ) of (super-selection) sectors [BDH15, Sec1.B] of A . We expect this generalizes a construction of M¨uger [M¨ug05], which produces a G -crossed braidedcategory from the action of global symmetries on a coordinatized conformal net. However, it is difficultto compare these two G -crossed braided categories, since it is not obvious how to construct a symmetricmonoidal 3-category of coordinatized conformal nets. Example 1.15 (Topological phases) . The collection of (2+1)D gapped topological phases is expected toform a 3-category [GJF19b, GJF19a]. Given a global, onsite symmetry, there is an associated G -crossedbraided category of twist defects [BBCW19]. Our construction can be understood as a direct generalizationof this heuristic. Indeed, our pictures and arguments can be viewed as a more mathematically precise versionof the arguments and structure given in the physical context (e.g., see [BBCW19, Fig. 7]). Example 1.16 (Homotopy quantum field theory) . Homotopy quantum field theories are topological fieldtheories on bordisms equipped with a map to a fixed target space. If this target space is the classifyingspace B G of a finite group G , such field theories are also known as G -equivariant field theories . Followingthe cobordism hypothesis [BD95, Lur09b], such a fully extended (framed) 3-dimensional G -equivariant topo-logical field theory valued in a fully dualizable symmetric monoidal 3-category corresponds to a 3-functorB G → C (i.e. a fully dualizable object A in C equipped with an ‘internal G -action’, given by a monoidal2-functor X : G → End C ( A )). It therefore follows from Theorem A that to any such field theory, there is anassociated G -crossed braided category.In particular, if Fus is the 3-category of fusion categories introduced in [DSPS13], we expect the G -crossed braided category constructed via Theorem A from a fully extended G -equivariant three-dimensionalfield theory valued in Fus to coincide with the G -crossed braided category constructed in [SW18] by evaluatingthe field theory on ( G -structured) circles. In particular, if G is trivial, this recovers the construction of theDrinfeld center of a fusion category A as FusCat ( A A A ⇒ A A A ). The notion of tricategory used in [DH12, BDH18], namely an internal bicategory in
Cat , is expected, but not proven to beequivalent to the notion of algebraic tricategory [Gur13] used in the present article. .4 Outline Section 2 contains basic definitions and a brief introduction to the graphical calculus of
Gray -monoids usedthroughout.Section 3 proves various strictification results for 1-surjective pointed 3-categories ( § § § § § Cat G (Definition 3.2) is equivalent to itsstrict sub-2-category 3 Cat st G (Corollary 3.5).Section 4 defines the 2-category G CrsBrd of G -crossed braided categories ( § G CrsBrd st , constructs the strict 2-functor 3 Cat st G → G CrsBrd st ( § § G -pointed 3-category, suchas linearity and rigidity, may be translated across the equivalence of Theorem A to the resulting G -crossedbraided category.Appendix A unpacks the definitions of (weak) 3-functors, transformations, modifications and perturba-tions between Gray -monoids in terms of the graphical calculus.Appendices B and C contain most of the coherence proofs from Sections 3 and 4, respectively.
Acknowledgements
The authors would like to thank Shawn Cui, Nick Gurski, Niles Johnson, and Andr´eHenriques for helpful conversations.This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the authors DP and DR were in residence at the Mathematical Sciences Research Institutein Berkeley, California, during the Spring 2020 semester. CJ was supported by NSF DMS grant 1901082.DP was supported by NSF DMS grant 1654159. DR is grateful for the financial support and hospitality ofthe Max Planck Institute for Mathematics where part of this work was carried out.
In this article, by a 3 -category we mean an algebraic tricategory in the sense of [Gur13, Def 4.1], and by functor , transformation , modification , and perturbation , we mean the corresponding notions of trihomomor-phism, tritransformation, trimodification, and perturbation of [Gur13, Def 4.10, 4.16, 4.18, 4.21]. We includeAppendix A below which unpacks the full definitions of these notions for Gray -monoids using the graphicalcalculus discussed in § Remark 2.1.
In this article, we use the term invertible as a property, i.e., the existence of a coherentinverse. Indeed, by [Gur12], every invertible 1-morphism (biequivalence) in a 3-category is part of a biadjointbiequivalence, and every invertible 2-morphism is part of an adjoint equivalence. Moreover, there is acontractible space of choices for these coherent inverses. Whenever we need to make such choices, we willrefer back to this remark.
Gray -categories and
Gray -monoids
In this section, we give a terse definition of
Gray -category and
Gray -monoid, and a brief discussion on thediagrammatic calculus for
Gray -monoids. We refer the reader to [Gur06] for a more detailed treatment of
Gray -categories and to [BMS12, § Definition 2.2.
The symmetric monoidal category
Gray is the 1-category of strict 2-categories and strict2-functors equipped with the Gray monoidal structure [Gur06, § Gray -category is a category enrichedin
Gray in the sense of [Kel05]. A
Gray -monoid is a monoid object in
Gray . Given a
Gray -monoid C , its delooping B C is the Gray -category with one object and endomorphisms C .We now unpack the notion of Gray -monoid from Definition 2.2.
Notation 2.3.
Given a
Gray -monoid C , we refer to its objects, 1-morphisms, and 2-morphisms as 0-cells,1-cells, and 2-cells respectively in order to distinguish these basic components of C from morphisms in anambient category in which C lives. 10he remarks and warning below are adapted directly from [DR18]. Remark 2.4.
Unpacking Definition 2.2, a
Gray - monoid consists of the following data:(D1) a strict 2-category C , where composition of 1-morphisms is denoted by ◦ and composition of 2-morphisms is denoted by ∗ ;(D2) an identity C ∈ C ;(D3) strict left and right tensor product L a = a ⊗ − and R a = − ⊗ a for each object a ∈ C : L a = a ⊗ − : C → C R a = − ⊗ a : C → C , (D4) an interchanger φ x,y for each pair of 1-cells x : a → b and g : c → d : φ x,y : ( x ⊗ id d ) ◦ (id a ⊗ y ) ⇒ (id b ⊗ y ) ◦ ( x ⊗ id c )subject to the following conditions:(C1) left and right tensor product agree: for all objects a, b ∈ C , L a b = R b a = a ⊗ b ;(C2) tensor product is strictly unital and associative: L C = id C = R C L a L b = L a ⊗ b R b R a = R a ⊗ b L a R b = R b L a ;(C3) the interchanger φ respects identities, i.e., for a 0-cell A ∈ C and a 1-cell f : C → D , φ f, id A = id f ⊗ A φ id A ,f = id A ⊗ f (C4) the interchanger φ respects composition, i.e., for x : a → a (cid:48) , x (cid:48) : a (cid:48) → a (cid:48)(cid:48) , y : b → b (cid:48) and y (cid:48) : b (cid:48) → b (cid:48)(cid:48) , φ x (cid:48) ◦ x,y = ( φ x (cid:48) ,y ◦ ( x ⊗ id b )) ∗ (( x (cid:48) ⊗ id b (cid:48) ) ◦ φ x,y ) φ x,y (cid:48) ◦ y = ((id a (cid:48) ⊗ y (cid:48) ) ◦ φ x,y ) ∗ ( φ x,y (cid:48) ◦ (id a ⊗ y ))(C5) the interchanger φ is natural, i.e., for 1-cells x, x (cid:48) : a → a (cid:48) , y, y (cid:48) : b → b (cid:48) and 2-cells α : x ⇒ x (cid:48) , β : y ⇒ y (cid:48) , φ x (cid:48) ,y ∗ (( α ⊗ id b (cid:48) ) ◦ (id a ⊗ y )) = ((id a (cid:48) ⊗ y ) ◦ ( α ⊗ id b )) ∗ φ x,y φ x,y (cid:48) ∗ (( x ⊗ id b (cid:48) ) ◦ (id a ⊗ β )) = ((id a (cid:48) ⊗ β ) ◦ ( x ⊗ id b )) ∗ φ x,y (C6) the interchanger φ respects tensor product, i.e., for x : a → a (cid:48) , y : b → b (cid:48) and z : c → c (cid:48) , φ id a ⊗ y,z = id a ⊗ φ y,z φ x ⊗ id b ,z = φ x, id b ⊗ z φ x,y ⊗ id c = φ x,y ⊗ id c A Gray -monoid is called linear if the underlying 2-category is linear and for all objects a the functors a ⊗ − and − ⊗ a are linear. 11 arning 2.5 (Horizontal composition of 1-morphisms) . We warn the reader that the tensor product in a
Gray -monoid does not provide a unique definition of the tensor product of two 1-cells. Given x : a → b and y : c → d , we define x ⊗ y := ( x ⊗ id d ) ◦ (id a ⊗ y ) ; (8)this convention is known as nudging [GPS95, § Gray -monoid C as described in Definition 2.4 givesrise to an ( opcubical cf [Gur13, § C [Gur13, Thm. 8.12]. Remark 2.6 (Strictification for monoidal 2-categories) . By the strictification for tricategories from [GPS95]or [Gur13, Cor. 9.16], every (linear) weakly monoidal weak 2-category admits a monoidal 2-equivalence to a(linear)
Gray -monoid of the form in Definition 2.4.
Gray -monoids
Gray -categories admit a graphical calculus of surfaces, lines, and vertices in three-dimensional space. Werefer the reader to [BMS12, § Gray -monoids. Our exposition below follows [Bar14].The 0-cells of our strict 2-category C (D1) are denoted by strands in the plane a and the identity 0-cell 1 C (D2) is denoted by the empty strand. The 1-cells are denoted by coupons betweenlabelled strands x : a → b abx The composition of 1-cells is denoted by vertical stacking of such diagrams.The strict tensor product ⊗ is denoted by horizontal juxtaposition. For example, the tensor productfunctors L a and R a (D3) are denoted by placing a strand labelled by a to the left or right respectively. L a ( x : b → c ) := id a ⊗ x = bca x R a ( x : b → c ) := x ⊗ id a = bc ax Given x : a → b and y : c → d , we define their tensor product using the nudging convention from Warning2.5. x ⊗ y := ( x ⊗ id d ) ◦ (id a ⊗ y ) = ab x cdy Observe that no two coupons ever share the same vertical height.The 2-cells are inherently 3-dimensional, and can be thought of as ‘movies’ between our 2-dimensionalstring diagrams. Rather than drawing 2-cells, we denote them by arrows ⇒ between diagrams correspondingto their source and target 1-cells. For example, the interchanger φ x,y from (D4) is simply denoted by ab x cdy φ x,y = ⇒ ab x cdy . otation 2.7. When working with
Gray -monoids, one often needs to whisker 2-cells between 1-cells, andthe notation can quickly become cumbersome. Instead, we use the convention of a dashed box when weapply a 2-cell locally to a 1-cell, and we simply label the whiskered 2-cell by the name of the locally applied2-cell. Later on, we will draw commutative diagrams whose vertices are 1-cells. When we want to apply two2-cells locally in different places to the same 1-cell, we will use two dashed boxes with different colors, usuallyred and blue. When one of these two 2-cells is applied to the entire diagram, we do not use a dashed box,and we only use one dashed box of another color, usually red. As an explicit example, the second equationin (C4) in string diagrams is given by: ab x cdyz ab x cdyzab x cdyzφ x,y φ x,y ◦ z φ x,z For the convenience of the reader, we have included Appendix A which unpacks the notions of 3-functor,transformation, modification, and perturbation for
Gray -monoids using this graphical calculus. G -pointed 3-categories Let G be a group. We recall from § G denoted the delooping of G , i.e., G considered as a 1-categorywith one object. As discussed at the beginning of §
2, the terms n - category and n - functor for n ≤ n -categories and weak n -functors. Observe that since a k -category may be viewed as an n -category for n ≥ k with only identity higher morphisms, we may talk about an n -functor from a k -categoryto an n -category. Recall from Remark 2.1 that we use the adjective invertible for (bi)adjoint (bi)equivalences. Definition 3.1.
A 3-functor A : C → D is 1- surjective if it is essentially surjective on objects and if for everypair of objects c , c of C , the 2-functors A c ,c : C ( c → c ) → D ( A ( c ) → A ( c )) are essentially surjectiveon objects. Definition 3.2.
Let G be a group. We define the 4-category Cat G of G -pointed 3-categories to be thefull sub-4-category of the under-category 3 Cat B G/ on the 1-surjective 3-functors B G → C . Explicitly, this4-category can be described as follows: • objects are 3-categories C equipped with a 1-surjective 3-functor π C : B G → C . • A, α ) : ( C , π C ) → ( D , π D ) are pairs where A : C → D is a 3-functor and α : π D ⇒ A ◦ π C is an invertible natural transformation; All results in this section can be stated and proved at the level of various 2-categories of ( k − k -morphismsand equivalence classes of ( k + 1)-morphisms of 3 Cat G ; we therefore will not show that 3 Cat G forms a 4-category — and in factwill not even choose any definition of 4-category. We only use the conceptual idea of a 4-category as an underlying organizationalprinciple for our results. η, m ) : ( A, α ) ⇒ ( B, β ) are pairs where η : A ⇒ B is a natural transformation and m isan invertible modificationB G CD . π C π D Bβ m (cid:86) B G CD π C π D Aα Bη . (9) • p, ρ ) : ( η, m ) (cid:86) ( ζ, n ) are modifications p : η (cid:86) ζ together with an invertible perturba-tion ρ : π D A ◦ π C B ◦ π C . αβ ζ ◦ π C n ρ π D A ◦ π C B ◦ π C αβ η ◦ π C ζ ◦ π C m p ◦ π C (10) • ξ : ( p, ρ ) ( q, δ ) are perturbations ξ : p q satisfying β ( η ◦ π C ) ∗ α ( ζ ◦ π C ) ∗ α. mn ( q ◦ π C ) ∗ αδ = β ( η ◦ π C ) ∗ α ( ζ ◦ π C ) ∗ α mn ( p ◦ π C ) ∗ αρ ( ξ ◦ π C ) ∗ α ( q ◦ π C ) ∗ α (11) Remark 3.3.
As stated, Definition 3.2 and Theorem 3.4 below assume the existence of a (weak) 4-category3
Cat of algebraic tricategories, trifunctors, tritransformations, modifications, and perturbations which hasthe appropriate homotopy bicategories between parallel k -morphisms. Assuming the existence of such a 4-category 3 Cat , we may define 3
Cat G as a certain full sub-4-category of the under-category as in Definition 3.2.In Theorem 3.4 we show, working a bicategory at a time, that this 4-category 3 Cat G is equivalent to a sub-4-category 3 Cat pt G with only identity 3- and 4-morphisms, and is hence equivalent to a bicategory. Afterhaving established Theorem 3.4, we will from then on only work with this bicategory 3 Cat pt G .Unfortunately, to the best of our knowledge, such a 4-category 3 Cat has not yet been constructed in anyof the established models of weak 4-category. However, none of the results in this article truly depend onthe specifics of 4-categories, and 4-categories only appear as a convenient conceptual organizing tool.The reader uncomfortable with this sort of model-independent argument may unpack the statement ofour main Theorem A to assert the following:(1) For a pair of parallel ‘1-morphisms’ as in Definition 3.2, the bicategory of 2-morphisms, 3-morphismsand 4-morphisms between them is equivalent to a set.(2) The bicategory of objects, 1-morphisms and 2-morphisms up to invertible 3-morphisms of Definition 3.2is equivalent to the 2-category of G -crossed braided categories. Theorem 3.4.
The 4-category Cat G is equivalent to the 4-subcategory Cat pt G where • objects (B C , π C ) are those objects of Cat G for which the -category is a Gray -category with one object,and hence given by B C for some Gray -monoid C , and for which π C : B G → B C is a strictly -bijective Gray -functor. Equivalently, an object is a
Gray -monoid C whose set of -cells is { g C := π C ( g ) } g ∈ G andcomposition of 0-cells given by group multiplication. • -morphisms ( A, α ) : (B C , π C ) → (B D , π D ) satisfy: A ( g C ) = g D for all g ∈ G , – the adjoint equivalence µ : ⊗ D ◦ ( A × A ) ⇒ A ◦ ⊗ C satisfies µ g C ,h C = id gh D : g D ⊗ h D ⇒ gh D , – the adjoint equivalence ι A = ( ι A ∗ , ι A ) : I D ⇒ A ◦ I C satisfies ι A ∗ = id e D , and ι A = A e , – the associators and unitors ω A , (cid:96) A , r A are identities, – α ∗ = e D and α g = id g D , and α id g = A g , – α = id id e D and α g,h = id id gh D for all g, h ∈ G . • -morphisms ( η, m ) : ( A, α ) ⇒ ( B, β ) satisfy η ∗ C = e D , η g C = id g D , η = id id e D and η g C ,h C = id id gh D and m ∗ = e D , m g = id id g D . That is, m is the identity modification. • ( p, ρ ) : ( η, m ) (cid:86) ( ζ, n ) satisfy p ∗ C = id e D , p g C = id id g D , and ρ ∗ = id id e D . That is, thereare only identity 3-morphisms. • ξ : ( p, ρ ) ( q, δ ) satisfy ξ ∗ C = id id e D . That is, the only 4-endomorphism of an identity3-morphism is the identity.Proof. In § Cat G is equivalent to one of the desired form in 3 Cat pt G . All proofs in thesefurther subsections amount to checking the appropriate coherences for 3-functors, 3-natural transformations,3-modifications, and 3-perturbations outlined in Appendix A and are deferred to Appendix B. We signifywhere the reader may find the deferred proof of a statement by including a small box with a link to theappropriate appendix after the statement.Since the only 3- and 4-morphisms of 3 Cat pt G are identities, it is evident that 3 Cat pt G — and hence byTheorem 3.4 also 3 Cat G — is 2-truncated and actually defines a 2-category. In the following corollary, wegive a streamlined description of this 2-category without the redundant data. Corollary 3.5.
The -category Cat pt G is isomorphic to the strict -category Cat st G , defined as follows: • An object is a
Gray -monoid C whose set of -cells is G (below, we will denote the elements of G seenas -cells in C by g C ) and composition of 0-cells is given by group multiplication. • A -morphism A : C → D is a -functor A : B C → B D such that – A ( g C ) = g D for all g ∈ G , – the adjoint equivalence µ : ⊗ D ◦ ( A × A ) ⇒ A ◦ ⊗ C satisfies µ g,h = id gh : g D ⊗ h D ⇒ gh D , – the adjoint equivalence ι A = ( ι A ∗ , ι A ) : I D ⇒ A ◦ I C satisfies ι A ∗ = id e D , and ι A = A e , – the associators and unitors ω A , (cid:96) A , r A are identities. • A -morphism η : A ⇒ B is a natural transformation such that η ∗ = e D , η g = id g D , η = id id e D and η g,h = id id gh D for all g, h ∈ G .Composition of 1- and 2-morphisms is the usual composition of -functors and natural transformations [Gur13].Proof. The natural transformation α , the modifications m and p and the perturbations ρ and ξ in the state-ment of Theorem 3.4 are completely determined by the imposed conditions on their coefficients. Moreover,the so defined coefficients always assemble into natural transformations, modifications, and perturbations,respectively, between the respective morphisms described in Corollary 3.5.We now show that 3 Cat st G is indeed a strict 2-category. Suppose we have two composable 1-morphisms( A, A , A , µ A , ι A ) ∈ Cat st G ( D → E ) and (
B, B , B , µ B , ι B ) ∈ Cat st G ( C → D ). Then the formulas for thecomponents for the composite ( A ◦ B, ( A ◦ B ) , ( A ◦ B ) , µ A ◦ B , ι A ◦ B ) are given by( A ◦ B ) g = A ( B g ) ∗ A g ∀ g ∈ G ( A ◦ B ) x,y = A ( B x,y ) ∗ A B ( x ) ,B ( y ) ∀ x ∈ C ( h C → k C ) , ∀ y ∈ C ( g C → h C ) µ A ◦ Bx,y = A ( µ Bx,y ) ∗ µ AB ( x ) ,B ( y ) ∀ x ∈ C ( g C → k C ) , ∀ y ∈ C ( h C → (cid:96) C ) ι A ◦ B = A ( ι B ) ∗ ι A , which are easily seen to be strictly associative and strictly unital. It is also straightforward to see thatcomposition of 2-morphisms is strictly associative and strictly unital as well.15 .1 Strictifying objects In the following section, we prove the ‘object part’ of Theorem 3.4 and show that every object π = π C :B G → C of the 4-category 3 Cat G is equivalent to a strictly 1-bijective Gray -functor π (cid:48) : B G → B C (cid:48) , where C (cid:48) is a Gray -monoid whose set of 0-cells is G with composition the group multiplication. The following lemmais a direct consequence of Gurski’s strictification of 3-categories [Gur13, Cor. 9.15]. Lemma 3.6.
Any -surjective -functor π : B G → C is equivalent, in Cat G , to a -surjective -functor π (cid:48) : B G → B C (cid:48) where C (cid:48) is a Gray -monoid.Proof.
By [Gur13, Cor. 9.15], there is a
Gray -category C (cid:48) and a 3-equivalence C → C (cid:48) . By 1-surjectivity of π ,it follows that the composite B G → C → C (cid:48) factors through the full endomorphism Gray -monoid C (cid:48) of C (cid:48) onthe single object in the image of the composite, resulting in a 3-functor π (cid:48) : B G → B C (cid:48) which is equivalentto π : B G → C in 3 Cat G .To further strictify π : B G → B C , we use the following direct consequence of a theorem of Buhn´e [Buh14].Recall that a 3-functor F : A → B between
Gray -categories A and B is locally strict if the 2-functors F a,b : A ( a → b ) → B ( F ( a ) → F ( b )) are strict. Proposition 3.7.
Given
Gray -monoids G , C and a locally strict 3-functor π : B G → B C , there exists a Gray -monoid C (cid:48) , an equivalence A : B C → B C (cid:48) , a Gray -functor π (cid:48) : B G → B C (cid:48) and a natural isomorphism π (cid:48) ⇒ A ◦ π .Proof. By [Buh14, Thm. 8], every locally strict 3-functor from a (small)
Gray -category into a cocomplete
Gray -category is equivalent to a
Gray -functor. Here, cocomplete is used in the sense of enriched categorytheory [Kel05, § Gray -categories A , B , we denote by [ A , B ] the Gray -category of
Gray -functors
A → B . Considerthe
Gray -enriched Yoneda embedding y : B C → [B C op , Gray ], where the target is cocomplete as
Gray iscocomplete [Kel05, § G π −→ B C y −→ [(B C ) op , Gray ]is a composite of a locally strict 3-functor with a
Gray -functor and hence itself locally strict. Therefore, thereis a
Gray -functor π (cid:48) : B G → [(B C ) op , Gray ] which is equivalent to the composite.Now we define B C (cid:48) to be the full sub- Gray -category of [(B C ) op , Gray ] on the object π (cid:48) ( ∗ ) and define π (cid:48) :B G → B C (cid:48) as the codomain-restriction of π (cid:48) to B C (cid:48) . Finally, observe that both the Gray -Yoneda embedding y : B C → [(B C ) op , Gray ] and the inclusion B C (cid:48) → [(B C ) op , Gray ] are fully faithful Gray -functors which mapthe single objects of B C and B C (cid:48) to equivalent objects. Hence, there is an equivalence A : B C → B C (cid:48) and anatural isomorphism α : π (cid:48) ⇒ A ◦ π . Remark 3.8.
In general, we cannot get rid of the local strictness assumption on π by the example given in[Buh15, Ex 2.2]. Theorem 3.9 (Strictifying objects) . Every object ( C , π ) ∈ Cat G is equivalent to an object (B C (cid:48) , π (cid:48) ) of thesubcategory Cat pt G where π (cid:48) : B G → B C (cid:48) is a strictly -bijective Gray -functor into a
Gray -monoid C (cid:48) whoseset of 0-cells is G with composition the group multiplication.Proof. Since B G is a 1-category, it follows from [Buh15, Cor 2.6] that every 3-functor B G → C is equivalentto a locally strict 3-functor. Applying Proposition 3.7, we obtain a Gray -monoid D and a Gray -functor π D : B G → B D such that (B D , π D ) is equivalent to ( C , π ) in 3 Cat G .Let D (cid:48) be the full 2-subcategory of D whose objects are exactly those in the image of π D . Since π D is a Gray -functor, D (cid:48) is a Gray -submonoid of D , which comes equipped with the corestricted Gray -functor π D (cid:48) : B G → B D (cid:48) which is strictly 1-surjective, i.e., onto Ob( D (cid:48) ). Since π D is 1-surjective, (B D , π D ) isequivalent in 3 Cat G to (B D (cid:48) , π D (cid:48) ).Since π D (cid:48) : B G → B D (cid:48) is a strictly 1-surjective Gray -functor, there is in particular a surjective homo-morphism φ : G → Ob( D (cid:48) ). We define a Gray -monoid C (cid:48) as follows. The 0-cells of C (cid:48) are the elements of G , Here, by a fully faithful
Gray -functor we mean a
Gray -functor F : A → B whose induced 2-functors F a,b : A ( a → b ) →B ( F ( a ) → F ( b )) are isomorphisms in Gray . C (cid:48) ( g → h ) := Hom D (cid:48) ( φ ( g ) → φ ( h )). Since φ is a homomorphism, C (cid:48) inherits a Gray -monoid structure from D (cid:48) together with an obvious strictly 1-bijective Gray -homomorphism π (cid:48) : B G → B C (cid:48) . Since φ is surjective, (B C (cid:48) , π (cid:48) ) is equivalent to (B D (cid:48) , π D (cid:48) ) in 3 Cat G . Given objects (B C , π C ) and (B D , π D ) in 3 Cat pt G comprised of Gray -monoids C and D and a strictly 1-bijective Gray -functor (
A, α ) ∈ Cat G (cid:0) (B C , π C ) → (B D , π D ) (cid:1) , we construct a 1-morphism ( B, β ) ∈ Cat pt G and anequivalence ( B, β ) ⇒ ( A, α ). As C , D are Gray -monoids, we make heavy use of the graphical calculusdiscussed in § A consists of the data from Definition A.1. The invertible natural transfor-mation α : π D ⇒ A ◦ π C is comprised of the data from Definition A.2. We depict α ∗ by an oriented redstrand: α ∗ . By the third unitality bullet point in (T-II), we have that α id g = A g since π C is strict. By Remark 2.1, thereis a contractible choice of ways to extend the invertible 0-cell α ∗ to a biadjoint biequivalence (BB); we do soarbitrarily.We now define B : B C → B D as follows. First, B ( g C ) := g D for all g ∈ G . Given x ∈ C ( g C → h C ), wedefine B g C h C x := g D h D A ( x ) = α − ∗ α ∗ g D h D A ( x ) α g α − h . Given x, y ∈ C ( g C → h C ) and f ∈ C ( x ⇒ y ), we define B ( f ) to be the following 2-cell in D : g D h D A ( x ) A ( f ) ⇒ g D h D A ( y ) . For g ∈ G , we define B g ∈ D (id g D ⇒ B (id g C )) to be the composite g D ⇒ g D ⇒ g D g D A g ⇒ g D g D A (id g ) . (12)17or x ∈ C ( g C → h C ) and y ∈ C ( h C → k C ), we define B x,y ∈ D ( B ( y ) ◦ B ( x ) ⇒ B ( y ◦ x )) to be the composite g D h D k D A ( y ) A ( x ) ⇒ g D h D k D A ( y ) A ( x ) ⇒ g D k D A ( y ) A ( x ) A x,y ⇒ g D k D A ( y ◦ x ) . (13) Lemma 3.10.
The data ( B, B , B ) : C → D defines a 2-functor. § B.1
We now endow B with the structure of a weak 3-functor B C → B D . Construction 3.11.
We define an adjoint equivalence µ B : ⊗ D ◦ ( B × B ) ⇒ B ◦ ⊗ C as follows. First wedefine µ Bg,h ∈ D ( g D ⊗ h D → gh D ) to be the identity. Next, for x ∈ C ( g C → h C ) and y ∈ C ( k C → (cid:96) C ), wedefine the natural isomorphism µ Bx,y ∈ D ( µ Bg,(cid:96) ◦ ( B ( x ) ⊗ B ( y )) ⇒ B ( x ⊗ y ) ◦ µ Bg,k ) to be the composite g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ⇒ g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ⇒ g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ⇒ g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ⇒ g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ⇒ g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ⇒⇒ g D k D µ Bh,(cid:96) h(cid:96) D A ( y ) A ( x ) ( α h,(cid:96) ) − ⇒ g D k D h(cid:96) D A ( y ) A ( x ) µ Ah,(cid:96) ⇒ g D k D h(cid:96) D A ( y ) A ( x ) µ Ah,(cid:96) µ Ax,y ⇒ g D k D h(cid:96) D A ( x ⊗ y ) µ Ag,k ( α g,k ) − ⇒ g D k D µ Bg,k h(cid:96) D A ( x ⊗ y ) . We define an adjoint equivalence ι B = ( ι B ∗ , ι B ) : I D ⇒ B ◦ I C by ι B ∗ = id e D , and ι B := B e ∈ D (id e D ⇒ B (id e C )) from (12). Finally, we define the associator ω B and unitors (cid:96) B , r B to be identities. Lemma 3.12.
The data ( µ B , ι B , ω B , (cid:96) B , r B ) endows B : B C → B D with the structure of a weak 3-functor. § B.1
Lemma 3.13.
The data β = ( β ∗ := e D , β g := id g D , β id g := B g , β := id id e D , β g,h := id id gh D , ) : π D ⇒ B ◦ π C defines a natural isomorphism. § B.1
18e now define for x ∈ C ( g C → h C ) the 2-cell γ x given by g D A ( h C ) A ( x ) ⇒ g D A ( h C ) A ( x ) ⇒ g D A ( h C ) A ( x ) ⇒ g D A ( h C ) A ( x ) ⇒ g D A ( h C ) A ( x ) ⇒ g D A ( h C ) A ( x ) . (14) Theorem 3.14.
The 1-morphisms ( A, α ) , ( B, β ) ∈ Cat G ((B C , π C ) → (B D , π D )) are equivalent via the 2-morphism ( γ, id) : ( B, β ) ⇒ ( A, α ) where γ = ( γ ∗ := α ∗ , γ g := α g , γ x , γ := α , γ g,h := α g,h ) : B ⇒ A is thenatural isomorphism where γ x is given in (14) above. § B.1
Remark 3.15.
Working a bit harder, we can actually make (
B, β ) strictly unital , i.e., B (id g C ) = id g D and B g = id g D for all g ∈ G . This has the following advantages: ι becomes trivial, µ B id e C ,x = id B ( x ) for all x ∈ C ( g C → h C ) by (F-V), π D = B ◦ π C on the nose, and β : π D ⇒ B ◦ π C is the identity transformation.Unfortunately, this would complicate our definition of the coherence data for B considerably, and it wouldfurther obfuscate the reasons why certain commuting diagrams commute in the sequel. Moreover, it has notyet been shown in the literature that every G -crossed braided functor is equivalent to a strictly unital one,although this would follow as a corollary of our main theorem. We are thus content to work with our ( B, β )with β completely determined by B . Suppose (B C , π C ) , (B D , π D ) ∈ Cat pt G and ( A, α ) , ( B, β ) : (B C , π C ) → (B D , π D ) are two 1-morphisms in3 Cat pt G . Since ( A, α ) , ( B, β ) are 1-morphisms in 3
Cat pt G , A ( g C ) = g D = B ( g C ) for all g ∈ G , and α ∗ = e D = β ∗ and α g = id g D = β g . Suppose ( η, m ) : ∈ Cat G (( A, α ) ⇒ ( B, β )). We prove that ( η, m ) is equivalent to a2-morphism ( ζ, id) ∈ Cat pt G (( A, α ) ⇒ ( B, β )).As in Defintion A.2, we denote the 0-cell η ∗ by an oriented green strand. The modification m = ( m ∗ , m g )as in Definition A.3 consists of an invertible 1-cell m ∗ : β ∗ ⇒ η ∗ ⊗ α ∗ together with coherent invertible 2-cells β ∗ g D η ∗ α ∗ g D m ∗ m g ⇒ β ∗ g D η ∗ α ∗ g D m ∗ . (15)Observe that since β ∗ = e D = α ∗ and β g = id g = α g , we may completely omit the dashed lines in (15). Asin Remark 2.1, we extend the invertible 1-cell m ∗ ∈ D to an adjoint equivalence arbitrarily.19or x ∈ C ( g C → h C ), we define an invertible 2-cell ζ x as the following composite: g D h D A ( x ) ⇒ g D h D A ( x ) m ∗ m − ∗ m − h ⇒ g D h D A ( x ) m ∗ m − ∗ φ ⇒ g D h D A ( x ) m ∗ m − ∗ η x ⇒ g D h D B ( x ) m ∗ m − ∗ m g ⇒ g D h D B ( x ) m ∗ m − ∗ φ ⇒ g D h D B ( x ) m ∗ m − ∗ ⇒ g D h D B ( x ) . (16)We define the unit map as in (T-III) by ζ := id id e D and the monoidal map as in (T-IV) by ζ g,h := id id gh D . Lemma 3.16.
The data ζ := ( ζ ∗ = e, ζ g = id g , ζ x , ζ := id id e D , ζ g,h := id id gh D ) together with the identitymodification defines a 2-morphism ( ζ, id) ∈ Cat pt G (( A, α ) ⇒ ( B, β )) . § B.2
Observe now that by the strictness properties of α and β , m ∗ : e D ⇒ η ∗ . Erasing the dotted lines from(15) for m g , we see that the same data as m = ( m ∗ , m g ) actually defines an invertible modification ζ (cid:86) η ! Theorem 3.17.
The 2-morphisms ( η, m ) , ( ζ, id) ∈ Cat G (( A, α ) ⇒ ( B, β )) are equivalent via the 3-morphism ( m, id) ∈ Cat G (( ζ, id) (cid:86) ( η, id)) . § B.2
Suppose now that ( η, m = id) , ( ζ, n = id) : ( A, α ) ⇒ ( B, β ) are two 2-morphisms in 3
Cat pt G and ( p, ρ ) :( η, id) (cid:86) ( ζ, id) is a 3-morphism in 3 Cat G .First, since ( η, id) , ( ζ, id) are 2-morphisms in 3 Cat pt G , we have that η ∗ = e D = ζ ∗ and η g = id g D = ζ g forall g ∈ G , and the modifications are identities. This means the perturbation ρ is a 2-cell= β ∗ = e D α ∗ = e D ζ ∗ = e D m ∗ =id e D ρ ⇒ β ∗ = e D α ∗ = e D ζ ∗ = e D n ∗ =id e D p ∗ η ∗ = e D = p ∗ ρ as an invertible 2-cell id id e D ⇒ p ∗ , under which (P-1)becomes g D ρ ∗ ⇒ g D p ∗ p g ⇒ g D p ∗ = g D ρ ∗ ⇒ g D p ∗ ∀ g ∈ G. (17) Lemma 3.18.
Any -morphism in Cat G between -morphisms in the subcategory Cat pt G is an endomor-phism. § B.3
Theorem 3.19.
Any 3-morphism in Cat G between -morphisms in the subcategory Cat pt G is isomorphicto the identity -morphism.Proof. First, by Lemma 3.18, every 3-morphism is a 3-endomorphism. Suppose ( η, id) is a 2-morphism in3 Cat pt G and ( p, ρ ) is a 3-endomorphism of ( η, id). As above, we may view ρ ∗ as an invertible 2-morphismid e D ⇒ p ∗ that satisfies (17). This is exactly saying that ρ ∗ is a perturbation id ( η, id) ( p, ρ ). Theorem 3.20.
The only 4-endomorphism in Cat G of an identity 3-morphism in the subcategory Cat pt G is the identity.Proof. Suppose ξ is a 4-endomorphism of an identity 3-morphism ( p = id , ρ = id) in 3 Cat pt G . Then ξ satisfiesthe criterion (11), which in diagrams is β ∗ = e D α ∗ = e D η ∗ = e D m ∗ =id e D ρ =id ⇒ n ∗ =id e D p ∗ =id e D ξ ⇒ n ∗ =id e D p ∗ =id e D = m ∗ =id e D ρ =id ⇒ n ∗ =id e D p ∗ =id e D . We conclude that ξ = id. G -crossed braided categories In § G CrsBrd of G -crossed braided categories. By [Gal17], G CrsBrd is equivalent to the full 2-subcategory G CrsBrd st of strict G -crossed braided categories. In this section, weprove our second main theorem. Theorem 4.1.
The 2-category Cat st G is equivalent to G CrsBrd st .Proof. In § Cat st G → G CrsBrd st . In § § ∼ and an isomorphism ∼ =, where thehooked arrows denote inclusions of full subcategories.3 Cat G Cat pt G Cat st G G CrsBrd st G CrsBrd ∼ Thm. 3.4 ∼ =Cor. 3.5 ∼ Thm. 4.1 ∼ [Gal17] .1 Definitions Let G be a group. We now give a definition of a (possibly non-additive) G -crossed braided category. Below,we give a definition in terms of the component categories C g . When each component C g is linear and thetensor product functors and G -action functors are linear, C := (cid:76) g ∈ G C g is an ordinary G -crossed braidedmonoidal category in the sense of [EGNO15, § Definition 4.2. A G - crossed braided category C consists of the following data: • a collection of categories ( C g ) g ∈ G ; • a family of bifunctors ⊗ g,h : C g × C h → C gh ; • an associator natural isomorphism α g,h,k : ⊗ gh,k ◦ ( ⊗ g,h × id C k ) ⇒ ⊗ g,hk ◦ (id C g ×⊗ h,k ); • a unit object 1 C ∈ C e ; • unitor natural isomorphisms λ : ⊗ e,g ◦ (1 C × − ) ⇒ id C g and ρ : ⊗ g,e ◦ ( − × C ) ⇒ id C g .Using the convention a g ⊗ b h := ⊗ g,h ( a g × b h ) ∀ a g ∈ C g and b h ∈ C h , this data should satisfy the obvious pentagon and triangle axioms of a monoidal category.Moreover, C is equipped with a G -action F g : C h → C ghg − together with an isomorphism i g : 1 C → F g (1 C ) and natural isomorphisms ψ g , µ g,h , and ι g C h × C k C ghkg − F g ◦⊗ h,k ⊗ ghg − ,gkg − ◦ ( F g × F g ) ⇑ ψ gh,k C k C ghkh − g − F gh F g ◦ F h ⇑ µ g,hk C g C gF e id C g ⇑ ι g which satisfy the following associativity and unitality conditions where we suppress whiskering:( ψ
1) (associativity) The following diagram commutes: ⊗ ghkg − ,g(cid:96)g − ◦ ( ⊗ ghg − ,gkg − ◦ ( F g × F g ) × F g ) ⊗ ghg − ,gk(cid:96)g − ◦ ( F g × ( ⊗ gkg − ,g(cid:96)g − ◦ ( F g × F g ))) ⊗ ghkg − ,g(cid:96)g − ◦ (( F g ◦ ⊗ h,k ) × F g ) ⊗ ghg − ,gk(cid:96)g − ◦ ( F g × ( F g ◦ ⊗ k,(cid:96) )) F g ◦ ⊗ h,k(cid:96) ◦ (id C h ×⊗ k,(cid:96) ) α ⊗ ghkg − ◦ ( ψ gh,k × id C g(cid:96)g − ) ⊗ ghg − ,gk(cid:96)g − ◦ (id C ghg − × ψ gk,(cid:96) ) ψ ghk,(cid:96) ψ gh,k(cid:96) ( ψ
2) (unitality) For every a h ∈ C h , the following diagram commutes:1 C ⊗ F g ( a h ) F g (1 C ) ⊗ F g ( a h ) F g ( a h ) F g (1 C ⊗ a h ) ⊗ e,ghg − ( i g × id C ghg − ) λ Fg ( ah ) ψ ge,h F g ( λ ah ) as does a similar diagram where 1 C appears on the right with ρ .22 µ
1) (monoidality) The following diagram commutes: ⊗ ghkg − h − ,gh(cid:96)g − h − ◦ (( F g ◦ F h ) × ( F g ◦ F h )) F g ◦ ⊗ hkh − ,h(cid:96)h − ◦ ( F h × F h ) ⊗ ghkg − h − ,gh(cid:96)g − h − ◦ ( F gh × F gh ) F g ◦ F h ◦ ⊗ k,(cid:96) F gh ◦ ⊗ k,(cid:96)ψ ghkh − ,h(cid:96)h − ⊗ ghkg − h − ,gh(cid:96)g − h − ( µ g,hk × µ g,h(cid:96) ) F g ( ψ hk,(cid:96) ) ψ ghk,(cid:96) µ g,hk(cid:96) ( µ
2) (associativity) The following diagram commutes: F g ◦ F h ◦ F k F g ◦ F hk F gh ◦ F k F ghkµ g,hk(cid:96)k − F g ( µ h,k(cid:96) ) µ g,hk(cid:96) µ gh,k(cid:96) ( ι
1) (monoidality) The following diagram commutes:id C hk ◦⊗ h,k ⊗ h,k ◦ (id C h × id C k ) F e ◦ ⊗ h,k ⊗ h,k ◦ ( F e × F e ) ι hk ⊗ h,k ( ι h × ι k ) ψ eh,k ( ι
2) (unitality) The following diagrams commute:id C ghg − ◦ F g F g F e ◦ F gι ghg − µ e,gh and F g F g ◦ id C h F g ◦ F e F g ( ι h ) µ g,eh . Finally, we have the G -crossed braiding natural isomorphism C h × C g C gh C g × C h ⊗ ghg − ,g ◦ ( F g × id C g ) ⇑ β g,h swap ⊗ g,h a g ⊗ b h β g,hag,bh −−−−→ F g ( b h ) ⊗ a g ∀ a g ∈ C G , b h ∈ C h . The G -action and G -crossed braiding are subject to the following coherence axioms taken from [EGNO15].For all a g ∈ C g , b h ∈ C h , and c k ∈ C k , the following diagrams commute, where suppress all labels. F g ( b h ) ⊗ F g ( c k ) F ghg − F g ( c k ) ⊗ F g ( b h ) F g ( b h ⊗ c k ) F gh ( c k ) ⊗ F g ( b h ) F g ( F h ( c k ) ⊗ b h ) F g F h ( c k ) ⊗ F g ( b h ) ( β a g ⊗ b h ) ⊗ c k a g ⊗ ( b h ⊗ c k ) ( F g ( b h ) ⊗ a g ) ⊗ c k F g ( b h ⊗ c k ) ⊗ a g F g ( b h ) ⊗ ( a g ⊗ c k )( F g ( b h ) ⊗ F g ( c k )) ⊗ a g F g ( b h ) ⊗ ( F g ( c k ) ⊗ a g ) ( β a g ⊗ ( b h ⊗ c k )( a g ⊗ b h ) ⊗ c k a g ⊗ ( F h ( c k ) ⊗ b h ) F gh ( c k ) ⊗ ( a g ⊗ b h ) ( a g ⊗ F h ( c k )) ⊗ b h F g F h ( c k ) ⊗ ( a g ⊗ b h ) ( F g F h ( c k ) ⊗ a g ) ⊗ b h ( β Definition 4.3.
Given two G -crossed braided categories C and D , a G - crossed braided functor ( A , a ) : C → D consists of a family of functors ( A g : C g → D g ) g ∈ G together with a unitor isomorphism A : 1 D → A (1 C )and a tensorator natural isomorphism A a g ,b h : A ( a g ) ⊗ A ( b h ) → A ( a g ⊗ b h ) for all a g ∈ C g and b h ∈ C h satisfying the obvious coherences. The monoidal functor A = ( A g , A , A ) comes equipped with a family a = { a g : F D g ◦ A ⇒ A ◦ F C g } g ∈ G of monoidal natural isomorphisms such that for all g, h ∈ G , the followingdiagrams commute, where we suppress whiskering from the notation. F D g ◦ F D h ◦ A F D gh ◦ A F D g ◦ A ◦ F C h A ◦ F C gh A ◦ F D g ◦ F C hµ D g,h a h a gh a g µ C g,h ( γ A ( a ) ⊗ A ( b ) A ( a ⊗ b ) F D h ( A ( b )) ⊗ A ( a ) A ( F C ( b ) ⊗ a ) A ( F C ( b )) ⊗ A ( a ) A a,b β D A ( β C ) a h ⊗ id A F C ( b ) ,a ( γ Definition 4.4.
If ( A , a ) , ( B , b ) : C → D are G -crossed braided functors, a G -crossed braided naturaltransformation h : ( A , a ) ⇒ ( B , b ) is a monoidal natural transformation h : A ⇒ B such that for all g ∈ G ,the following diagram commutes. F D g ◦ A F D g ◦ BA ◦ F C g B ◦ F C gF D g ( h ( · ) ) a g b g h F C g ( · ) (18)24t is straightforward to verify that G -crossed braided categories, functors, and natural transformationsassemble into a strict 2-category called G CrsBrd with familiar composition formulas similar to those from thestrict 2-category of monoidal categories. (See the proof of Proposition 4.13 in Appendix C for full details.)
Definition 4.5 (Adapted from [Gal17, p.6]) . A G -crossed braided category is called strict if α, λ, ρ are allidentities, and all i g , ψ g , µ g,h , and ι g are identities. Observe this implies that F e is the identity as well.By the main theorem of [Gal17], every G -crossed braided category is equivalent (via a G -crossed braidedfunctor which is an equivalence of categories) to a strict G -crossed braided category. In particular, the2-category G CrsBrd is equivalent to the full subcategory G CrsBrd st of strict G -crossed braided categories. Cat st G to G CrsBrd st In this section, we construct a strict 2-functor 3
Cat st G → G CrsBrd st . We begin by explaining how to obtain astrict G -crossed braided category C from an object C ∈ Cat st G , i.e., C is a Gray -monoid with 0-cells { g C } g ∈ G with 0-composition the group multiplication. Construction 4.6.
For each g ∈ G , we define the category C g := C (1 C → g C ). We denote 1-cells in C g bysmall disks. For better readability, we distinguish different 1-cells in a given diagram by different shadingsof the corresponding disks. We will use the shorthand notation that white, green, and blue shaded diskscorrespond to 1-cells into g C , h C , and k C , respectively: g C h C k C . We define the bifunctor ⊗ g,h : C g × C h → C gh by − ⊗ − : g C h C × (cid:55)−→ g C h C . The associator ⊗ gh,k ◦ ( ⊗ g,h × C k ) ⇒ ⊗ g,hk ◦ ( C g × ⊗ h,k ) is the identity. The unit object 1 C := id e ∈ C e ,which we denote by a univalent vertex attached to a dashed string. The unitors ⊗ e,g ◦ ( i × − ) ⇒ id C g and ⊗ g,e ◦ ( − × i ) ⇒ id C g are also identities e C g C = g C = e C g C . Clearly the associators and unitors satisfy the obvious pentagon and triangle axioms of a G -crossed braidedcategory. Construction 4.7 ( G -action) . We define a G -action F g : C h → C ghg − by F g (cid:32) h C (cid:33) := g C g − C h C =: ghg − C . On the right hand side, we abbreviate this ‘cup’ action by a single g -labelled red cup drawn under therespective node. The functors F g are strict tensor functors, i.e., the tensorators ψ g : ⊗ ghg − ,gkg − ◦ ( F g × F g ) ⇒ F g ◦ ⊗ h,k are identity natural isomorphisms. The tensorator µ g,h : F g ◦ F h ⇒ F gh and the unit map ι h : id C h → F e are also both identities. It is straightforward to see that these identity natural isomorphisms ψ g , µ g,h , and ι h satisfy ( ψ ψ µ µ ι ι Construction 4.8 ( G -crossed braiding) . The G -crossed braiding natural isomorphisms β g,h are given byinterchangers in C : g C h C φ ⇒ g C h C = g C g − C h C g C = ghg − C h C . heorem 4.9. The data ( C , ⊗ g,h , F g , β g,h ) from Constructions 4.6, 4.7, and 4.8 forms a strict G -crossedbraided category. § C.1
Now suppose that C , D ∈ Cat st G and A ∈ Cat st G ( C → D ) This means A ( g C ) = g D on the nose for all g ∈ G , the adjoint equivalence µ A : ⊗ D ◦ ( A × A ) ⇒ A ◦ ⊗ C satisfies µ Ag,h = id gh ∈ D ( g D ⊗ h D → gh D ), theadjoint equivalence ι A : ( ι A ∗ , ι A ) : I D ⇒ A ◦ I C satisfies ι A ∗ = id e D , and ι A := A e ∈ D (id e D ⇒ B (id e C )), andthe associators and unitors ω, (cid:96), r are identities.Let C and D be the strict G -crossed braided categories obtained from C and D respectively from Theorem4.9. We now define a G -crossed braided functor ( A , a ) : C → D . Construction 4.10.
First, for a ∈ C g := C ( e C → g C ), we define A ( a ) := A ( a ) ∈ D ( e D → g D ) = D g . For x ∈ C g ( a → b ), we define A ( x ) := A ( x ) ∈ D g ( A ( a ) → A ( b )). It is straightforward to verify A is a functor.We now endow A with a tensorator. For a ∈ C g and b ∈ C h , we define A a,b ∈ D ( A ( a ) ⊗ A ( b ) → A ( a ⊗ b ))to be µ Aa,b : ( A ( a ) ⊗ id h ) ◦ (id e ⊗ A ( b )) ⇒ A (( a ⊗ id h ) ◦ (id e ⊗ b )) . We define the unitor by A := A e ∈ D (1 D → A (1 C )) = D (id e D → A (id e C )). Lemma 4.11.
The data ( A , A , A ) : C → D is a G -graded monoidal functor. § C.1
We now construct the compatibility a between the G -actions on C and D . For a ∈ C h = C (1 C → h C ),we define a ag : F D g ( A ( a )) ⇒ A ( F C g ( a )) using the tensorator µ A : F D g ( A ( a )) = h D g A ( a ) A g ⇒ h D g D g − D A ( a ) A (id g C ) µ A id g C ,a ⇒ h D g D g − D A (id g C ⊗ a ) A g − C ⇒ h D g D g − D A (id g C ⊗ a ) A (id g − C ) µ A id g C ⊗ a, id g − C ⇒ h D gA (id g C ⊗ a ⊗ id g − C ) = A h C g a = A ( F C g ( a )) . (19) Theorem 4.12.
The data ( A , A , A , a ) is a G -crossed braided monoidal functor. § C.1
Proposition 4.13.
The map ( A, µ A , ι A ) (cid:55)→ ( A , a ) strictly preserves identity 1-morphisms and compositionof 1-morphisms. § C.1
Suppose C , D ∈ Cat st G , A, B ∈ Cat st G ( C → D ), and η ∈ Cat st G ( A ⇒ B ). This means that η ∗ = e D and η g = id g D for all g ∈ G . Let C , D be the G -crossed braided categories obtained from C , D respectively fromTheorem 4.9. Let ( A , a ) , ( B , b ) : C → D be the G -crossed braided functors obtained from A, B respectivelyfrom Theorem 4.12,
Construction 4.14.
We define h : ( A , a ) ⇒ ( B , b ) by h a := η a ∈ D ( A ( a ) ⇒ B ( a )) for a ∈ C g = C (1 C → g C ). Theorem 4.15.
The data h defines a G -crossed braided natural transformation ( A , a ) ⇒ ( B , b ) . § C.1
Theorem 4.16.
The map
C (cid:55)→ ( C , ⊗ g,h , F g , β g,h ) , ( A, µ A , ι A ) (cid:55)→ ( A , a ) , η (cid:55)→ h is a strict 2-functor Cat st G → G CrsBrd st .Proof. By Proposition 4.13, we saw that this candidate 2-functor strictly preserves identity 1-morphisms andcomposition of 1-morphisms. It remains to prove that the map η (cid:55)→ h preserves identities and 2-composition.This is immediate from Construction 4.14 as ( η ∗ ζ ) a = η a ∗ ζ a as 3 Cat st G is strict.26 .3 The 2-functor is an equivalence We now show our 2-functor 3
Cat st G → G CrsBrd st constructed in § Essential surjectivity on objects.
We begin by showing essential surjective, applying the techniquesfrom [Cui19]. Suppose C is a strict G -crossed braided category. We define a Gray -monoid
C ∈ Cat st G asfollows. The 0-cells of C are simply the elements of G , and 0-composition ⊗ is the group multiplication.For g, h ∈ G , we define the hom category C ( g → h ) := C hg − . These hom categories comprise a strict2-category by defining vertical composition by the tensor product in C , i.e., if a ∈ C ( g → h ) = C hg − and b ∈ C ( h → k ) = C kh − , we define b ◦ C a := b ⊗ C a = ⊗ kh − ,hg − ( b × a ) . It is straightforward to verify that C is a strict 2-category by strictness of the associator and unitor of C .We now endow C with a monoidal product and interchanger. We define the monoidal product withidentity 1-morphisms as follows. Given a ∈ C ( g → h ) = C hg − and k ∈ G , we set a ⊗ id k := a ∈ C hg − = C hkk − g − = C ( gk → hk ) , i.e., tensoring on the right with id k does nothing. Tensoring on the left, however, implements the G -action:id k ⊗ a := F k ( a ) ∈ C ( kgh − k − ) = C ( hg → kh ) . The interchanger φ is given by the G -crossed braiding. In more detail, given a ∈ C ( g → h ) = C hg − and b ∈ C ( k → (cid:96) ) (cid:96)k − , we define φ a,b := β hg − ,g(cid:96)k − (cid:96) − a,b ∈ C (( a ⊗ id (cid:96) ) ◦ (id g ⊗ b ) ⇒ (id h ⊗ b ) ◦ ( a ⊗ id k ))Indeed, since C is strict, F hg − = F h F g − on the nose, and β hg − ,g(cid:96)k − g − is a natural isomorphism C g(cid:96)k − g − × C hg − C h(cid:96)k − g − C hg − × C g(cid:96)k − g − ⊗ hg − g(cid:96)k − g − gh − ,hg − ◦ ( F hg − × id C hg − ) ⇑ β hg − ,g(cid:96)k − g − σ ⊗ hg − ,g(cid:96)k − g − . Notation 4.17.
In the graphical calculus, one can think of a 1-cell in C ( g → h ) as a 1-cell in C (1 C → hg − )with a g -strand on the right hand side, which does nothing. gh := hg − g Vertical composition is then given by g hk := kh − hg − g ∈ ⊗ kh − ,hg − ( C kh − × C hg − ) = C kg − . G -action. gh ⊗ kk := gh kk := hg − gk ∈ C hg − gg ⊗ hk := kh − g := kh − ghg ∈ C gkh − g − . That the interchanger is given by the G -crossed braiding can now be represented graphically by k(cid:96)gh = hg − (cid:96)k − kg = hg − (cid:96)k − gkg β hg − ,g(cid:96)k − g − −−−−−−−−−−→ (cid:96)k − hg − gkh = k(cid:96)gh . One then checks that C defined as above is a Gray -monoid and thus defines an object of 3
Cat st G . Indeed,the verification is entirely similar to [Cui19] (see also [DR18, Construction 2.1.23]). Moreover, applying ourconstruction from Theorem 4.9 to the so defined C , recovers the G -crossed braided category C on the nose.Hence, the strict 2-functor 3 Cat st G → G CrsBrd st is in fact strictly surjective on objects. Essential surjectivity on -morphisms. Let C , D be the G -crossed braided categories obtained from C , D ∈ Cat st G respectively from Theorem 4.9, and suppose ( A , a ) : C → D is a G -crossed braided functor.We now construct an A ∈ Cat st G ( C → D ) which maps to ( A , a ) under Construction 4.10 and (19). Construction 4.18.
First, we must have A ( g C ) = g D for all g ∈ G . Recall that we have an isomorphismof categories C ( g C → h C ) ∼ = C (1 C → hg − C ) given by the strict 2-functor R g − C = − ⊗ g − C . For a 1-cell x ∈ C ( g C → h C ), we define A ( x ) := A ( x ⊗ g − C ) ⊗ g D , and similarly for 2-cells f ∈ C ( x ⇒ y ). We define theunitor A g := A e ⊗ g D ∈ D (id g D ⇒ A (id g C ) = A (id e C ) ⊗ g D ) , and for x ∈ C ( g C → h C ) and y ∈ C ( h C → k C ), the compositor A y,x as the composite A ( y ) ◦ A ( x ) = ( A ( y ⊗ h − C ) ⊗ h D ) ◦ ( A ( x ⊗ g − C ) ⊗ g D )= ( A ( y ⊗ h − C ) ⊗ hg − D ⊗ g D ) ◦ ( A ( x ⊗ g − C ) ⊗ g D )= (( A ( y ⊗ h − C ) ⊗ hg − D ) ◦ ( A ( x ⊗ g − C )) ⊗ g D − ⊗ g D strict= (( A ( y ⊗ h − C ) ⊗ A ( x ⊗ g − C )) ⊗ g D Nudging (8) A −−→ ( A (( y ⊗ h − C ) ⊗ ( x ⊗ g − C )) ⊗ g D = ( A (( y ⊗ h − C ⊗ hg − C ) ◦ ( x ⊗ g − C )) ⊗ g D Nudging (8)= ( A (( y ⊗ g − C ) ◦ ( x ⊗ g − C )) ⊗ g D = ( A (( y ◦ x ) ⊗ g − C ) ⊗ g D − ⊗ g − C strict= A ( y ◦ x ) . Lemma 4.19.
The data ( A, A , A ) defines a 2-functor C → D such that A ( g C ) = g D for all g ∈ G . § C.2
Construction 4.20.
The adjoint equivalence µ A : ⊗ D ◦ ( A × A ) ⇒ A ◦ ⊗ C is defined as follows. First, µ Ag,h := id gh D ∈ D ( g D ⊗ h D ⇒ gh D ). For x ∈ C ( g C → h C ) and y ∈ C ( k C → (cid:96) C ), we define the natural This graphical calculus is analogous to diagrams for endomorphisms of a von Neumann algebra or a Cuntz C ∗ -algebra[Izu17, §
2] where adding a strand labelled by an endomorphisms of a von Neumann algebra on the right does nothing, andadding a strand on the left implements the action of that endomorphism. µ Ax,y ∈ D ( A ( x ) ⊗ A ( y ) ⇒ A ( x ⊗ y )) by the composite A ( x ) ⊗ A ( y ) = A ( x ⊗ g − C )) ⊗ g D ⊗ A ( y ⊗ k − C )) ⊗ k D = A ( x ⊗ g − C )) ⊗ F D g ( A ( y ⊗ k − C ))) ⊗ g D ⊗ k D = A ( x ⊗ g − C )) ⊗ F D g ( A ( y ⊗ k − C ))) ⊗ gk D a −→ A ( x ⊗ g − C )) ⊗ A ( F C g ( y ⊗ k − C ))) ⊗ gk D A −−→ A ( x ⊗ g − C ⊗ F C g ( y ⊗ k − C )) ⊗ gk D = A ( x ⊗ F C g − F C g ( y ⊗ k − C ) ⊗ g − C ) ⊗ gk D = A ( x ⊗ F C e ( y ⊗ k − C ) ⊗ g − C ) ⊗ gk D = A ( x ⊗ y ⊗ k − C ⊗ g − C ) ⊗ gk D = A ( x ⊗ y ⊗ ( gk ) − C ) ⊗ gk D = A ( x ⊗ y ) . The adjoint equivalence ι A = ( ι ! ∗ , ι A ) : I D ⇒ A ◦ I C is defined by ι A ∗ := id e D , ι A := A e . The associator ω A and the unitors (cid:96) A , r A are all defined to be identities. Theorem 4.21.
The data ( A, µ A , ι A ) defines a 1-morphism in Cat st G ( C → D ) . § C.2
Finally, we observe that the G -crossed braided functor constructed from A in Theorem 4.12 is exactly( A , a ) by construction. Indeed, the strict 2-functors − ⊗ e C and − ⊗ e D are the identity on the nose. Hence,3 Cat st G → G CrsBrd st is in fact surjective on 1-morphisms on the nose. Fully faithfulness on -morphisms. For C , D ∈ Cat st G , A, B ∈ Cat st G ( C → D ), and η ∈ Cat st G ( A ⇒ B ),let C , D be the G -crossed braided categories obtained from C , D respectively from Theorem 4.9, and let( A , a ) , ( B , b ) : C → D be the G -crossed braided functors obtained from A, B respectively from Theorem 4.12.In Construction 4.14 we defined h : ( A , a ) ⇒ ( B , b ) by h a := η a ∈ D ( A ( a ) ⇒ B ( a )) for a ∈ C g = C (1 C → g C ). Theorem 4.22.
The map η (cid:55)→ h is a bijection Cat st G ( A ⇒ B ) → G CrsBrd st ( A ⇒ B ) . § C.2
Theorem A constructs an equivalence between G -crossed braided categories and 1-surjective G -pointed 3-categories. In this section, we investigate how various additional structures and properties of 3-categories,such as linearity and dualizability translate into the corresponding properties of G -crossed braided categories.Let π C : B G → C be a 1-surjective G -pointed 3-category and let { C g } g ∈ G be the corresponding G -crossedbraided category constructed via Theorem A.The first result below is immediate. Proposition 5.1 (Linearity) . If C is a linear -category, then C := (cid:76) g ∈ G C g is a G -crossed braided categoryin these sense of [EGNO15, § Following the conventions in [DSPS13, Defs. 2.1.1, 2.1.2, 2.1.4], given a 1-morphism f : c → d in a2-category, we write ( f L : d → c, ev f : f L ◦ f ⇒ id c , coev f : id d ⇒ f ◦ f L ) for the left adjoint of f and( f R : d → c, f ev : f ◦ f R ⇒ id d , f coev : id c ⇒ f R ◦ f ) for its right adjoint. Given an object x in amonoidal category M , we write ( x ∨ , ev x : x ∨ ⊗ x → M , coev x : 1 M → x ⊗ x ∨ ) for the right dual of x , and( ∨ x, x ev : x ⊗ ∨ x → M , x coev : 1 M → ∨ x ⊗ x ) for the left dual of x .Recall that a braided monoidal category has right duals if and only if it has left duals. Lemma 5.2. A G -crossed braided monoidal category C = (cid:76) g ∈ G C g has right duals if and only if it has leftduals. roof. We prove that having right duals implies having left duals; the other direction is analogous. Suppose x ∈ C g has a right dual ( x ∨ ∈ C g − , ev x : x ∨ ⊗ x → C , coev b : 1 C → x ⊗ x ∨ ). Then, ∨ x := g − ( x ∨ ) is a leftdual with the following evaluation and coevaluation morphism: x ev := ev x ◦ ( µ xg,g − ⊗ id x ∨ ) ◦ ( β x,g − ( x ∨ ) ) : x ⊗ g − ( x ∨ ) → C x coev := β − g − ( x ∨ ) ,x ◦ ψ g − x,x ∨ ◦ g − (coev x ) : 1 C → ∨ x ⊗ x That these maps satisfy the zig-zag/snake equations is straightforward.
Remark 5.3.
Similar to Lemma 5.2, every 2-morphism between invertible 1-morphisms (or more generally,between fully dualizable 1-morphisms) in a 3-category has a right adjoint if and only if it has a left adjoint[Reu19, Prop. A.2].
Proposition 5.4 (Rigidity) . Suppose C is linear so that Proposition 5.1 holds. If every -morphism in C ( π C ( e ) → π C ( g )) has either a right or a left adjoint (and thus necessarily both by Remark 5.3) for all g ∈ G ,then the G -crossed braided linear monoidal category C is rigid.Proof. As the statement and the assumptions in this proposition are invariant under equivalences in 3
Cat G and G CrsBrd respectively, we may assume that C is an object of 3 Cat st G , and hence the delooping B A of a Gray -monoid A whose set of 0-cells is { g A } g ∈ G with 0-composition ⊗ the group multiplication, and that C is the strict G -crossed braided category obtained from Constructions 4.6–4.8.By Lemma 5.2 it suffices to prove that for every g ∈ G , every object x ∈ C g (given by a 1-cell x : 1 A → g A in the strict 2-category A ) has either a right dual or a left dual. We assume the underlying 1-cell x : 1 A → g A has a left adjoint x L : g A → A in the 2-category A and prove that the corresponding object x ∈ C g has aright dual in the monoidal category C . We use the shorthand notation:= g C x := g C x L Setting x ∨ := g − C = g C g − C ∈ C g − , it is a direct consequence of the adjunction between x and x L (here denoted ε : x L ◦ x ⇒ id e C and η : id g C ⇒ x ◦ x L ) that the following evaluation and coevaluation morphisms exhibit x ∨ as a right dual of x :ev x : x ∨ ⊗ x = g C g − C g C = e C ε = ⇒ e C = 1 C coev x : 1 C = e C = g C g − C η = ⇒ g C g C g − C = x ⊗ x ∨ .
30e explicitly prove the relation (id x ⊗ ev x ) ◦ (coev x ⊗ id x ) = id x ; the other relation is left to the reader. g C g C g C g C g − C g C g C g − C g C η id εη In the diagram above, the composite (id x ⊗ ev x ) ◦ α ◦ (coev x ⊗ id x ) is the path going down and then to theright. The square commutes as both maps are identical. The triangle commutes by the adjunction. Remark 5.5.
There is a version of Proposition 5.4 that holds in the non-linear setting; one must be carefulto define the correct notion of duals.The following propostion is also immediate.
Proposition 5.6 (Multifusion) . Suppose C is as in the hypotheses of Proposition 5.4 so that C is rigid linearmonoidal. If each -morphism category C ( π C ( e ) → π C ( g )) is semisimple, then C is multifusion. If moreoverthe -morphism id π C ( e ) : π C ( e ) → π C ( e ) is simple, then C is fusion. Since the fusion 2-categories of [DR18] satisfy the hypotheses of Proposition 5.6, we get the followingcorollary.
Corollary 5.7. If C is a fusion -category in the sense of [DR18] and π : G → C is a monoidal -functorwhich is essentially surjective on objects, then C is a G -crossed braided fusion category. Remark 5.8 (Unitarity) . We define a dagger structure on a
Gray -monoid C in terms of the unpackedDefinition 2.4 above. We require the strict 2-category C to be a dagger 2-category, all 2-functors to bedagger 2-functors, and all isomorphisms to be unitary. Similarly, one can define the notion of a C ∗ or W ∗ Gray -monoid. Given a dagger
Gray -monoid C and an appropriately compatible G action on C (all actionsare by dagger functors and all isomorphisms are unitary), we expect our construction will yield a G -crossedbraided dagger category. We expect analogous results in the C ∗ and W ∗ settings. However, the notionof dagger Gray -monoid is not compatible with weak equivalences and
Gray -ification. These notions meritfurther study.
A Functors and higher morphisms between
Gray -monoids
In this section, we unpack the definitions of trihomomorphism, tritransformation, trimodiciation, and per-turbation of [Gur13, Def 4.10, 4.16, 4.18, 4.21] between two
Gray -monoids in terms of the graphical calculus.We remind the reader that as in Notation 2.3, given a
Gray -monoid C , we refer to its objects, 1-morphisms,and 2-morphisms as 0-cells, 1-cells, and 2-cells respectively in order to distinguish these basic components of C from morphisms in an ambient category in which C lives. The notion of adjoint equivalence in a 2-categoryis well-known, so we will not unpack it further. A biadjoint biequivalence [Gur12] 0-cell α in a Gray -monoidconsists of 0-cells α ∗ , α − ∗ which we depict in the graphical calculus as oriented red strands: α ∗ α − ∗ , ∼ = ∼ = α ∗ ∼ = ∼ = α ∗ ∼ = ∼ = ∼ = ∼ = (BB)fulfilling certain coherence conditions; see [Gur12, Def. 2.1, Rem. 2.2, Def. 2.3]. A.1 3-functors between
Gray -monoids
Definition A.1.
Suppose C , D are Gray -monoids. A 3-functor A : B C → B D consists of:(F-I) A 2-functor ( A, A , A ) : C → D . That is, a function on globular sets A , an invertible 2-cell A c :id A ( c ) ⇒ A (id c ), and an invertible 2-cell A y,x : A ( y ) ◦ A ( x ) ⇒ A ( y ◦ x ), which satisfy the followingcoherence conditions:(F-I).i For all x ∈ C ( a → b ), y ∈ C ( b → c ), and z ∈ C ( c → d ), the following diagram commutes: A ( a ) A ( x ) A ( y ) A ( z ) A ( d ) A ( b ) A ( c ) A ( a ) A ( y ◦ x ) A ( z ) A ( d ) A ( c ) A ( a ) A ( x ) A ( z ◦ y ) A ( d ) A ( b ) A ( a ) A ( z ◦ y ◦ x ) A ( d ) A y,x A z,y A z,y ◦ x A z ◦ y,x x ∈ C ( a → b ), the following two triangles commute: A ( a ) A ( x ) A ( b ) A ( a ) A (id a ) A ( x ) A ( b ) A ( a ) A ( a ) A ( x ) A (id b ) A ( b ) A ( b ) A ( a ) A ( x ) A ( b ) A a A b id A ( x ) A x, id a A b,x (F-II) An adjoint equivalence µ A : ⊗ D ◦ ( A × A ) ⇒ A ◦ ⊗ C in the 2-category of 2-functors C × C → D .Explicitly, this is given by, for each pair of 0-cells ( a, b ) ∈ C × C , an adjoint equivalence 1-cell µ Aa,b : A ( a ) ⊗ A ( b ) → A ( a ⊗ b ) and for each pair of 1-cells ( x, y ) : ( a, b ) → ( c, d ), an invertible 2-cell A ( a ) A ( b ) A ( c ⊗ d ) A ( f ) A ( g ) µ Aa,b A ( c ) A ( d ) µ Af,g ⇒ A ( a ) A ( b ) A ( c ⊗ d ) µ Aa,b A ( f ⊗ g ) A ( a ⊗ b ) . That µ A is a 2-transformation means we have the following cohrences.(F-II).i For all x, x (cid:48) : a → c and y, y (cid:48) : b → d and all f : x ⇒ x (cid:48) and g : y ⇒ y , the following squarecommutes: A ( a ) A ( b ) A ( c ⊗ d ) A ( x ) A ( y ) µ Aa,b A ( c ) A ( d ) A ( a ) A ( b ) A ( c ⊗ d ) µ Aa,b A ( x ⊗ y ) A ( a ⊗ b ) A ( a ) A ( b ) A ( c ⊗ d ) A ( x (cid:48) ) A ( y (cid:48) ) µ Aa,b A ( c ) A ( d ) A ( a ) A ( b ) A ( c ⊗ d ) µ Aa,b A ( x (cid:48) ⊗ y (cid:48) ) A ( a ⊗ b ) µ Ax,y A ( f ) ⊗ A ( g ) A ( f ⊗ g ) µ Ax (cid:48) ,y (cid:48) x ∈ C ( a → a ), x ∈ C ( a → a ), y ∈ C ( b → b ), and y ∈ C ( b → b ), A ( a ) A ( b ) A ( a ⊗ b ) A ( x ) A ( y ) A ( x ) A ( y ) A ( a ) A ( b ) µ Aa,b A ( a ) A ( b ) A ( a ) A ( b ) A ( a ⊗ b ) A ( x ) A ( y ) A ( x ⊗ y ) µ Aa,b A ( a ) A ( b ) A ( a ) A ( b ) A ( a ⊗ b ) µ Ax ,y A ( x ⊗ y ) A ( x ⊗ y ) A ( a ⊗ b ) A ( a ⊗ b ) A ( a ) A ( b ) A ( a ⊗ b ) µ Aa,b A (( x ⊗ y ) ◦ ( x ⊗ y )) A ( a ⊗ b ) A ( a ) A ( b ) A ( a ⊗ b ) A ( x ) A ( y ) A ( x ) A ( y ) µ Aa,b A ( a ) A ( b ) A ( a ) A ( b ) A ( a ⊗ b ) A ( x ◦ x ) A ( y ◦ y ) µ Aa,b A ( a ) A ( b ) A ( a ) A ( b ) A ( a ⊗ b ) µ Aa,b A (( x ◦ x ) ⊗ ( y ◦ y )) A ( a ⊗ b ) µ Ax ,y φ − µ Ax ,y A x ⊗ y ,x ⊗ y A x ,x ⊗ A y ,y µ Ax ◦ x ,y ◦ y A ( φ ) (F-II).iii For all 0-cells a, b ∈ C , the following diagram commutes: A ( a ) A ( b ) A ( a ⊗ b ) µ Aa,b A ( a ) A ( b ) A ( c ⊗ d ) µ Aa,b A (id x ⊗ y ) A ( a ⊗ b ) A ( a ) A ( b ) A ( a ⊗ b ) A (id a ) A (id b ) µ Aa,b A ( a ) A ( b ) A a ⊗ A b A a ⊗ b µ A id a, id b (F-III) An adjoint equivalence ι A : I D ⇒ A ◦ I C (in the 2-category of 2-functors ∗ → D ) where I C : ∗ → C is theinclusion of the trivial 2-category into C which picks out 1 C , id C , id id C , and similarly for D . Explicitly,34his is given by an adjoint equivalence 1-cell ι A ∗ : 1 D → A (1 C ) and an invertible 2-cell A (1 C ) ι A ∗ ι A ⇒ A (1 C ) ι A ∗ A (id C ) A (1 C ) = A (1 C ) ι A ∗ A C ⇒ A (1 C ) ι A ∗ A (id C ) A (1 C ) . That ι A is a 2-transformation implies that ι A equals the map on the right hand side above, which is awhiskering with A e . This means ι A is automatically natural and compatible with A .(F-IV) An invertible associator 2-modification ω A . Explicitly, this is given by, for each a, b, c ∈ C , an invertible2-cell A ( a ) A ( b ) A ( c ) A ( a ⊗ b ⊗ c ) µ Aa,b µ Aa ⊗ b,c A ( a ⊗ b ) ω Aa,b,c ⇒ A ( a ) A ( b ) A ( c ) A ( a ⊗ b ⊗ c ) µ Ab,c µ Aa,b ⊗ c A ( b ⊗ c ) and the fact that ω is a 2-modification means that for all x ∈ C ( a → a ), y ∈ C ( b → b ), and z ∈ C ( c → c ), A ( a ) A ( b ) A ( c ) A ( c ⊗ b ⊗ c ) A ( x ) A ( y ) A ( z ) µ Aa ,b µ Aa ⊗ b ,c A ( a ) A ( b ) A ( c ) A ( a ⊗ b ⊗ c ) A ( x ⊗ y ) A ( z ) µ Aa ,b µ Aa ⊗ b ,c A ( a ) A ( b ) A ( c ) A ( a ⊗ b ⊗ c ) A ( x ⊗ y ) A ( z ) µ Aa ,b µ Aa ⊗ b ,c A ( a ) A ( b ) A ( c ) A ( a ⊗ b ⊗ c ) µ Aa ,b µ Aa ⊗ b ,c A ( x ⊗ y ⊗ z ) A ( a ) A ( b ) A ( c ) A ( c ⊗ b ⊗ c ) A ( x ) A ( y ) A ( z ) µ Ab ,c µ Aa ,b ⊗ c A ( a ) A ( b ) A ( c ) A ( c ⊗ b ⊗ c ) A ( x ) A ( y ) A ( z ) µ Ab ,c µ Aa ,b ⊗ c A ( a ) A ( b ) A ( c ) A ( c ⊗ b ⊗ c ) A ( x ) A ( y ⊗ z ) µ Ab ,c µ Aa ,b ⊗ c A ( a ) A ( b ) A ( c ) A ( a ⊗ b ⊗ c ) µ Ab ,c µ Aa ,b ⊗ c A ( x ⊗ y ⊗ z ) µ Ax,y ω Aa ,b ,c φ µ Ax ⊗ y,z ω Aa ,b ,c φ µ Ay,z µ Ax,y ⊗ z (cid:96) A and r A , i.e., for each c ∈ C , invertible 2-cells A ( c ) A ( c ) ι A ∗ µ A C ,c A (1 C ) (cid:96) Ac ⇒ A ( c ) A ( c ) r Ac ⇐ A ( c ) A ( c ) ι A ∗ µ A C ,c A (1 C ) The fact that (cid:96) and r are 2-modifications means that for all x ∈ C ( a → b ), the following diagramcommutes: A ( a ) A ( b ) ι A ∗ A ( x ) µ A C ,b A ( a ) A ( b ) A ( x ) A ( a ) A ( b ) ι A ∗ A (id C ) A ( x ) µ A C ,b A ( a ) A ( b ) ι A ∗ A (id C ) A ( x ) µ A C ,b A ( a ) A ( b ) A ( x ) ι A ∗ µ A C ,a (cid:96) b ι A φ µ A id1 C ,x (cid:96) a and a similar condition for r .This data is subject to the additional two coherence conditions c.f. [Gur13, Def. 4.10]:36F-1) For all a, b, c, d ∈ C , the following diagram commutes: A ( a ) A ( b ) A ( c ) A ( d ) A ( a ⊗ b ⊗ c ⊗ d ) µ Aa,b µ Aa ⊗ b,c µ Aa ⊗ b ⊗ c,d A ( a ) A ( b ) A ( c ) A ( d ) A ( a ⊗ b ⊗ c ⊗ d ) µ Ab,c µ Aa,b ⊗ c µ Aa ⊗ b ⊗ c,d A ( a ) A ( b ) A ( c ) A ( d ) A ( a ⊗ b ⊗ c ⊗ d ) µ Ab,c µ Ab ⊗ c,d µ Aa,b ⊗ c ⊗ d A ( a ) A ( b ) A ( c ) A ( d ) A ( a ⊗ b ⊗ c ⊗ d ) µ Aa,b µ Ac,d µ Aa ⊗ b,c ⊗ d A ( a ) A ( b ) A ( c ) A ( d ) A ( a ⊗ b ⊗ c ⊗ d ) µ Aa,b µ Ac,d µ Aa ⊗ b,c ⊗ d A ( a ) A ( b ) A ( c ) A ( d ) A ( a ⊗ b ⊗ c ⊗ d ) µ Ac,d µ Ab,c ⊗ d µ Aa,b ⊗ c ⊗ d ω Aa,b,c ω Aa ⊗ b,c,d ω Aa,b ⊗ c,d ω Ab,c,d φ − ω Aa,b,c ⊗ d (F-2) For all a, b, c ∈ C , the following diagram commutes: A ( a ) A ( b ) A ( a ⊗ b ) ι A ∗ µ Aa, C µ Aa,b A ( a ) A ( b ) A ( a ⊗ b ) µ A C ,b ι A ∗ µ Aa,b A ( a ) A ( b ) A ( a ⊗ b ) µ Aa,b ω Aa, C ,b r a (cid:96) b A.2 Transformations between functors of
Gray -monoids
Definition A.2.
Suppose C , D are Gray -monoids and
A, B : B
C → B D are 3-functors. A transformation η : A ⇒ B consists of:(T-I) An object η ∗ ∈ D , which we depict by an oriented green strand: η ∗ (T-II) An adjoint equivalence η : D (id , η ∗ ) ◦ A ⇒ D ( η ∗ , id) ◦ B in the 2-category of 2-functors C → D .Explicitly, this is given by, for each c ∈ C , an adjoint equivalence 1-cell η c : η ∗ ⊗ A ( c ) ⇒ B ( c ) ⊗ η ∗ η c = η ∗ A ( c ) η ∗ B ( c ) η c =: η ∗ A ( c ) η ∗ B ( c ) , together with, for each x ∈ C ( a → b ), invertible 2-cells η ∗ A ( a ) B ( b ) η ∗ A ( x ) A ( b ) η x ⇒ η ∗ A ( a ) B ( b ) η ∗ B ( x ) A ( b ) . The fact that η is a 2-natural transformation means that:(T-II).i For every x, y ∈ C ( a → b ) and f ∈ C ( x ⇒ y ), the following diagram commutes: η ∗ A ( a ) B ( b ) η ∗ A ( x ) η ∗ A ( a ) B ( b ) η ∗ B ( x ) η ∗ A ( a ) B ( b ) η ∗ A ( y ) η ∗ A ( a ) B ( b ) η ∗ B ( y ) η x A ( f ) B ( f ) η y x ∈ C ( a → b ) and y ∈ C ( b ⇒ c ), the following diagram commutes: η ∗ A ( a ) B ( c ) η ∗ A ( y ) A ( x ) η ∗ A ( a ) B ( b ) η ∗ B ( y ) A ( x ) η ∗ A ( a ) B ( b ) η ∗ B ( y ) B ( x ) η ∗ A ( a ) B ( c ) η ∗ A ( y ◦ x ) η ∗ A ( a ) B ( b ) η ∗ B ( y ◦ x ) η y A y,x η x B y,x η y ◦ x (T-II).iii For every c ∈ C , the following diagram commutes: η ∗ η ∗ A ( c ) B ( c ) η ∗ A ( c ) B ( c ) η ∗ A (id c ) η ∗ A ( c ) B ( c ) η ∗ B (id c ) A c B c η id c Observe this immediately implies that η id c = B c ∗ ( A c ) − for all c ∈ C .(T-III) A unit coherence invertible 2-modification η ∗ B (1 C ) η ∗ ι A ∗ A (1 C ) η ⇒ η ∗ B (1 C ) η ∗ ι B ∗ The 2-modification criterion for η is automatically satisfied by (F-III) (which says ι A = A C and ι B = B C ) and (T-II).iii above. 39T-IV) For every a, b ∈ C , a monoidality coherence invertible 2-modification η ∗ A ( a ) A ( b ) B ( a ⊗ b ) η ∗ µ Aa,b A ( a ⊗ b ) η a,b ⇒ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) η ∗ µ Ba,b B ( a ) B ( b ) That η is a 2-modification means that for all x ∈ C ( a → a ) and y ∈ C ( b → b ), η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) A ( x ) A ( y ) µ Aa ,b η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) A ( x ⊗ y ) µ Aa ,b η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) B ( x ⊗ y ) µ Aa ,b η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) B ( x ⊗ y ) µ Ba ,b η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba ,b A ( x ) A ( y ) η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba ,b B ( x ) A ( y ) η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba ,b B ( x ) A ( y ) η ∗ η ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba ,b B ( x ) B ( y ) η a ,b µ Ax,y η x ⊗ y η a ,b η x φ − ◦ φ η y µ Bx,y
This data is subject to the following additional three coherences c.f. [Gur13, Def. 4.16]40T-1) For all a, b, c ∈ C , the following diagram commutes: A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Aa,b µ Aa ⊗ b,c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Aa,b µ Ba ⊗ b,c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Ba,b µ Ba ⊗ b,c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Ba,b µ Ba ⊗ b,c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Ab,c µ Aa,b ⊗ c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Ab,c µ Ba,b ⊗ c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Ab,c µ Ba,b ⊗ c A ( a ) A ( b ) A ( c ) B ( a ⊗ b ⊗ c ) η ∗ η ∗ µ Bb,c µ Ba,b ⊗ c ω Aa,b,c η a ⊗ b,c η a,b φ − ω Ba,b,c η a,b ⊗ c φ η b,c (T-2) For all c ∈ C , the following diagram commutes: η ∗ A ( c ) B ( c ) η ∗ µ A C ,c ι A ∗ η ∗ A ( c ) B ( a ⊗ b ) η ∗ µ B C ,c ι A ∗ η ∗ A ( c ) B ( a ⊗ b ) η ∗ µ B C ,c ι B ∗ η ∗ A ( c ) η ∗ B ( c ) η ∗ A ( c ) B ( a ⊗ b ) η ∗ µ B C ,c ι B ∗ (cid:96) Ac η C ,c η φ − (cid:96) Bc and a similar coherence equation holds for r Ac as well.Observe that (T-2) completely determines η in terms of lower data. This means that one needs only verifythe existence of some η satisfying (T-2) to verify (T-III) above. A.3 Modifications between transformations of
Gray -monoids
Definition A.3.
Suppose C , D are Gray -monoids,
A, B : B
C → B D are 3-functors, and η, ζ : A ⇒ B aretransformations. A modification m : η (cid:86) ζ consists of:41M-I) a 1-cell m ∗ : η ∗ → ζ ∗ depicted η ∗ ζ ∗ m ∗ (M-II) For each c ∈ C , an invertible 2-modification η ∗ A ( c ) B ( c ) ζ ∗ m ∗ m c ⇒ η ∗ A ( c ) B ( c ) ζ ∗ m ∗ Explicitly, this means that m c satisfies the following coherence condition for all x : a → b : η ∗ A ( a ) B ( b ) ζ ∗ A ( x ) m ∗ η ∗ A ( a ) B ( b ) ζ ∗ A ( x ) m ∗ η ∗ A ( a ) B ( b ) ζ ∗ B ( x ) m ∗ η ∗ A ( a ) B ( b ) ζ ∗ A ( x ) m ∗ η ∗ A ( a ) B ( b ) ζ ∗ B ( x ) m ∗ η ∗ A ( a ) B ( b ) ζ ∗ B ( x ) m ∗ m b φ ζ x m a η x φ − This data is subject to the following two conditions c.f. [Gur13, Def. 4.18]:42M-1) For all a, b ∈ C , η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Aa,b m ∗ η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Aa,b m ∗ η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba,b m ∗ η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Aa,b m ∗ η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba,b m ∗ η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba,b m ∗ η ∗ ζ ∗ A ( a ) A ( b ) B ( a ⊗ b ) µ Ba,b m ∗ φ m a ⊗ b η a,b φ − ζ a,b m a m b (M-2) The following diagram commutes: η ∗ ζ ∗ B (1 C ) m ∗ ι A ∗ η ∗ ζ ∗ B (1 C ) m ∗ ι A ∗ η ∗ ζ ∗ B (1 C ) m ∗ ι B ∗ η ∗ ζ ∗ B (1 C ) m ∗ ι A ∗ η ∗ ζ ∗ B (1 C ) m ∗ ι B ∗ φ m C η φ − ζ Observe this coherence completely determines m C in terms of η and ζ . A.4 Perturbations between modifications of
Gray -monoids
Definition A.4.
Suppose C , D are Gray -monoids,
A, B : B
C → B D are 3-functors, η, ζ : A ⇒ B aretransformations, and m, n : η (cid:86) ζ are modifications. A perturbation ρ : m n consists of a 2-cell ρ ∗ : m ∗ ⇒ n ∗ satisfying the following coherence condition c.f. [Gur13, Def. 4.21]:43P-1) For each c ∈ C , the following square commutes: η ∗ A ( c ) B ( c ) ζ ∗ m ∗ η ∗ A ( c ) B ( c ) ζ ∗ m ∗ η ∗ A ( c ) B ( c ) ζ ∗ n ∗ η ∗ A ( c ) B ( c ) ζ ∗ n ∗ m c ρ ∗ ρ ∗ n c B Coherence proofs for strictification
This appendix contains all proofs from § B.1 Coherence proofs for Strictifying 1-morphisms § We remind the reader that (B C , π C ) , (B D , π D ) are two objects in 3 Cat pt G , so that C , D are Gray -monoids, and(
A, α ) ∈ Cat G ((B C , π C ) → (B D , π D )). We specified data for ( B, β ) : (B C , π C ) → (B D , π D ) above in § γ, id) : ( B, β ) ⇒ ( A, α ). Notation B.1.
In this section, we will use a shorthand notation for 1-cells in D for proofs using commutativediagrams. For x ∈ C ( a → b ), y ∈ C ( b → c ), and z ∈ C ( c → d ), we will denote their corresponding image in D under A as a small shaded square, e.g.,:= A ( a ) A ( b ) A ( x ) := A ( b ) A ( c ) A ( y ) := A ( c ) A ( d ) A ( z ) While the 1-composition ◦ in D is stacking of diagrams, we denote A applied to a 1-composite in C byvertically joining the shaded squares::= A ( a ) A ( c ) A ( y ) ◦ A ( x ) := A ( a ) A ( c ) A ( y ◦ x ) . Since 1-composition in C and D are both strict, we denote a triple 1-composite by stacking three boxes, and44e denote A applied to a triple 1-composite by vertically joining three boxes::= A ( b ) A ( d ) A ( z ) ◦ A ( y ) ◦ A ( x ) := A ( a ) A ( d ) A ( z ◦ y ◦ x ) . For the ⊗ composite of 1-cells, we use the nudging convention as in (8). We denote A applied to a ⊗ compositeof 1-cells in C by joining the shaded boxes along corners. For the following example, given x ∈ C ( a → b ), y ∈ C ( b → c ), x ∈ C ( a → b ), and y ∈ C ( b → c ), we write:= A ( a ) A ( b ) A ( x ) := A ( b ) A ( c ) A ( y ) := A ( a ) A ( b ) A ( x ) := A ( b ) A ( c ) A ( y ) We then write := A ( a ) A ( b ) A ( a ) A ( b ) A ( x ) ⊗ A ( x ) := A ( a ⊗ a ) A ( b ⊗ b ) A ( x ⊗ x ) . In this notation, we would write the following diagram for A applied to the following composites::= A ( a ⊗ a ) A ( c ⊗ c ) A ( y ⊗ y ) ◦ A ( x ⊗ x ) := A ( a ⊗ a ) A ( c ⊗ c ) A ( y ◦ x ) ⊗ A ( y ◦ x ) . Proof of Lem. 3.10: ( B, B , B ) is a 2-functor. We must check (F-I) for B . We provide a complete prooffor (F-I).i, and leave most of (F-I).ii as an exercise for the reader.(F-I).i For x ∈ C ( g C → h C ), y ∈ C ( h C → k C ), and z ∈ C ( k C → (cid:96) C ), we use the following shorthand as inNotation B.1: := A ( g C ) A ( h C ) A ( x ) := A ( h C ) A ( k C ) A ( y ) := A ( k C ) A ( (cid:96) C ) A ( z ) y,x A z,y (F-I).i A z,y ◦ x A z ◦ y,x Every square except for the bottom right square commutes by functoriality of 1-cell composition ◦ in a Gray -monoid, that is, applying two 2-cells locally to non-overlapping regions in a 1-cell commutes. The bottomright square commutes by (F-I).i applied to the underlying 2-functor of A .(F-I).ii Follows from the properties of the adjoint equivalence α (see Remark B.2 below) together with (F-I).iifor the underlying 2-functor of A . Remark B.2.
In subsequent proofs, we will freely combine squares that commute by functoriality of 1-cellcomposition ◦ when the involved 2-cells applied locally are part of the biadjoint biequivalence α ∗ (BB) orthe adjoint equivalences α g . We will then simply state this larger face commutes by the properties of theadjoint equivalence α , i.e., the properties of the biadjoint biequivalence α ∗ (BB) and the properties of theadjoint equivalences α g . Proof of Lem. 3.12: ( B, µ B , ι B , ω, (cid:96), r ) is a weak 3-functor B C → B D . (F-II).i Every component which makes up µ B , in Construction 3.11, especially µ A , is natural.(F-II).ii For x ∈ C ( a → b ), y ∈ C ( b → c ), x ∈ C ( a → b ), and y ∈ C ( b → c ), we use the followingshorthand as in Notation B.1::= A ( g C ) A ( h C ) A ( x ) := A ( h C ) A ( k C ) A ( y ) := A ( p C ) A ( q C ) A ( x ) := A ( q C ) A ( r C ) A ( y ) For the following diagram to fit on one page, we compress the definition of µ Bx,y from Construction 3.11 intofour steps µ Bx,y := ⇒ ( α h,q ) − ⇒ µ Ah,q µ A ⇒ µ Ag,p ( α g,p ) − ⇒ , (20)46e suppress as many interchangers as possible, and we combine commuting squares involving only α ∗ andthe α g as in Remark B.2. µ Ak,r µ Ah,q µ Ak,r µ Ah,q µ Ah,q µ Ak,r (F-II).ii for A µ Ah,q µ Ag,p µ Ak,r µ Ag,p µ Ag,p µ Ak,r µ Ag,k φ − ( α k,r ) − µ Ay ,y ( α h,q ) − ( α h,q ) − φ − ( α k,r ) − ( α k,r ) − µ Ay ,y ( α h,q ) − µ Ax ,x A y ,x ⊗ A y ,x A y ,x ⊗ A y ,x ( α k,r ) − φ − µ Ax ,x ( α g,p ) − A y ,x ⊗ A y ,x ( α g,p ) − A y ⊗ y ,x ⊗ x ( α g,p ) − ( α k,r ) − µ Ay ◦ x ,y ◦ x A ( φ )( α g,p ) − A ( φ ) A y ⊗ y ,x ⊗ x Non-labelled faces either commute by functoriality of 1-cell composition ◦ , axioms (C4) and (C5) of theinterchanger, or Remark B.2.(F-II).iii This follows by Remark B.2 and functoriality of 1-cell composition ◦ , together with (F-II).iii appliedto A .(F-III) This part is automatic as ι B := B e .(F-IV) This follows by Remark B.2 and functoriality of 1-cell composition ◦ , together with (F-IV) applied to A and two instances of (T-1) for the transformation α : π D ⇒ A ◦ π C .47F-V) This follows by Remark B.2 and functoriality of 1-cell composition ◦ , together with (F-V) applied to A and two instances of (T-2) for the transformation α : π D ⇒ A ◦ π C .(F-1) Every map is the identity map.(F-2) Every map is the identity map. Proof of Lem. 3.13: ( β ∗ , β g , β id g , β , β ) is a 2-natural transformation π D ⇒ B ◦ π C . (T-II).i This condition is immediate as the only 1-cells and 2-cells in B G are identities.(T-II).ii This step amounts to checking B g , id g ∗ ( B g ◦ B g ) = B g . Using Remark B.2 and functoriality of1-cell composition ◦ , this reduces to the identity A g , id g ∗ ( A g ◦ A g ) = A g .(T-II).iii This condition is immediate as ( π D ) g = id id g D and β g = B g .(T-III) This condition is automatically satisfied.(T-IV) This condition is immediate as µ π D g,h = id gh D = µ B ◦ π C g,h and β g = id g for all g ∈ G .(T-1) Every map is the identity map.(T-2) Every map is the identity map. Proof of Thm. 3.14: ( γ, id) : ( B, β ) ⇒ ( A, α ) is an invertible 2-morphism in Cat G . It suffices to prove that γ defines a 2-transformation γ : B ⇒ A , as it clearly invertible.(T-II).i Every component which makes up γ x in (14) is natural in x .(T-II).ii This follows by Remark B.2.(T-II).iii This follows by Remark B.2.(T-III) This condition is automatically satisfied.(T-IV) For x ∈ C ( g C → h C ) and y ∈ C ( k C → (cid:96) C ), we use the following shorthand as in Notation B.1::= A ( g C ) A ( h C ) A ( x ) := A ( k C ) A ( (cid:96) C ) A ( y ) For the following diagram to fit on one page, we compress the definition of µ Bx,y from Construction 3.11 intofour steps as in (20), we suppress as many interchangers as possible, and we combine commuting squares48nvolving only α ∗ and the α g as in Remark B.2. µ Ah,(cid:96) µ Ag,k µ Ah,(cid:96) µ Ah,(cid:96) µ Ah,(cid:96) µ Ah,(cid:96) µ Ag,k α h,(cid:96) ( α h,(cid:96) ) − α h,(cid:96) µ Ax,y γ ( α g,k ) γγ x γ γ x ⊗ y γ y µ Ax,y α g,k Every square here commutes by properties of the biadjoint biequivalence α ∗ (BB), the adjoint equivalences α g , and functoriality of 1-cell composition ◦ .(T-1) Since ω Bg,h,k is the identity, it is equal to ω π D g,h,k . Since (T-1) holds for α : π D ⇒ A ◦ π C , we conclude(T-1) holds for ζ : B ⇒ A .(T-2) Since (cid:96) Bg , r Bg are identities, they are equal to (cid:96) π D g , r π D g respectively. Since (T-2) holds for α : π D ⇒ A ◦ π C ,we conclude (T-2) holds for ζ : B ⇒ A . B.2 Coherence proofs for Strictifying 2-morphisms § We remind the reader that (B C , π C ) , (B D , π D ) are two objects in 3 Cat pt G , ( A, α ) , ( B, β ) ∈ Cat pt G ((B C , π C ) → (B D , π D )), and ( η, m ) ∈ Cat G (( A, α ) ⇒ ( B, β )). We specified data for ( ζ, id) : ( A, α ) ⇒ ( B, β ) above in § Proof of Lem. 3.16: ( ζ, id) : ( A, α ) ⇒ ( B, β ) is a 2-morphism in Cat G . It suffices to check that ζ : A ⇒ B is a 2-natural transformation.(T-II).i Every component which makes up ζ x in (16) is natural in x .(T-II).ii This follows by Remark B.2 and functoriality of 1-cell composition ◦ , together with (T-II).ii appliedto α : π D ⇒ A ◦ π C .(T-II).iii This follows by Remark B.2 and functoriality of 1-cell composition ◦ , together with (T-II).iii appliedto α : π D ⇒ A ◦ π C .(T-III) This condition is automatically satisfied.(T-IV) For x ∈ C ( g C → h C ) and y ∈ C ( k C → (cid:96) C ), we use the following shorthand as in Notation B.1::= g D h D A ( x ) := k D (cid:96) D A ( y ) := g D h D B ( x ) := k D (cid:96) D B ( y ) := η ∗ m ∗ := η ∗ m − ∗ . ζ x from (16) into three steps ζ x := ⇒ η x ⇒ ⇒ , and we combine and suppress as many interchangers as possible, simply writing φ . µ A (M-1) φ φµ A η η x ⊗ y (T-IV) for η η (M-1) η x φ η x η x φ η y µ B µ B η x φ η y φ φη y φη y Non-labelled faces commute by either functoriality of 1-cell composition ◦ or by properties of a (bi)adjoint(bi)equivalence. 50T-1) Every map is the identity map.(T-2) Every map is the identity map.We remind the reader that by the strictness properties for α, β as components of 1-morphisms in 3 Cat pt G , m ∗ : e D ⇒ η ∗ , and m g is an invertible 2-cell g D m ∗ m g ⇒ g D m ∗ . (15) Proof of Thm. 3.17: ( m, id) : ( ζ, id) (cid:86) ( η, m ) is an invertible 3-morphism in Cat G . It suffices to check that m : ζ (cid:86) η is an invertible 3-modification.(M-II) This condition corresponds for m : ζ (cid:86) η corresponds to the outside of the following commutativediagram, where we use the following shorthand notation for x ∈ C ( g C → h C ) and m ∗ , m − ∗ ::= g D h D A ( x ) := g D h D B ( x ) := η ∗ m ∗ := η ∗ m − ∗ m h φ η x m g φm − h φ η x m g φ All inner faces in the above diagram are squares which commute by functoriality of 1-cell composition ◦ .(M-1) This is exactly (M-1) applied to m viewed as a modification m : β (cid:86) ( η ◦ id π C ) ∗ α as in (9) above(M-2) By the strictness properties of ( A, α ) and (
B, β ), (M-2) for the modification m : β (cid:86) ( η ◦ id π C ) ∗ α asin (9) above tells us that m C = η on the nose. This exactly gives the coherence (M-2) for γ . B.3 Coherence proofs for Strictifying 3-morphisms § We remind the reader that in this section, ( η, m = id) , ( ζ, n = id) : ( A, α ) ⇒ ( B, β ) are two 2-morphismsin 3
Cat pt G and ( p, ρ ) : ( η, id) (cid:86) ( ζ, id) is a 3-morphism in 3 Cat G . This means ρ ∗ is an invertible 2-cellid id e D ⇒ p ∗ satisfying the coherence g D ρ ∗ ⇒ g D p ∗ p g ⇒ g D p ∗ = g D ρ ∗ ⇒ g D p ∗ ∀ g ∈ G. (17) Proof of Lem. 3.18: η x = ζ x for all x ∈ C ( g C → h C ) . For x ∈ C ( g C → h C ), we use the following shorthand asin Notation B.1. := g D h D A ( x ) := g D h D B ( x ) := e D e D p ∗ η x and one arrow ζ x ; hence η x = ζ x : η x ζ x ρ ∗ ρ ∗ (C5) ρ ∗ p h (17) φ ζ x (M-II) (17) p g η x φ − ρ ∗ ρ ∗ (C5) ρ ∗ The unlabeled faces commute by functoriality of 1-cell composition ◦ . C Coherence proofs for G -crossed braided categories This appendix contains all proofs from § C.1 Coherence proofs for the 2-functor Cat st G to G CrsBrd st from § We now supply the proofs for statements in § C , ⊗ g,h , F g , β g,h ) is the dataconstructed from C ∈ Cat st G in Constructions 4.6, 4.7, and 4.8. Proof of Thm. 4.9: ( C , ⊗ g,h , F g , β g,h ) forms a strict G -crossed braided category. We remind the reader thatwe use the shorthand notation that white, green, and blue shaded disks correspond to 1-morphisms into g C , h C , and k C , respectively: g C h C k C . It remains to check the commutativity of ( β β β β
1) in detail. Going around theoutside of the diagram below corresponds to ( β ghg − C g gkg − C g g g ghg − ,g gg g gh gg h g,h g ∼ = = == ∼ = = == = ( β β
2) and ( β
3) are similar. In the two diagrams below, the outside 7 diagrams are thevertices in the heptagons ( β
2) and ( β
3) respectively. There is only one non-trivial face in each the twodiagrams below corresponding to these two coherences, and this face commutes by the axiom (C4) of theinterchanger in a
Gray -monoid. g C h C k C gg g gg g g g = ∼ = ∼ = (C4) ∼ = = = ∼ = = = ∼ == = ∼ == ( β C h C k C hgh h hg,h g,h = ∼ = ∼ = (C4) ∼ = == = ∼ = == = ∼ == ( β C , D ∈ Cat st G , A, B ∈ Cat st G ( C → D ), and η ∈ Cat st G ( A ⇒ B ), let C , D be the G -crossed braidedcategories obtained from C , D respectively from Theorem 4.9. In Construction 4.10 and (19), we defined thedata ( A , a ) : C → D . Proof of Lem. 4.11: ( A , A , A ) : C → D is a G -graded monoidal functor. That each A g is a functor followsimmediately from the fact that A is a functor. The data A satisfies associativity by property (F-IV) of( A, µ A , ι A ), and the data A and A satisfies unitality by property (F-V) of ( A, µ A , ι A ). (Observe that in(F-IV) and (F-V), all instances of φ , ω A , (cid:96) A , and r A are identities, so these reduce to the usual associativityand unitality conditions for a monoidal functor.) Proof of Thm. 4.12: ( A , A , A , a ) : C → D is a G -crossed braided monoidal functor. Naturality of a followsby naturality of A and (F-II).i of µ A . It remains to prove the coherences ( γ
1) and ( γ γ
1) Observe that since C and D are strict, the coherence condition ( γ
1) is actually a triangle. For a ∈ C k ,we use the shorthand notation a small shaded box for A ( a ). For g, h ∈ G and a ∈ C k = C (1 C → k C ), we usethe following shorthand as in Notation B.1::= id g D := g D g D A (id g ) := id h D := h D h D A (id h ) := k D A ( a ) Observe that since C is Gray , we have an equality id g C ⊗ id h C = id gh C := = gh D gh D A (id gh C ) Expanding (19), we see that ( γ
1) follows from the following commuting diagram. (Recall that the cups on54he bottom in (19) are really identity maps, and do not need to be drawn.) A gh C A h C (F-II).iii µ A (F-II).iii A h − C A gh ) − C µ A A g C µ A µ A µ A (F-IV) µ A µ A (F-IV) A g C A h − C (F-II).iii µ A A h − C µ A µ A (F-IV) A g − C µ A µ A A g C A g − C µ A A g − C µ A Each square above is labelled by the property for A which makes it commute. Unlabelled squares commuteby functoriality of 1-cell composition ◦ .( γ
2) For g, h ∈ G , a ∈ C g = C (1 C → g C ), and b ∈ C h = C (1 C → h C ), we use the following shorthand as inNotation B.1::= e D e D A (id e ) := g D g D A (id g ) := g D A ( a ) := h D A ( b ) := h D A ( F C g ( b )) Recall that by Construction 4.8 of the G -crossed braiding in C , we have the identities= = . γ = µ A φ A h ⊗ A e A ( φ ) (F-II).ii A h ⊗ A e (F-I).ii A h,a ⊗ A b, id e φ µ A µ A A g ⊗ A e A g µ A µ A (F-II).ii (F-V) A e A A e (C5) A e µ A A ⊗ A φ (F-I).ii A e µ A µ A µ A (F-IV) µ A (F-II).iii µ A A e A g − C ⊗ A g (F-V) (F-V) A e A g − C A g µ A A g µ A Again, each square above is labelled by the property for A which makes it commute. Remark C.1.
By an argument similar to the right half of the commutative diagram in the proof of ( γ A ∈ Cat st G , x ∈ C (1 C → g C ), and y ∈ C ( h ) C → k C ), the following square commutes: g D h D k D A ( x ) A (id k C ) A ( y ) h D gk D A ( x ⊗ id k C ) A ( y ) g D h D k D A ( x ) A ( y ) gk D h D A ( x ⊗ y ) µ A A k C ,y A x ⊗ id k C ,y µ Ax,y (21)
Proof of Prop. 4.13: ( A, µ A , ι A ) (cid:55)→ ( A , a ) is strict.
56t is straightforward to see that if (
A, A , A , µ A , ι A ) ∈ Cat st G ( C → C ) is the identity 3-functor, then so is( A , A , A , a ) ∈ G CrsBrd st ( C → C ). Suppose now we have two composable 1-morphisms ( A, A , A , µ A , ι A ) ∈ Cat st G ( D → E ) and (
B, B , B , µ B , ι B ) ∈ Cat st G ( C → D ). We now calculate the composition formu-las for the composite G -crossed braided monoidal functor ( A ◦ B , ( A ◦ B ) , ( A ◦ B ) , a ◦ b ) associatedto ( A ◦ B, ( A ◦ B ) , ( A ◦ B ) , µ A ◦ B , ι A ◦ B ). The unitor ( A ◦ B ) and tensorator ( A ◦ B ) are straightforward:( A ◦ B ) = ( A ◦ B ) e = A ( B e ) ∗ A e = A ( B ) ∗ B e ( A ◦ B ) x,y = µ A ◦ Bx,y = A ( µ Bx,y ) ∗ µ AB ( x ) ,B ( y ) = A ( B x,y ) ∗ A B ( x ) , B ( y ) . To compute ( a ◦ b ) x for x ∈ C (1 C → g C ), we use the following shorthand as in Notation B.1, whereblack rectangles and strings corresponds to 1-cells in E after applying A , and blue rectangles and strandscorresponds to 1-cells in D after applying B . We also draw red strands to denote id g in both D and E . Forexample: := h D B ( x ) := h E A ( B ( x )) := gh E g E A (id g D ⊗ B ( x )) We draw unshaded boxes on red strands to denote B (id g C ) , B (id g − D , A (id g D ) , A (id g − D .:= g D g D B (id g C ) := g E g E A ( B (id g C )) := g E g E A (id g D ) The composite along the diagonal in the commuting diagram below is the definition of ( a ◦ b ) x . Each facewithout a label above commutes by functoriality of 1-cell composition ◦ . A g by def’n.( A ◦ B ) g µ A id g D ,B ( x ) A ( B g ) (F-II).i A g − A ( B g ⊗ id B ( x ) ) µ A id g D ⊗ B ( x ) , id g − D A ( B g ⊗ id B ( x ) ) A ( B g ⊗ id B ( x ) ⊗ id g D )(F-II).i µ AB (id g C ) ,B ( x ) by def’n. µ A ◦ B id g C ,x A g − A ( µ B id g C ,x ) µ AB (id g C ) ⊗ B ( x ) , id g − D A ( µ B id g C ,x ) A ( µ B id g C ,x ⊗ id g D )(F-II).i A g − by def’n.( A ◦ B ) g − µ AB (id g C ⊗ x ) , id g − D A ( B g − ) A (id B (id g C ⊗ x ) ⊗ B g − )(F-II).i µ AB (id g C ⊗ x ) ,B (id g − C ) by def’n. µ A ◦ B id g C ⊗ x, id g − C µ AB (id g C ⊗ x ) ,B (id g − C ) As the above diagram commutes, ( a ◦ b ) x = A ( a F D g ( B ( x ) ) ∗ a B ( x ) .Finally, we observe this data agrees with the composite data for the data for the composite of the G -crossed braided monoidal functors ( A , A , A , a ) and ( B , B , B , b ) in G CrsBrd .57or C , D ∈ Cat st G , A, B ∈ Cat st G ( C → D ), and η ∈ Cat st G ( A ⇒ B ), let C , D be the G -crossed braidedcategories obtained from C , D respectively from Theorem 4.9, and let ( A , a ) , ( B , b ) : C → D be the G -crossed braided functors obtained from A, B respectively from Theorem 4.12. In Construction 4.14 wedefined h : ( A , a ) ⇒ ( B , b ) by h a := η a ∈ D ( A ( a ) ⇒ B ( a )) for a ∈ C g = C (1 C → g C ). Proof of Thm. 4.15: h : ( A , a ) ⇒ ( B , b ) is a G -crossed braided monoidal transformation. Naturality: This is immediate by the definition h x := η x for x ∈ C g = C (1 C → g C ) and (T-II).i.Unitality: By (T-II).iii. h C = η id e C = B e ∗ ( A e ) − = B ∗ ( A ) − .Monoidality: That B x,y ∗ ( h x ⊗ h y ) = h x ⊗ y ∗ A x,y follows immediately by (T-IV).(18) For x ∈ C h = C (1 C → h C ), we use the following shorthand as in Notation B.1. We also draw red strandsto denote id g in D , and we draw unshaded boxes on red strands to denote each of A (id g C ) , A (id g − C and B (id g C ) , B (id g − D , For example::= h D A ( x ) := h D B ( x ) := g D g D A (id g C ) := g D g D B (id g C ) The outside of the commuting diagram below corresponds to (18). A g η x (T-II).iii µ A id g C ,x η id g ⊗ η x (T-IV) A g − η id g C ⊗ x (T-II).iii (T-IV) µ A id g C ⊗ x, id g − C η ⊗ η ηB g µ B id g C ,x B g − µ B id g C ⊗ x, id g − C This completes the proof.
C.2 Coherence proofs for the equivalence § In this section, we supply the proofs from § Cat st G → G CrsBrd st from Theorem 4.16 is an equivalence. We begin by expanding on Notation B.1. Notation C.2.
In this section, we use an expanded shorthand notation for 1-cells in D and D for proofs usingcommutative diagrams. For x ∈ C ( g C → h C ), x ∈ C ( h C → k C ), y ∈ C ( p C → q C ), and y ∈ C ( q C → r C ),we will denote the image under A after tensoring with the identity of the source object using small shadedsquares with one strand coming out of the top, e.g.,:= hg − D A ( x ⊗ g − C ) := kh − D A ( y ⊗ h − C ) := qp − D A ( x ⊗ p − C ) := rq − D A ( y ⊗ q − C ) We denote the G -actions F D g and F D h as in Construction 4.8 by a red strand underneath the 1-morphism in C , where red corresponds to g and green corresponds to h . We denote A applied to the G -actions F C g and F C by outlining the shaded square with red or green respectively, e.g.,:= gqp − g − D F D g ( A ( x ⊗ p − C )) := gqp − g − D A ( F C g ( x ⊗ p − C )) := hrq − h − D F D h ( A ( y ⊗ q − C )) := hrq − h − D A ( F C h ( y ⊗ q − C ))
58e use a similar convention for the ⊗ composite of 1-cells as in Notation B.1. For example, if x ∈ C g = C (1 C → g C ) and y ∈ C h = C (1 C → h C ), we write:= hg − D A ( x ) := qp − D A ( y ) := g C h C A ( x ) ⊗ A ( y ) := g C ⊗ h C A ( x ⊗ y ) . This means that by Construction 4.8 of the G -crossed braiding in C , we have that A (cid:32) β hg − ,grq − g − x ⊗ g − C ,F C g ( y ⊗ q − C ) (cid:33) −−−−−−−−−−−−−−−−−→ . Proof of Lem. 4.19: ( A, A , A ) : C → D is a 2-functor. (F-I).i For x ∈ C ( g C → h C ), y ∈ C ( h C → k C ), and z ∈ C ( k C → (cid:96) C ), using the nudging convention (8), thesquare for A is exactly A ( z ⊗ k − C ) ⊗ A ( y ⊗ h − C ) ⊗ A ( x ⊗ g − C ) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A ( z ) ◦ A ( y ) ◦ A ( x ) A ( z ⊗ k − C ) ⊗ A ( y ⊗ h − C ⊗ x ⊗ g − C ) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A ( z ) ◦ A ( y ◦ x ) A ( z ⊗ k − C ⊗ y ⊗ h − C ) ⊗ A ( x ⊗ g − C ) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A ( z ◦ y ) ◦ A ( x ) A (( z ⊗ k − C ) ⊗ ( y ⊗ h − C ) ⊗ ( x ⊗ g − C )) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A ( z ◦ y ◦ x )id A ( z ⊗ k − C ) ⊗ A ⊗ id g D A ⊗ id A ( x ⊗ g − C ) ⊗ id g D A ⊗ id g D A ⊗ id g D which commutes by strictness of C , D and associativity of A .(F-I).ii For x ∈ C ( g C → h C ), using the nudging convention (8), the lower triangle for A and A is exactly e D ⊗ A ( x ⊗ g − C ) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A ( x ) A ( e C ) ⊗ A ( x ⊗ g − C ) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A (id h ) ◦ A ( x ) A ( e C ⊗ x ⊗ g − C ) ⊗ g D (cid:124) (cid:123)(cid:122) (cid:125) A ( x ) A e ⊗ id A ( x ⊗ g − C ) ⊗ id g D id x A ⊗ id g D which commutes by unitality of A , A . The other triangle is similar. Proof of Thm. 4.21: ( A, µ A , ι A ) ∈ Cat st G ( C → D ) . (F-II).i Each component in the definition of µ Ay,x is natural in x and y .(F-II).ii For g, h ∈ G , x ∈ C ( g C → h C ), x ∈ C ( h C → k C ), y ∈ C ( p C → q C ), and y ∈ C ( q C → r C ), we use thefollowing shorthand as in Notation B.1::= hg − D A ( x ⊗ g − C ) := kh − D A ( y ⊗ h − C ) := qp − D A ( x ⊗ p − C ) := rq − D A ( y ⊗ q − C ) A from A and the nudging convention (8), we have A ( x ) A ( x ) A ( y ) A ( y ) = A ( y ⊗ h − C ) ⊗ F D h ( A ( y ⊗ q − C )) ⊗ A ( x ⊗ g − C ) ⊗ F D g ( A ( x ⊗ p − C )) ⊗ gp D = h g gp D A (( y ⊗ y ) ◦ ( x ⊗ x )) = A (( y ⊗ h − C ) ⊗ F C h ( y ⊗ q − C ) ⊗ ( x ⊗ g − C ) ⊗ F C g ( x ⊗ p − C )) ⊗ gp D = gp D Going around the outside of the diagram below corresponds to (F-II).ii, except we leave off the extra gp D strand on the right hand side of each string diagram. h g g gg g h , g − ,g g h , g − gg g g g gg φ − = β − a h ( γ A A a g A a g a g a hg − ( γ A a g monoidal β A a g A ( β ) a g A a g a g A A a g A A ⊗ A A A A ( β ) A A A A A A (id ⊗ β ) A A A ( φ )= A ( β ) The faces without labels above commute either by naturality or by associativity of A .(F-II).iii This follows since each F D g is strictly unital, and thus for all g ∈ G , A e = A e , id e ∗ ( A e ⊗ A e ) = A e ,F D g (id e ) ∗ ( A e , F D g ( A e ) . (F-III) This part is automatic as ι A = A e .(F-IV) This follows by monoidality of a g and associativity of A . We omit the full proof as it is much easierthan (F-II).ii above.(F-V) This reduces to unitality of A and A , i.e., for all x ∈ C ( g C → h C ), A e ,x ⊗ g − C ∗ ( A ⊗ id A ( x ⊗ g − C ) ) = id A ( x ⊗ g − C ) . Proof of Thm. 4.22: the map Cat st G ( A ⇒ B ) → G CrsBrd ( A ⇒ B ) is bijective. Suppose that η, ζ ∈ Cat st G ( A ⇒ B ) satisfy η x = ζ x for every x ∈ C (1 C → g C ) for all g ∈ G . Since η, ζ are2-morphisms in 3 Cat st G , we have η ∗ = e D = ζ ∗ , η g = id g D = ζ g for all g ∈ G . For an arbitrary y ∈ C ( g C → h C ),we have η y ⊗ g − C = ζ y ⊗ g − C . By (T-IV) for η : A ⇒ B , the following diagram commutes: hg − D g D g D A ( y ⊗ g − C ) A ( g C ) g D g D A ( y ) hg − D g D g D B ( y ⊗ g − C ) B ( g C ) g D g D B ( y ) µ A η y ⊗ g − C ⊗ η id g C η y µ B (22)as does a similar diagram for ζ replacing η . Since η y ⊗ g − C = ζ y ⊗ g − C by assumption, η id g = B g ∗ ( A g ) − = ζ id g by (T-II).iii, and µ A , µ B are invertible 2-cells, we conclude that η x = ζ x .Now suppose h : A ⇒ B is a G -monoidal natural transformation. We define η : A ⇒ B by η ∗ = e D , η g = id g D for all g ∈ G , and for y ∈ C ( g C → h C ), we use (22) above to define η y := µ By ⊗ g − C ,g C ∗ ( h y ⊗ g − C ⊗ ( B g ∗ ( A g ) − )) ∗ ( µ Ay ⊗ g − C ,g C ) − . By construction, provided η is a transformation η (cid:55)→ h . It remains to verify that η : A ⇒ B is indeed atransformation. We prove one of the coherences below, and we give a hint as how to proceed for the othercoherences.(T-II).i Every composite step in the definition of η is natural.(T-II).ii This follows from functoriality of 1-cell composition ◦ together with the fact that h is monoidal, andtwo instances each (one for each of A and B ) of (F-I).ii, (21), and (F-IV).(T-II).iii This follows using 2 instances of (F-II).iii (one for each of A and B ) together with the fact that h id e = B ⊗ ( A ) − = B e ∗ ( A e ) − which is unitality for a monoidal natural transformation.(T-III) This condition is automatically satisfied.(T-IV) For x ∈ C ( g C → h C ) and y ∈ C ( k C → (cid:96) C ), we use the following shorthand as in Notation B.1 and61otation C.2::= g D h D A ( x ) := hg − D A ( x ⊗ g − C ) := k D (cid:96) D A ( y ) := (cid:96)k − D A ( y ⊗ k − C ) := j D j D A (id j C ) ∀ j ∈ G. := g D h D B ( x ) := hg − D B ( x ⊗ g − C ) := k D (cid:96) D B ( y ) := (cid:96)k − D B ( y ⊗ k − C ) := j D j D B (id j C ) ∀ j ∈ G. Suppose x ∈ C ( g C → h C ) and y ∈ C ( k C → (cid:96) C ). We begin with the following observation that the following62iagram commutes: (F-IV) µ A µ A ( µ A ) − (F-IV) ( A ) − (F-IV) µ A µ A µ A (F-IV) µ A (F-IV) µ A µ A (F-II).iii µ A µ A µ A µ A µ A µ A µ A µ A A A A µ A µ A A A µ A A A µ A A (F-IV) µ A µ A A A ⊗ A Defintion of a µ A (F-II).iii(F-V) µ A A a A µ A A ⊗ A A (23)Observe that (23) above also holds with ( A, A , µ A , A , a ) replaced by ( B, B , µ B , B , b ).Going around the outside of the diagram below corresponds to (T-IV), where we also use the abuse of63otation of h for B ∗ ( A ) − . µ A µ A ( µ A ) − ( A ) − h ⊗ h µ A h ⊗ h (23) for AA µ B µ A µ B h ⊗ h h ⊗ B ⊗ id ⊗ A A ⊗ A A h ⊗ h a h ⊗ h (18) µ A h ⊗ h h monoidal h ( µ A ) − B ⊗ B B b µ B B h ⊗ h µ B µ B (23) for BB µ B (F-IV) µ B µ B ( µ B ) − The faces without labels above commute by functoriality of 1-cell composition ◦ or by the shorthand h = B ∗ ( A ) − .(T-1) Every map is the identity map.(T-2) Every map is the identity map. References [Bar14] Bruce Bartlett,
Quasistrict symmetric monoidal 2-categories via wire diagrams , 2014, arXiv:1409.2148 .[BBCW19] Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang,
Symmetry, defects, and gauging of topo-logical phases , PHYSICAL REVIEW B (2019), 115147, arXiv:1410.4540 DOI:10.1103/PhysRevB.100.115147 . BD95] John C. Baez and James Dolan,
Higher-dimensional algebra and topological quantum field theory , J. Math. Phys. (1995), no. 11, 6073–6105, MR1355899 arXiv:q-alg/9503002 DOI:10.1063/1.531236 .[BDH15] Arthur Bartels, Christopher L. Douglas, and Andr´e Henriques,
Conformal nets I: Coordinate-free nets , Int. Math.Res. Not. IMRN (2015), no. 13, 4975–5052,
MR3439097 DOI:10.1093/imrn/rnu080 arXiv:1302.2604 .[BDH17] ,
Conformal nets II: Conformal blocks , Comm. Math. Phys. (2017), no. 1, 393–458,
MR3656522DOI:10.1007/s00220-016-2814-5 arXiv:1409.8672 .[BDH18] ,
Conformal nets IV: The 3-category , Algebr. Geom. Topol. (2018), no. 2, 897–956, MR3773743DOI:10.2140/agt.2018.18.897 arXiv:1605.00662 .[BDH19] ,
Fusion of defects , Mem. Amer. Math. Soc. (2019), no. 1237, vii+100,
MR3927541 arXiv:1310.8263 .[BGM19] Eugenia Bernaschini, C´esar Galindo, and Mart´ın Mombelli,
Group actions on 2-categories , Manuscripta Math. (2019), no. 1-2, 81–115,
MR3936135 DOI:10.1007/s00229-018-1031-2 arXiv:1702.02627 .[BMS12] John W. Barrett, Catherine Meusburger, and Gregor Schaumann,
Gray categories with duals and their diagrams ,2012, arXiv:1211.0529 .[Bra20] Oliver Braunling,
Quinn’s formula and abelian 3-cocycles for quadratic forms , 2020, arXiv:2005.05243 .[BS10] John C. Baez and Michael Shulman,
Lectures on n -categories and cohomology , Towards higher categories, IMAVol. Math. Appl., vol. 152, Springer, New York, 2010, MR2664619 arXiv:math/0608420 , pp. 1–68.[Buh14] Lukas Buhn´e,
Homomorphisms of Gray-categories as pseudo algebras , 2014, arXiv:1408.3481 .[Buh15] ,
Topics in three-dimensional descent theory , Ph.D. thesis, Fakult¨at f¨ur Mathematik, Informatik undNaturwissenschaften Fachbereich Mathematik der Universit¨at Hamburg, 2015, Available at https://d-nb.info/1072553694/34 .[CG11] Eugenia Cheng and Nick Gurski,
The periodic table of n -categories II: Degenerate tricategories , Cah. Topol. G´eom.Diff´er. Cat´eg. (2011), no. 2, 82–125, MR2839900 arXiv:0706.2307 .[CGPW16] Shawn X. Cui, C´esar Galindo, Julia Yael Plavnik, and Zhenghan Wang,
On gauging symmetry of modular categories ,Comm. Math. Phys. (2016), no. 3, 1043–1064,
MR3555361 DOI:10.1007/s00220-016-2633-8 .[Cui19] Shawn X. Cui,
Four dimensional topological quantum field theories from G -crossed braided categories , QuantumTopol. (2019), no. 4, 593–676, MR4033513 DOI:10.4171/qt/128 arXiv:610.07628 .[DGP +
20] Colleen Delaney, C´esar Galindo, Julia Plavnik, Eric C. Rowell, and Qing Zhang,
Braided zesting and its applications ,2020, arXiv:2005.05544 .[DH12] Chris Douglas and Andr´e Henriques,
Internal bicategories , 2012, arXiv:1206.4284 .[DR18] Christopher L. Douglas and David J. Reutter,
Fusion 2-categories and a state-sum invariant for 4-manifolds , 2018, arXiv:1812.11933 .[DSPS13] Chris Douglas, Chris Schommer-Pries, and Noah Snyder,
Dualizable tensor categories , 2013, arXiv:1312.7188 .[EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik,
Tensor categories , Mathematical Surveys andMonographs, vol. 205, American Mathematical Society, Providence, RI, 2015,
MR3242743 DOI:10.1090/surv/205 .[EM54] Samuel Eilenberg and Saunders MacLane,
On the groups H (Π , n ) . III , Ann. of Math. (2) (1954), 513–557, MR65163 DOI:10.2307/1969849 .[ENO10] Pavel Etingof, Dmitri Nikshych, and Victor Ostrik,
Fusion categories and homotopy theory , Quantum Topol. (2010), no. 3, 209–273, With an appendix by Ehud Meir, .[Gal17] C´esar Galindo, Coherence for monoidal G -categories and braided G -crossed categories , J. Algebra (2017),118–137, MR3671186 DOI:10.1016/j.jalgebra.2017.05.027 arXiv:1604.01679 .[GJ19] Terry Gannon and Corey Jones,
Vanishing of Categorical Obstructions for Permutation Orbifolds , Comm. Math.Phys. (2019), no. 1, 245–259,
MR3959559 DOI:10.1007/s00220-019-03288-9 arXiv:1804.08343 .[GJF19a] Davide Gaiotto and Theo Johnson-Freyd,
Condensations in higher categories , 2019, arXiv:1905.09566 .[GJF19b] Davide Gaiotto and Theo Johnson-Freyd,
Symmetry protected topological phases and generalized cohomology , J.High Energy Phys. (2019), no. 5, 007, 34,
MR3978827 arXiv:1712.07950 .[GMP +
18] Pinhas Grossman, Scott Morrison, David Penneys, Emily Peters, and Noah Snyder,
The Extended Haagerup fusioncategories , 2018, arXiv:1810.06076 .[GNN09] Shlomo Gelaki, Deepak Naidu, and Dmitri Nikshych,
Centers of graded fusion categories , Algebra Number Theory (2009), no. 8, 959–990, MR2587410 DOI:10.2140/ant.2009.3.959 arXiv:0905.3117 .[GPS95] R. Gordon, A. J. Power, and Ross Street,
Coherence for tricategories , Mem. Amer. Math. Soc. (1995), no. 558,vi+81,
MR1261589 DOI:10.1090/memo/0558 .[Gre10] Justin Greenough,
Monoidal 2-structure of bimodule categories , J. Algebra (2010), no. 8, 1818–1859,
MR2678824DOI:10.1016/j.jalgebra.2010.06.018 arXiv:0911.4979 .[Gur06] Michael Nicholas Gurski,
An algebraic theory of tricategories , ProQuest LLC, Ann Arbor, MI, 2006, Thesis (Ph.D.)–The University of Chicago,
MR2717302 , available at https://gauss.math.yale.edu/~mg622/tricats.pdf .[Gur12] Nick Gurski,
Biequivalences in tricategories , Theory Appl. Categ. (2012), No. 14, 349–384, MR2972968arXiv:1102.0979 . Gur13] ,
Coherence in three-dimensional category theory , Cambridge Tracts in Mathematics, vol. 201, CambridgeUniversity Press, Cambridge, 2013,
MR3076451 DOI:10.1017/CBO9781139542333 .[Gut19] Peter Guthmann,
The tricategory of formal composites and its strictification , 2019, arXiv:1903.05777 .[Hau17] Rune Haugseng,
The higher Morita category of E n -algebras , Geom. Topol. (2017), no. 3, 1631–1730, MR3650080DOI:10.2140/gt.2017.21.1631 .[HPT16] Andr´e Henriques, David Penneys, and James E. Tener,
Planar algebras in braided tensor categories , 2016, arXiv:1607.06041 , to appear Mem. Amer. Math. Soc.[Izu17] Masaki Izumi,
A Cuntz algebra approach to the classification of near-group categories , Proceedings of the 2014Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday, Proc. Centre Math.Appl. Austral. Nat. Univ., vol. 46, Austral. Nat. Univ., Canberra, 2017,
MR3635673 arXiv:1512.04288 , pp. 222–343.[JFS17] Theo Johnson-Freyd and Claudia Scheimbauer, (Op)lax natural transformations, twisted quantum field theories,and “even higher” Morita categories , Adv. Math. (2017), 147–223,
MR3590516 DOI:10.1016/j.aim.2016.11.014arXiv:1502.06526 .[Jon99] Vaughan F. R. Jones,
Planar algebras I , 1999, arXiv:math.QA/9909027 .[Kel05] G. M. Kelly,
Basic concepts of enriched category theory , Repr. Theory Appl. Categ. (2005), no. 10, vi+137,
MR2177301 , Reprint of the 1982 original [Cambridge Univ. Press, Cambridge;
MR0651714 ].[Kir01] Alexander Kirillov, Jr.,
Modular categories and orbifold models II , 2001, arXiv:math/0110221 .[Kir02] ,
Modular categories and orbifold models , Comm. Math. Phys. (2002), no. 2, 309–335,
MR1923177DOI:10.1007/s002200200650 .[Lac11] Stephen Lack,
A Quillen model structure for Gray-categories , J. K-Theory (2011), no. 2, 183–221, MR2842929DOI:10.1017/is010008014jkt127 arXiv:1001.2366 .[Lur09a] Jacob Lurie,
Higher topos theory , Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton,NJ, 2009,
MR2522659 DOI:10.1515/9781400830558 .[Lur09b] ,
On the classification of topological field theories , Current developments in mathematics, 2008, Int. Press,Somerville, MA, 2009,
MR2555928 arXiv:0905.0465 , pp. 129–280.[Mac52] Saunders MacLane,
Cohomology theory of Abelian groups , Proceedings of the International Congress of Mathe-maticians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R. I., 1952,
MR0045115 , pp. 8–14.[M¨ug05] Michael M¨uger,
Conformal orbifold theories and braided crossed G -categories , Comm. Math. Phys. (2005),no. 3, 727–762, MR2183964 DOI:10.1007/s00220-005-1291-z arXiv:math/0403322 .[MW19] Lukas M¨uller and Lukas Woike,
The little bundles operad , 2019, arXiv:1901.04850 .[Reu19] David Reutter,
Higher linear algebra in topology and quantum information theory , Ph.D. thesis, University ofOxford, 2019, available at https://ora.ox.ac.uk/objects/uuid:0b26d9f5-0e6e-4b81-b02d-a92820a3803a .[SW18] Christoph Schweigert and Lukas Woike,
Extended homotopy quantum field theories and their orbifoldization , 2018, arXiv:1802.08512 , to appear J. Pure Appl. Algebra.[Tur10] Vladimir Turaev,
Homotopy quantum field theory , EMS Tracts in Mathematics, vol. 10, European MathematicalSociety (EMS), Z¨urich, 2010, Appendix 5 by Michael M¨uger and Appendices 6 and 7 by Alexis Virelizier,
MR2674592DOI:10.4171/086 ..