Lifting bicategories through the Grothendieck construction
aa r X i v : . [ m a t h . C T ] O c t Lifting bicategories through the Grothendieck construction
Juan OrendainAbstract:
We treat the problem of lifting bicategories into double categoriesthrough categories of vertical morphisms. We make use of a specific instanceof the Grothendieck construction to provide, for every bicategory equippedwith a possible vertical category, together with a suitable monoidal pre-cosheaf relating these two structures, a double category lifting the decoratedbicategory along the category of vertical morphisms provided as set of initialconditions. We prove in particular that every decorated bicategory admits alift to a double category. We study relations of instances of our constructionto foldings, cofoldings, framed bicategories and globularily generated doublecategories.
Contents
Double categories were introduced by Ehresmann in [10, 11]. 2-categoriesand bicategories were later introduced by Benabeu in [3] and [4] respec-tively. Both concepts model 2-dimensional categorical structures, each withits advantages and disadvantages. Double categories accomodate more flexi-ble structures, see [22, 23] while bicategories are better behaved with respect1o pasting, see [19] as opposed to [20], for example. There are many classicalways of relating double categories and bicategories. Every double categoryhas an underlying horizontal bicategory and every bicategory can be consid-ered as a trivial double category. More interestingly the Ehresmann doublecategory of quintets construction [12] and the category of adjoints construc-tion [17] non-trivially associate double categories to bicategories.We consider the following problem: Given a bicategory B and a category B ∗ such that the collection of 0-cells of B and the collection of objects of B ∗ are equal, we wish to construct interesting double categories C having B as horizontal bicategory and having B ∗ as category of objects. We expressthis through the equation H ∗ C = (B ∗ , B) . In that case we call C an inter-nalization of the pair (B ∗ , B) . Put pictorially, if we are given a collection ofdiagrams of the form: ● ● αβϕ forming a bicategory B and a collection of vertical arrows: ●● f, g forming a category B ∗ , such that these diagrams are related by the fact thatthe 0-cells above and below are the same, we wish to understand ways ofcoherently combining these structures into collections of squares: ● ●● ● αψβf g forming double categories C , such that the arrows of the left and right edgesof the the above squares are precisely the arrows of B ∗ and such that thesquares of the form: 2 ●● ● αψβid ● id ● can be identified with the diagrams defining B . Solutions to this problemare easily seen to exist in specific cases, e.g. the case in which B ∗ is trivialis solved by the trivial double category construction and the case in which B ∗ and the horizontal category of B are equal is solved by the Ehresmanndouble category of quintets construction for B . An interesting instance of thisproblem is the case in which B ∗ is the category of von Neumann algebras andtheir morphisms and B is the bicategory of Hilbert bimodules, see [1, 2, 13].We provide solutions to this problem for general B ∗ and B .The Grothendieck construction establishes an equivalence between thecategorical notion of Grothendieck fibration and the algebraic notion ofpresheaf on Cat . Intuitively Grothendieck fibrations are categorical struc-tures capturing the notion of category parametrized by categories. TheGrothendieck construction puts this notion into algebraic terms. We makeuse of a specific instance of the Grothendieck construction to provide solu-tions to the problem presented above. Given a bicategory B and a category B ∗ as above, we construct, for every pre-cosheaf Φ of B ∗ on Cat satisfyingcertain conditions related to B , a double category C Φ solving the equation H ∗ C Φ = (B ∗ , B) posed above. In particular we prove that such equationadmits solutions for every pair (B ∗ , B) . The Grothendieck construction hasalready been studied in the context of limits in double category in [14].Recent techniques in the theory of double categories use additional struc-ture to reduce questions regarding the theory of double categories to ques-tions of associated bicategories. These techniques include the theory ofholonomies of Brown and Spencer [7], the theory foldings and cofoldings ofBrown and Mosa [6] and the theory of framed bicategories of Schulman [21]among others. These structures relate vertical and horizontal structures ofdouble categories and establish correspondences between certain squares ondouble categories with simpler squares in a coherent way. We study rela-tions of double categories constructed through our methods and the condi-tions mentioned above. Our techniques are closely related to the techniquesintroduced by Schulman in his treatise of monoidal fibrations and framed bi-categories. We prove that not all double categories constructed through ourmethods are framed bicategories. The exact relation between double cate-3ories constructed through our methods and framed bicategories constructedvia monoidal fibrations is yet to be explored.The problem of finding solutions to the equation H ∗ C = (B ∗ , B) has al-ready been studied by the author in [15, 16] where the notion of globularilygenerated double category, or GG double category for short, and verticallength were presented. GG double categories are minimal solutions to theabove problem and we thus study relations of this condition and double cate-gories constructed through the methods presented in this paper. The verticallength of a double category is meant to serve as a measure of how intricatethe the relation between vertical and horizontal composition of squares in adouble category is. We study the vertical length of double categories con-structed through our methods.We present the contents of this paper. In section 2 we establish the no-tational and pictorial conventions used throughout the paper. In section3 we provide a detailed account of the construction of double categories ofthe form C Φ and we prove that equations H ∗ C = (B ∗ , B) as above alwaysadmit solutions. In section 4 we present examples of double categoriesconstructed through the methods presented in section 3. We provide in par-ticular the construction of a linear double category C such that H ∗ C is thepair formed by the category of von Neumann algebras and the bicategoryof Hilbert bimodules. In section 5 we study relations between double cate-gories constructed through our methods and the conditions of having foldingsand cofoldings. In section 6 we prove that double categories constructedin section 3 all have vertical length 1. In section 7 we study conditions fordouble categories constructed in section 2 that guarantee that these doublecategories are GG. Finally, in section 8 we study the problem of universalityof the construction presented in section 3 in the specific case of bicategoriesdecorated by groups. In this first section we establish the notational and pictorial conventions usedthroughout the paper. We briefly recall the Grothendieck construction, andwe present a brief introduction to GG double categories and vertical length.4 icategories
Given a bicategory B we will write B , B , B for the collections of 0-,1-, and2-cells of B . We will write dom B , codom B for both the 1- and 2-dimensionaldomain and codomain functions of B , we will write id B for both the 0- and 1-dimensional identity functions of B and we will write ○ , ⊛ for the vertical andthe horizontal composition operations in B . We call dom B , codom B , id B , ○ , ⊛ the structure data of B . We call the identity transformations and associatorthe coherence data for B . 0- and 1-cells in a double category are pictoriallyrepresented by points and horizontal arrows respectively and 2-morphismsare represented by globe diagrams read from top to bottom, i.e. a diagramas: a bβαϕ represents a 2-cell from α to β . The vertical and horizontal composition op-erations in B are pictorially represented by vertical and horizontal concatena-tion of diagrams as above, see [5]. We will adopt the following non-standardconvention for identitiy 1-and 2-cells: Given a 0-cell a in a bicategory B wewill represent the identity endomorphism of a in B by a red arrow from a to a . The following diagrams thus represent 2-cells from id a to β , from ψ to id a and an endomorphism η of id a : a a a a a aβϕ αψ η Other conventions for the pictorial representation of identity endomorphismsin bicategories are [8]. We write bCat for the category of bicategories andpseudofunctors. Given a 0-cell a in B the horizontal composition operation ⊛ in B provides the category of endomorphisms End B ( a ) of a in B with thestructure of a monoidal category. The monoidal unit in End B ( a ) is the iden-tity id B a . Conversely, every monoidal category D defines a bicategory, whichwe will denote as D , with a single object ∗ . The horizontal composition5peration ⊛ in D is the tensor product operation in D and the identity1-cell id B∗ of the single 0-cell in B is the monoidal identity D .We will further adopt the following convention: Given a bicategory B wewill write E B for the category whose collection of objects is the collectionof 1-cells α in B satisfying the equation dom B ( α ) = codom B ( α ) and whosemorphisms are 2-cells in B . Further, we will write ˜ B for the category whosecollection of objects is the compliment in B of the collection of objects in E B and whose morphisms are 2-cells in B . The composition operation inboth E B and ˜ B is the vertical composition operation of B . Double categories
Given a double category C we will write C , C for the category of objectsand the category of morphisms of C respectively, we will write s, t, i forthe source, target and horizontal identity functors of C , and following [22]we will write ⊟ for the horizontal composition functor of C and we willwrite q for the vertical composition operation of both vertical morphismsand 2-morphisms in C . We call s, t, i, ⊟ the structure data for C and wecall the identity transformations and associator the coherence data for C .We write dCat for the category of double categories and double functors.We represent objects in C pictorially as verteces, horizontal and verticalmorphisms as horizontal and vertical arrows and we represent 2-morphismsas squares. We read squares from top to bottom and from left to right, i.e.a diagram as: a bc dαϕβf g represents a 2-morphism ϕ such that s ( ϕ ) = f, t ( ϕ ) = g and such that thevertical domain and codomain of ϕ are α and β respectively. We will adopta similar non standard convention as the one considered in the previoussection when pictorially representing identities. We will represent horizontalidentities pictorially as horizontal red arrows. Thus diagrams of the form:6 a a a a ab b b b b bϕβf g αψf g ηf g represent squares ϕ, ψ, η with vertical domain i a , vertical codomain i b andvertical domain and codomain i a and i b respectively. We represent the hor-izontal identity i f of a vertical morphism f ∶ a → b pictorially as: a ab bi f f f Further, we represent vertical identities pictorially with blue vertical arrows.Thus diagrams of the form: a b a c a ba c b c a bαϕβ αψβ αηβ represent squares ϕ, ψ, η such that s ( ϕ ) = id a , t ( ψ ) = id c and s ( η ) = id a , t ( η ) = id b . We represent the vertical and horizontal composition operations q , ⊟ pictorially as vertical and horizontal concatenation respectively. With ourconventions globular squares are precisely those 2-morphisms admitting pic-torial representations as: a ba bαϕβ ecorated bicategories Given a double category C we write HC for the bicategory whose 0-, 1-,and 2-cells are objects, horizontal morphisms, and globular squares of C respectively. The operation C ↦ HC ’flattens’ the double category C into abicategory by interpreting diagrams in C of the form: a ba bαϕβ as diagrams of the form: a bβαϕ Given a bicategory B we will say that a category B ∗ is a decoration of B ifthe collection of objects of B ∗ is the collection of 0-cells of B . In this case wewill say that the pair (B ∗ , B) is a decorated bicategory. The pair ( C , HC ) is a decorated bicategory for every double category C . We write H ∗ C forthis decorated bicategory. We call H ∗ C the decorated horizontalization of C . We write bCat ∗ for the category of decorated bicategories and deco-rated pseudofunctors, where given decorated bicategories (B ∗ , B) , (B ′∗ , B ′ ) we understand for a decorated pseudofunctor from (B ∗ , B) to (B ′∗ , B ′ ) a pair ( F ∗ , F ) where F ∗ ∶ B ∗ → B ′∗ and F ∶ B → B ′ and such that F ∗ , F coincidein B . We are interested in the following problem: Problem 2.1.
Let (B ∗ , B) be a decorated bicategory. Find double categories C satisfying the equation H ∗ C = (B ∗ , B) .We understand problem 2.1 as the problem of lifting a bicategory B to adouble category through an orthogonal direction, provided by B ∗ . We callsolutions to this problem internalizations of (B ∗ , B) . Pictorially, solutions toproblem 2.1 are to be understood as ways to formally understand diagramsof the form: 8 bβαϕ as diagrams of the form: a ba bαϕβ within a double category, taking non-identity, i.e. non-blue vertical mor-phims into account. See [1, 2] for applications of problem 2.1 in the case ofdecorated bicategories of von Neumann algebras to the theory of invariantsof manifolds of dimension 3. The Grothendieck construction
We use the Grothendieck construction to provide solutions to problem 2.1.The initial input of the Grothendieck construction is a category C and afunctor Φ ∶ C → Cat . The Grothendieck construction associates to Φ a newcategory ∫ C Φ which we now describe. Definition 2.2.
Let C be a category. Let Φ ∶ C → Cat be a functor. Thecategory ∫ C Φ is defined as follows:1. Objects:
The collection of objects of ∫ C Φ is the collection of all pairs ( x, a ) where x is a object in C and a is an object in Φ ( x ) .2. Morphisms:
Let ( x, a ) , ( x ′ , a ′ ) be objects in ∫ C Φ . The collection ofmorphisms, in ∫ C Φ , from ( x, a ) to ( x ′ , a ′ ) is the collection of all pairs ( α, β ) where α is a morphism, in C , from x to x ′ and where β is amorphism, in Φ ( x ′ ) , from Φ ( α )( x ) to x ′ .3. Composition:
Let ( x, a ) , ( x ′ , a ′ ) , ( x ′′ , a ′′ ) be objects in ∫ C Φ . Let ( α, β ) and ( α ′ , β ′ ) be morphisms, in ∫ C Φ , from ( x, a ) to ( x ′ .a ′ ) andfrom ( x ′ , a ′ ) → ( x ′′ , a ′′ ) respectively. The composition ( α ′ , β ′ )( α, β ) isdefined by the equation 9 α ′ , β ′ )( α, β ) = ( α ′ α, β ′ Φ ( α ′ )( β )) The GG piece and vertical length
We briefly recall the basics of the theory of globularily generated doublecategories and their vertical length. We refer the reader to [15, 16] for anaccount on the subject. We say that a double category C is globularilygenerated or GG for short, if it is generated, as a double category, by itscollection of globular squares, i.e. a double category C is GG if C is generatedby squares of the form: a b a aa b b bϕf g αψβ The category of morphisms of a GG category C admits a presentation asa limit C = lim Ð→ V kC where the categories V kC are defined inductively by set-ting V C = HC and by making V kC be the category generated by horizontalcompositions of morphisms in V k − C for every k ≥ . We define the verticallength ℓC of C as the minimal k ≥ such that C = V kC if such k exists.We make ℓC = ∞ otherwise. The vertical length of a GG categories is to beunderstood as a measure of complexity on the relations between the verticaland horizontal composition operations q , ⊟ in C .For a general i.e. non-GG double category we write γC for the sub-double category of C generated by 2.morphisms admitting a diagramaticrepresentation of the form: a b a aa b b bϕf g αψβ Thus defined γC is such that H ∗ γC = H ∗ C . Moreover γC is contained inevery sub-double category of C satisfying this condition. Double categories10f the form γC are always GG and every GG double category is of this form.GG categories are thus considered as minimal solutions to problem 2.1. Weregard the theory of GG categories as lying ’in between’ the theory of doublecateogries and the theory of bicategories. We define the vertical length ℓC of a non-GG double category C by ℓγC . We use a special instance of the Grothendieck construction in order to pro-duce internalizations of decorated bicategories. The special situation weconsider is as follows:Let (B ∗ , B) be a decorated bicategory. We consider the Grothendieck con-struction with input data given by a functor Φ ∶ B ∗ → Cat such that forevery object a of B ∗ the category Φ ( a ) is equal to End B ( a ) and such thatfor every pair of objects a, b in B ∗ and for every α ∶ a → b in B ∗ thefunctor Φ α ∶ End B ( a ) → End B ( b ) is monoidal. That is, we consider theGrothendieck construction ∫ B ∗ Φ for functorial extensions Φ ∶ B ∗ → Cat ⊗ ofthe function associating to every object a in B ∗ the endomorphism category End B ( a ) . For our purposes we consider the following extension of ∫ B ∗ Φ inthe situation described above: We write ∫ ∗B ∗ Φ for the category obtained asthe disjoint union of ∫ B ∗ Φ and ˜ B . We prove the following theorem. Theorem 3.1.
In the situation above the pair (B ∗ , ∫ ∗B ∗ Φ ) admits the struc-ture of a double category. Writing C Φ for this double category, C Φ satisfiesthe equation: H ∗ C Φ = (B ∗ , B) Proof.
Let (B ∗ , B) be a decorated bicategory. Let Φ ∶ B ∗ → Cat ⊗ be afunctor such that Φ ( a ) is equal to End B ( a ) for every object a in B ∗ . Wewish to prove that in this case the pair C Φ = (B ∗ , ∫ ∗B ∗ Φ ) admits a structureof double category such that with this structure C Φ internalizes (B ∗ , B) . Webegin by defining the structure functors of C Φ .First observe that the collection of objects of ∫ B ∗ Φ is the collection of allpairs of the form ( a, α ) where a is a 0-cell of B and where α is an endomor-phism of a in B . Identifying every pair ( a, α ) with the endomorphism α weidentify the collection of objects of ∫ B ∗ Φ with the collection of 1-cells α of B sα = tα . That is, we identify the collection of objects of ∫ B ∗ Φ withthe compliment in B of the collection of objects of ˜ B . The collection ofobjects of ∫ ∗B ∗ Φ can thus be identified with B . We assume this identificationhas been performed and we thus say that the collection of objects of ∫ ∗B ∗ Φ is equal to B .We define source and target functors s, t ∶ ∫ ∗B ∗ Φ → B ∗ for C Φ . We makethe object functions of s, t to be equal to the restrictions, to B , of thedomain and codomain functions dom B , codom B functions of B . Let ( f, ϕ ) bea morphism in ∫ B ∗ Φ . In that case we make both s ( f, ϕ ) and t ( f, ϕ ) to beequal to the morphism f in B ∗ . Let now ϕ be a morphism in ˜ B . Supposethat ϕ admits a pictorial representation, in B , as: a bβαϕ In that case we make s ( ϕ ) = id a and t ( ϕ ) = id b . The fact that thus defined s, t indeed define functors from ∫ ∗B ∗ Φ to B ∗ follows from the way compositionin ∫ B ∗ Φ is defined, see section 2, and by the fact that B is a bicategory. Wethus represent morphisms ϕ in ˜ B and morphisms ( f, ϕ ) in ∫ B ∗ Φ pictoriallyas: a b b ba b a aβϕα β ( f, ϕ ) αf f respectively. We now define a horizontal identity funtor for C Φ .Let a be an object of B ∗ . We make the horizontal identity i a of a , in C Φ , to be equal to the horizontal identity id B a of a in B . Let f ∶ a → b bea morphism in B ∗ . In that case we make i f to be equal to the morphism ( f, id B i b ) of ∫ ∗B ∗ Φ . The fact that thus defined i indeed defines a functor from B ∗ to ∫ ∗B ∗ Φ again easily follows from the way the composition operationin ∫ B ∗ Φ is defined. An easy check proves that i is compatible with thefunctors s, t defined above. We represent the horizontal identity ( f, id B i b ) ofa morphism f ∶ a → b in B ∗ pictorially as:12 ba ai f f f In order to define a horizontal composition functor on C Φ we first intro-duce a notational modification on the collection of morphisms of ∫ ∗B ∗ Φ . Wewill write ( f, f, ϕ ) for every morphsim ( f, Φ ) in ∫ B ∗ Φ and we will write ( id a , id b , ϕ ) for every 2-cell ϕ in ˜ B such that the 0-dimensional domain andcodomain of ϕ are equal to a and b respectively. Under this convention thecollection of morphisms of ∫ ∗B ∗ Φ is the collection of all triples ( f, g, ϕ ) where f, g are morphisms in B ∗ and where ϕ is a 2-cell in B . If the morphisms f, g are not the identities, in B ∗ , of the 0-dimensional domain and codomain of ϕ respectively, then f and g are equal.With this notational convention in place we now define a horizontal com-position functor ⊟ for C Φ . We make ⊟ to be defined as the horizontal com-position operation of B on the collection of objects of ∫ ∗B ∗ Φ . In order todefine an extension of this operation to a functor ⊟ ∶ ∫ ∗B ∗ Φ × B ∗ ∫ ∗B ∗ Φ → ∫ ∗B ∗ Φ we first analize the morphism of ∫ ∗B ∗ Φ × B ∗ ∫ ∗B ∗ Φ . Let ( f i , g i , ϕ i ) with i = , be a morphism in ∫ ∗B ∗ Φ × B ∗ ∫ ∗B ∗ Φ . Observe that by the way the functors s, t were defined if one of f , f , g , g is not an identity morphism of B ∗ then none of f , f , g , g are identity morphisms in B ∗ and moreover in thatcase f , f , g , g are all equal. A pair ( f i , g i , ϕ i ) of morphisms in ∫ ∗B ∗ Φ isthus a morphism in ∫ ∗B ∗ Φ × B ∗ ∫ ∗B ∗ Φ if either all of the f i , g i with i = , are equal or all of the f i , g i with i = , are identity morphisms in B ∗ , inwhich case the pair ( f i , g i , ϕ i ) with i = , is actually horizontally compos-able in B , i.e. is a 2-cell in B × B B . We define the horizontal compositionof pairs satisfying this last condition as their horizontal composition in B .In order to define a horizontal composition functor ⊟ we now need onlyto define the horizontal composition of morhisms in ∫ B ∗ Φ × B ∗ ∫ B ∗ Φ . Let ( f, ϕ ) , ( f, ψ ) be a pair in ∫ B ∗ Φ × B ∗ ∫ B ∗ Φ . We make the horizontal composi-tion ( f, ϕ )⊟( f, ψ ) to be the pair ( f, ϕ ⊛ ψ ) where φ ⊛ ψ denotes the horizontalcomposition, in B , of the composable par of 2-cells ( ϕ, ψ ) . We prove thatthus defined the operation of horizontal composition ⊟ does indeed define afunctor ∫ ∗B ∗ Φ × B ∗ ∫ ∗B ∗ Φ → ∫ ∗B ∗ Φ .In order to prove the functoriality of the operation ⊟ defined above weneed to prove that the two possible compositions of compatible squares asin the following diagram: 13 b b ca ′ b ′ b ′ c ′ a ′ b ′ b ′ c ′ a ′′ b ′′ b ′′ c ′′ ϕα f ψβfαϕ ′ g βψ ′ g are equal. From the fact that B is a bicategory and thus satisfies the ex-change lemma the above diagrammatic equation is true whenever the mor-phisms involved are 2-cells in B , i.e. when all the f i , g i with i = , , areidentity morphisms in B ∗ . We now prove that the above equation is true formorphisms in ∫ B ∗ Φ . Let ( f, ϕ ) , ( f, ψ ) and ( f ′ , ϕ ′ ) , ( f ′ , ψ ′ ) be morphisms in ∫ B ∗ Φ fitting in the diagram: a a a ab b b bb b b bc c c cα ( f, ϕ ) βf f γ ( f, ψ ) ηf fβ ( f ′ , ϕ ′ ) νf ′ f ′ η ( f ′ ψ ′ ) ǫf ′ f ′ We wish to prove that in this case the two possible compositions of thediagramatic scheme above are equal. That is, we wish to prove that thefollowing equation holds: [( f ′ , ϕ ′ ) ⊟ ( f ′ , ψ ′ )] q [( f, ϕ ) ⊟ ( f, ψ )] = [( f ′ , ϕ ′ ) q ( f, ϕ )] ⊟ [( f ′ , ψ ′ ) q ( f, ψ )] ( f ′ , ϕ ′ ⊛ ψ ′ ) q ( f, ϕ ⊛ ψ ) which in turn is equal to the pair ( f ′ f, ( ϕ ′ ⊛ ψ ′ ) Φ f ′ ( ϕ ⊛ ψ )) Now the right hand side of the above equation is equal to the horizontalcomposition ( f ′ f, ϕ ′ Φ f ′ ϕ ) ⊟ ( f ′ f, ψ ′ Φ f ′ ψ ) which in turn is equal to the pair ( f ′ f, [ ϕ ′ Φ f ′ ϕ ] ⊛ [ ψ ′ Φ f ′ ψ ]) In order to prove the desired equation we thus need to prove the equality: ( ϕ ′ ⊛ ψ ′ ) Φ f ′ ( ϕ ⊛ ψ ) = [ ϕ ′ Φ f ′ ϕ ] ⊛ [ ψ ′ Φ f ′ ψ ] but this follows from the fact that Φ is a functor on Cat ⊗ . The horizontalcomposition operation ⊟ defined above is thus functorial not only on ˜ B butis also functorial on ∫ B ∗ Φ . We conclude that the horizontal compositionoperation ⊟ defined above is a functor from ∫ ∗B ∗ Φ × B ∗ ∫ ∗B ∗ Φ to ∫ ∗B ∗ Φ .From the way they were defined it is easily seen that the surce and targetfunctors s and t of C Φ are compatible with the horizontal composition functor ⊟ of C Φ . Moreover, the left and right unit transformations and the associatorof C are easily seen to lift to compatible left and right horizontal identitytransformations and associators for C Φ . We conclude that as defined above C Φ is a double category.Finally, the category of objects C Φ0 of C Φ is equal to B ∗ , the collectionof objects of the category of morphisms C Φ1 of C Φ is the collection of 1-cells B of B . Now the collection of globular squares of C Φ is equal to theset of squares ( f, ϕ ) in ∫ ∗B ∗ Φ such that both s ( f, ϕ ) and t ( f, ϕ ) are identitiymorphisms in B ∗ . This set is precisely the collection of 2-cells B of B . By theway the double category C Φ was constructed the restriction of the structurefunctors defining C Φ , i.e. source, target, horizontal identity and horizontalcomposition functors, to cells in B are precisely the corresponding structurefunctions for B . This proves that C Φ satisfies the equation H ∗ C Φ = ( B ∗ , B ) .This concludes the proof. ∎ ( B ∗ , B ) . Corollary 3.2.
Let ( B ∗ , B ) be a decorated bicategory. There exists a doublecategory C satisfying the equation H ∗ C = ( B ∗ , B ) . Proof.
Let ( B ∗ , B ) be a decorated bicategory. We wish to prove that thereexists a double cateogry C satisfying the equation H ∗ C = ( B ∗ , B ) .We prove that F un ( B ∗ , Cat ⊗ ) B is non-empty. We do this by exhibiting a Φ in F un ( B ∗ , Cat ⊗ ) B . We make Φ be such that Φ ( a ) = End B ( a ) for everyobject a in B ∗ . Let a, b be objects in B ∗ . Let f ∶ a → B be a morphismin B ∗ . We make Φ f ∶ End B ( a ) → End B ( b ) be the constant functor on theidentity endomorphism id B b . Thus defined Φ f is clearly monoidal. Moreover,the assignment f ↦ Φ f is clearly compatible with the category structure of B ∗ . The double category C Φ associated to Φ by theorem 3.1 satisfies theequation H ∗ C Φ = ( B ∗ , B ) . This concludes the proof. ∎ Given a decorated bicategory ( B ∗ , B ) we will write F un ( B ∗ , Cat ⊗ ) B for thecategory whose collection of objects is the collection of all functors satisfyingthe conditions in the statement of theorem 3.1, i.e. the collection of objectsof F un ( B ∗ , Cat ⊗ ) B is the collection of all functors Φ ∶ B ∗ → Cat ⊗ such that Φ ( a ) = End B ( a ) for every 0-cell a of B , and whose collection of morphismsis the collection of natural transformations between such functors. Further,we will write dCat (B ∗ , B) for the category whose objects are internalizationsof ( B ∗ , B ) , i.e. the objects of dCat (B ∗ , B) are double categories C satisfyingthe equation H ∗ C = ( B ∗ , B ) , and whose morphisms are double functors F such that H ∗ F = id (B ∗ , B) . The Grothendieck construction admits anobvious functorial extension (a 2-functorial extension in fact) to a categoryof Grothendieck fibrations. It is not diffucult to see that this functor extendsthe construction of the double categories C Φ presented in theorem 3.1 to afunctor C ● ∶ F un ( B ∗ , Cat ⊗ ) B → dCat (B ∗ , B) . We make use of this only insection 6 in a specific case. We are interested in the construction of C Φ when both B ∗ and B are linear. With this in mind we make the followingobservation: Observation 3.3.
Let k be a field. We will say that a bicategory B is linearover k if for every α, β ∈ B with the same source and target, the collec-tion of 2-cells Hom B ( α, β ) from α to β in B is endowed with the structureof a vector space over k in such a way that the vertical and the horizontalcomposition operations in B are k -bilinear and such that the coherence dataof B is linear. We will say that a decorated bicategory ( B ∗ , B ) is linear if16oth B ∗ and B are linear. Observe that given a linear bicategory B the hor-izontal composition operation in B provides the category of endomorphisms End B ( a ) of any 0-cell a in B with the structure of a linear tensor category.We will say that a double category C is linear if C , C are both linear and ifthe structure data s, t, i, ⊟ of C are linear. We write Cat ⊗ k for the categorywhose objects are linear tensor categories and whose morphisms are linearstrict tensor functors. Given a linear decorated bicategory ( B ∗ , B ) we write dCat k (B ∗ , B) for the category whose objects are linear double categories C satisfying H ∗ C = ( B ∗ , B ) and whose morphisms are double functors F withlinear objects and morphism functors F , F such that H ∗ F = id (B ∗ , B) . Itis easily seen that the construction presented in theorem 3.1 extends to afunctor C ● ∶ F un ( B ∗ , Cat ⊗ k ) → dCat k (B ∗ , B) . In this section we provide examples of double categories obtained throughthe methods provided by theorem 3.1. We provide non-equivalent solutionsto problem 2.1 in certain cases, we provide interpretations whithin the theoryof double categories of classic constructions in the theory of tensor categoriesand we provide a linear double category of von Neumann algebras.
Single object decorated bicategories
Let M be a monoid. We write Ω M for the delooping category of M . By theEckman-Hilton argument [9] Ω M admits a strict monoidal structure only inthe case in which M is commutative, in which case the tensor product oper-ation on Ω M is provided by the product operation of M . Given a monoidalcategory D the pair ( Ω M, D ) is a decorated bicategory and every strictsingle object decorated bicategory is of this form. We consider double cate-gories of the form C Φ for functors Φ in F un ( Ω M, Cat ⊗ ) D for a monoidalcategory D . We first consider the case in which D of the form Ω N for acommutative monoid N .Let Φ be a functor in F un ( Ω M, Cat ⊗ ) N for monoids M, N where N is commutative. Any such Φ associates End N ( ∗ ) to the only object ∗ in Ω M and associates a monoidal endofunctor Φ m of End N ( ∗ ) to every m ∈ M . That is, every functor Φ in F un ( Ω M, Cat ⊗ ) N associates Ω N tothe only object ∗ of Ω M and defines a morphism, which we keep denotingby Φ , from M to the monoid of monoid endomorphisms End ( N ) of N . Thecategory of objects C Φ0 of C Φ is equal to Ω M . To compute C Φ we thus only17eed to compute the category of morphisms C Φ1 of C Φ . The category ˜2Ω N is empty and thus C Φ1 is equal to ∫ Ω M Φ which in this case is equal to thedelooping category Ω ( N ⋊ Φ M ) . The squares of C Φ are thus all of the form: ∗ ∗∗ ∗ ( m, n ) m m with m ∈ M, n ∈ N . Vertical and horizontal composition of such squaresis defined by composition in N ⋊ Φ M and by the product operation in M respectively. Analogous computations apply in the case in which either M or N is a group and in the case M and N are algebras over a field k or vonNeumann algebras. We have the following corollary of theorem 3.1. Corollary 4.1.
Let
M, N be monoids such that N is commutative. Problem2.1 for the decorated bicategory ( Ω M, N ) admits at least as many solu-tions, up to double equivalence, as there are isomorphism types of extensions N ⋊ Φ N . Proof.
The result follows by theorem 3.1, by the observations above andby the easy observation that any double equivalence F ∶ C Φ → C Ψ for Φ , Ψ in F un ( Ω M, Cat ⊗ ) N induces an isomorphism from N ⋊ Φ M to N ⋊ Ψ M .This concludes the proof of the lemma. ∎ Corollary 4.1 can be refined as follows: Suppose that
M, N are groups. Inthat case observe that both the category of objects and the category ofmorphisms of the double cateogry C Φ presented in the proof of corollary 4.1are groupoids. In that case the decorated bicategory ( Ω M, N ) not onlyadmits solutions but admits solutions internal to the category of groupoidsand groupoid morphisms. A similar comment applies in the case in which M, N are algebras over a field k . In that case the double category ( Ω M, N ) not only admits internalizations but admits k -linear internalizations. Westudy a stronger version of corollary 4.1 in section 6. A non-strict case ofthe linear version of corollary 4.1 would imply a study of extensions of E algebras. Graded vector spaces
G, H be groups. Suppose H is abelian. We write C G ( H ) for themonoidal category whose objects are of the form δ g with g ∈ G and suchthat Hom C G ( H ) ( δ g , δ g ′ ) is equal to H when g = g ′ and empty otherwise. Ten-sor product in C G ( H ) is defined by δ g ⊗ δ g ′ = δ gg ′ on objects and by a ⊗ a ′ = aa ′ for endomorphisms a, a ′ of δ g and δ g ′ . The pair ( Ω G, C G ( H )) is a decorated2-category.Let Φ be a functor in F un ( Ω G, Cat ⊗ ) C G ( H ) . Such a functor is such that Φ ( ∗ ) = C G ( H ) and such that Φ g is a monoidal endofunctor of C G ( H ) forevery g ∈ G . Assume that Φ g is the identity on the collection of objects of C G ( H ) and that the restriction of Φ g to endomorphisms of δ g , δ g ′ is equalfor every g, g ′ ∈ G . Such a functor Φ defines a morphism, which we keepdenoting by Φ , from G to Aut ( H ) . In this case the category of objects C Φ0 of C Φ is equal to Ω G , the category ˜2 C G ( H ) is empty and thus the categoryof morphisms C Φ1 of C Φ is equal to ∫ Ω G Φ . The collection of objects of ∫ Ω G Φ is equal to G . Given g, g ′ ∈ G the collection of morphisms Hom ∫ Ω G Φ ( g, g ′ ) isthe collection of all pairs ( g, h ) with g ∈ G and h ∈ H whenever g = g ′ and isempty otherwise. That is, the squares of C Φ are all of the form: ∗ ∗∗ ∗ δ g ′ ( g, h ) δ g ′ g g where g, g ′ ∈ G and h ∈ H . The vertical composition of any two such pairs ( g ′ , h ′ ) q ( g, h ) is equal to ( g ′ g, h ′ Φ g ′ ( h )) and the horizontal composition isgiven by ( g, h ) ⊟ ( g, h ′ ) = ( g, hh ′ ) . Identifying every g ∈ G with the object δ g in C G ( H ) and identifying every pair ( g, h ) with g ∈ G and h ∈ H with h as an endomorphism of δ g it is easily seen that the vertical monoidalcategory of C Φ , i.e. the category whose objects are vertical morphisms of C Φ , whose morphisms are 2-morphisms in C Φ , whose composition operationis the horizontal composition of C Φ and whose tensor product is the verticalcomposition of C Φ , is isomorphic to the category C G ( H, Φ ) obtained from C G ( H ) by twisting by Φ , see [18]. The obvious linear extension of thisexample also holds. That is, if k is a field, G, H are groups, H is abelian and Φ is a functor in F un ( G, Cat ⊗ k ) Vec Gk ( H ) then C Φ is k -linear and its verticalcategory is isomorphic, as k -linear tensor categories to Vec Gk ( H, Φ ) .19 lgebras and von Neumann algebras Let k be a field. We write Alg k for the bicategory with unital k -algebras,representations, and intertwining operators as 0-, 1-, and 2-dimensional cells.The horizontal composition operation ⊛ on Alg k is the relative tensor prod-uct of representations. The tensor category of endomorphisms End
Alg k ( A ) of an algebra A is the category of A -bimodules M od A . We write Alg k forthe category whose objects are k -algebras and whose morphisms are unitalalgebra morphisms. The pair ( Alg k , Alg k ) is a linear decorated bicategory.Let Φ be the functor in F un ( Alg k , Cat ⊗ ) Alg k such that for every f ∶ A → B the functor Φ f is the constant tensor functor on B B B . The linear doublecategory C Φ is such that H ∗ C Φ = ( Alg k , Alg k ) . Non-globular squares A AB BMϕNf f in C Φ are formed by pairs ( f, ϕ ) where f ∶ A → B and ϕ is a morphism of B -modules from the trivial bimodule B to N .Analogous constructions apply for any flavor of algebra, representationand intertwiner operator. Of special relevance is the case of von Neumannalgebras, Hilbert bimodules, and bounded intertwiner operators. We write [ W ∗ ] for the bicategory whose 0-,1- and 2-cells are von Neumann algebras,Hilbert bimodules, and intertwiner operators respectively. The horizontalcomposition operation ⊟ is provided by Connes fusion ⊠ and the horizon-tal identity i A of a von Neumann algebra A is provided by the Haagerupstandard form L A of A , see [13, 24]. Given a von Neumann algebra A thecategory of endomorphisms End [ W ∗ ] ( A ) is the category of Hilbert bimod-ules M od A . We write vN for the category whose objects are von Neumannalgebras and their morphisms. The pair ( vN , [ W ∗ ]) is a linear decoratedbicategory. Wite Φ for the functor in F un ( vN , Cat ⊗ ) [ W ∗ ] such that Φ f is the constant functor on L B for every von Neumann algebra morphism f ∶ A → B . The double category C Φ is such that H ∗ C Φ = ( vN , [ W ∗ ]) . Anon-globular square 20 AB BMϕNf f in C Φ is now formed by a pair ( f, ϕ ) where f ∶ A → B is a morphism ofvon Neumann algebras and where ϕ ∶ L B → N is an intertwiner operatorof B -bimodules. The horizontal identity functor i in C Φ associates to everymorphism of von Neumann algebras f ∶ A → B the pair ( f, id L B ) and thehorizontal composition functor ⊟ in C Φ is defined as ( f, ϕ ) ⊟ ( f, ψ ) = ( f, ϕ ⊠ ψ ) for every horizontally compatible pair ( f, ϕ ) , ( f, ψ ) in C Φ . Induction
The examples provided above are rather unsatisfactory solutions to problem2.1 for the decorated bicategories of algebras and von Neumann algebras. Wepresent an alternative solution for proper subategories of the correspondingdecorations.Let
A, B be k -algebras. Let f ∶ A → B be a morphism. We say that f is tensor inductive if the induced representation functor f ∗ ∶ M od A → M od B associated to f is a tensor functor. Examples of tensor inductivemorphisms are isomorphism and morphisms between commutative algebras.The composition of tensor unductive morphisms is again tensor inductive andthe identity id A of an algebra A is always tensor inductive. We write Alg
T Ik for the category whose objects are unital k -algebras and whose morphisms areunital tensor inductive morphisms. Thus defined Alg
T Ik has the category ofcommutative algebras and the underlying groupoid of
Alg k as subcategories.The pair ( Alg
T Ik , Alg k ) is a linear decorated bicaegory.Write Φ ∶ Alg
T Ik → Cat ⊗ for the functor such that Φ ( A ) = M od A forevery algebra A and such that Φ ( f ) = f ∗ for every morphism f ∶ A → B in Alg
T Ik . Thus defined Φ is an object in F un ( Alg
T Ik , Cat ⊗ ) Alg k . The doublecategory C Φ is a linear solution to the equation H ∗ C Φ = ( Alg
T Ik , Alg k ) . Anon-globular square in C Φ of the form21 AB BMϕNf f is now formed by a pair ( f, ϕ ) where f is a tensor inductive morphism from A to B , and where ϕ is a morphism from f ∗ M to N in M od B . The horizontalidentity i f of a tensor inductive morphism f ∶ A → B is the pair ( f, id B ) .A contravariant version of the above construction applies for morphisms f such that the restriction functor f ∗ is a tensor functor.An analogous construction applies in the case of von Neumann algebras.We say that a morphism of von Neumann algebras f ∶ A → B is ⊠ -inductive if f ∗ ∶ M od A → M od B is a tensor functor. We write vN ⊠ for the category of vonNeumann algebras and ⊠ -inductive morphisms. Again the pair ( vN ⊠ , [ W ∗ ]) is a decorated bicategory. Write Φ for the functor from vN ⊠ to Cat ⊗ suchthat Φ ( A ) = M od A for every A and such that Φ f = f ∗ for every morphism f ∶ A → B in vN ⊠ . Thus defined Φ is a functor in F un ( vN ⊠ , Cat ⊗ ) [ W ∗ ] .The double category C Φ is a linear double category satisfying the equation H ∗ C Φ = ( vN ⊠ , [ W ∗ ]) . A non-globular square A AB BMϕNf f in C Φ is now defined by a pair ( f, ϕ ) where f ∶ A → B is a ⊠ -inductivemorphism and where ϕ is an intertwining operator from f ∗ M to N in M od B .The horizontal identity i f of a von Neumann algebra morphism f ∶ A → B isthe pair ( f, id L B ) . The horizontal composition functor ⊟ of C Φ again actson pairs of morphisms ( f, ϕ ) , ( f, ψ ) as ( f, ϕ ) ⊟ ( f, ψ ) = ( f, ϕ ⊠ ψ ) . Comparethis to [2]. A similar construction applies when substituting induction byrestriction and when substituting von Neumann algebras with C ∗ -algebras. In this section we analyze possible folding and cofolding structures on certaindouble categories constructed through the methods of theorem 3.1. We refer22he reader to [25] for a treatment of holonomies, foldings, and cofoldings ondouble categories.We consider the following situation: Let
M, A be monoids. Suppose A iscommutative. In that case the pair ( Ω M, A ) is a decorated 2-category.Let Φ be a functor in F un ( Ω M, Cat ⊗ ) . A holonomy on C Φ is a 2-functorfrom C Φ0 to H A , i.e. a holonomy on C Φ is a morphism from M to thecategory of horizontal morphisms of C Φ which are the identity on objects.This category is trivial since Ω A has a single object. Likewise C Φ admits asingle trivial coholonomy. Foldings are extensions of holonomies establish-ing compatible bijections between certain squares. Cofoldings are definedsimilarly. We prove the following proposition. Proposition 5.1.
Let
M, A be monoids. Suppose A is commutative. Let Φ be a functor in F un ( Ω M, Cat ⊗ ) A such that for Φ m = id Ω A for every m ∈ M . In that case C Φ admits a folding and a cofolding. Proof.
Let
M, A be monoids. Suppose that A is commutative. Let Φ bea functor in F un ( Ω M, Cat ⊗ ) A , such that Φ m = id Ω A for every m ∈ M .We wish to prove in this case that the trivial holonomy on C Φ extends to afolding on C Φ .We wish to prove that the trivial holonomy on C Φ extends to a 2-functor Λ ∶ C Φ0 → Q A such that H Λ = id A and such that Λ is fully faithful onsquares. For every square boundary: ∗ ∗∗ ∗ m m in C Φ we construct a bijection Λ ,mm, between the collection of squares, in C Φ ,of the following two forms ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ m m ( m, a ) ( , a ) Λ of these bijection forms the desired 2-functor. Wedefine Λ ,mm, as the function associating a 2-morphism ( m, a ) of the form onthe left-hand side above, the square ( , a ) . Thus defined Λ ,mm, is clearly abijection from the set of diagrams of the form on the left-hand side aboveto the set of diagrams of the form on the right-hand side above. It is easilyseen that the collection Λ of bijections of the form Λ ,mm, , with m ∈ M , isthe identity on globular 2-morphisms of C Φ and that is compatible withidentities and horizontal composition in C Φ . We prove that Λ is compatiblewith vertical composition in C Φ . To this end consider a vertical compositionof squares of the form: ∗ ∗∗ ∗∗ ∗ m m ( m, a ) m ′ m ′ ( m ′ , a ′ ) by the way Φ was chosen, the bove diagram is equal to the square: ∗ ∗∗ ∗ m ′ m m ′ m ( m ′ m, a ′ a ) Applying Λ to the above square we obtain the square: ∗ ∗∗ ∗ ( , a ′ a ) which in turn is equal to the square: 24 ∗ ∗∗ ∗ ∗∗ ∗ ∗ ( , a )( , ) ( , )( , a ′ ) We conclude that Λ , as defined above, is compatible with vertical composi-tion in C Φ and thus that Λ is a folding on C Φ . a similar argument proves thatthe trivial coholonomy on C Φ extends to a cofolding on C Φ . This concludesthe proof. ∎ From proposition 5.1 and from [21] we have the following corollary.
Corollary 5.2.
Let
M, A be monoids. Suppose A is commutative. Let Φ bea functor in F un ( Ω M, Cat ⊗ ) A such that Φ m = id Ω A for every m ∈ M . Inthat case C Φ is a framed 2-category. The following is an example of a double category of the form C Φ with M, D as in propostion 5.1 but with non-identity Φ such that C Φ does not admit afolding or a cofolding and is thus not a framed bicategory. Example 5.3.
In the notation of proposition 5.1 let M = Z , let D = Ω Z ,and let Φ be the functor in F un ( Ω Z , Cat ⊗ ) Z such that Φ a ( m ) = am forevery a ∈ Z , m ∈ Z where we write Z multiplicatively and Z additively.The double category C Φ has a single horizontal morphism with endomorphismgroup isomorphic to D . We prove that C Φ does not admit foldings. Suppose Λ is a folding of C Φ . Λ establishes a bijection between squares of the followingtwo forms: ∗ ∗ ∗ ∗∗ ∗ ∗ ∗− − oreover Λ is compatible with vertical and horizontal composition in C Φ .Compatibility of Λ with horizontal composition says that ϕ ∶ Z → Z suchthat Λ ( − , a ) = ( , ϕ ( a )) for every a ∈ Z is such that ϕ ∈ Aut ( Z ) = Z . Now, the compatibility of Λ with vertical composition in C Φ says that Λ ( − , a ) ⊟ Λ ( − , a ′ ) = Λ ( , a ′ − a ) for every a, a ′ ∈ Z , but the left hand side ofthe above equation is equal to ( , ϕ ( a ) + ϕ ( a ′ )) . Thus ϕ is an automorphismof Z satisfying the equation ϕ ( a ) + ϕ ( a ′ ) = a − a ′ for every a, a ′ ∈ Z . Such ϕ does not exist and thus C Φ does not admit foldings. In this section we articulate the idea that relations between horizontal andvertical compositions of globular and horizontal identity squares in doublecategories constructed through the methods of theorem 3.1 should be rela-tively simple to understand. We do this through the notion of vertical length.We refer the reader to [16] for the precise definitions.For the convenience of the reader we rephrase the definition of double cate-gories of vertical length 1 in a way specific to the pourpuses of this section.Given a double category C we write V γC for the first vertical category of C , i.e. V γC denotes the subcategory of C generated by the globular andhorizontal identity squares of C . The second vertical category V γC of C isthe subcategory of C generated by horizontal compositions of morphismsin V γC . We say that a double category C has vertical length 1, ℓC = insymbols, if V γC = V γC or equivalently if V γC is closed under the operation oftaking horizontal compositions. Intuitively a double category C has verticallength 1 whenever given two squares of the form: a a a ab b b bαf β fϕ α ′ f β ′ fψ in C and factorizations of ϕ and ψ as vertical compositions of squares of theform: 26 ● ● ●● ● ● ● the terms of these factorizations of ϕ and ψ can be re-arranged in such away that ϕ and ψ can be composed horizontally by composing terms of thecorresponding factorizations horizontally one by one. An explicit descriptionof 2-morphisms in V γC for any double category C is provided in [16]. Thefollowing lemma uses this result to provide an explicit description of squaresin V γC for every double category C . Lemma 6.1.
Let ( B ∗ , B ) be a decorated bicategory. Let Φ be an objectof F un ( B ∗ , Cat ⊗ ) B . Let ( f, ϕ ) be a morphism in ∫ B ∗ Φ with domain andcodomain, in B , the 1-cells α, β of 0-cells a, b in B respectively. The followingthree conditions for ( f, ϕ ) are equivalent:1. ( f, ϕ ) is a morphism in V γC Φ .2. ( f, ϕ ) admits a factorization in C Φ as: α βi a i b ( id a , ψ ) ( f, ϕ )( f, id i a ) ( id β , η ) for 2-cells ψ, η in B from α to i a and from i b to β respectively.3. ( f, ϕ ) admits a pictorial representation in C Φ as: aa ab bb bαψf f ( f, ϕ ) βη for 2-cells ψ, η in B from α to i a and from i b to β respectively. Proof.
Let ( B ∗ , B ) be a decorated bicategory. Let Φ be an object in thecategory F un ( B ∗ , Cat ⊗ ) B . Let ( f, ϕ ) be a morphism in ∫ B ∗ Φ with domainand codomain the endomorphisms α, β of 0-cells a, b in B respectively. Wewish to prove that the three conditions in the statement of the lemma areequivalent.Condition 3 is a pictorial interpretation of condition 2 above and theseconditions are easily seen to be equivalent. We thus wish to prove thatconditions 2 and 3 are equivalent to condition 1. Observe first that if ( f, ϕ ) satisfies conditions 2 and 3 then ( f, ϕ ) admits a representation as a verticalcomposition of globular squares of C Φ and horizontal identities of verticalmorphisms of C Φ . Thus, if ( f, ϕ ) satisfies conditions 2 and 3 then ( f, ϕ ) isa morphism in V γC Φ . We prove that if ( f, ϕ ) is a morphism in V γC Φ then ( f, ϕ ) admits a pictorial representation as in condition 3 above.Observe first that given objects a, b in B ∗ , an endomorphism ψ of i a in B and a morphism f ∶ a → b in B ∗ the two squares of C Φ represented by thediagrams: 28 a a ab b b bb b b bf fi f Φ f ( ψ ) ψf fi f are equal if and only if the equation ( f, id i a ) q ( id a , ψ ) = ( id b , Φ f ( ψ ))( f, id i a ) holds. It is easily see that the left hand side and the right hand side ofthe above equation are both equal to ( f, Φ f ( ψ )) . The two squares in C Φ represented above are thus always equal. By iterating this equation we obtainthe fact that for any 2-morphism of C Φ admitting a factorization as a verticalcomposition of 2-morphisms admitting pictorial representations as: a aa ab bb bf fi f admits a pictorial representation as: 29 aa ab bf fi f From this and from the obvious fact that every 2-morphism in C Φ admittinga diagrammatic representation as: a aa aa ab bαf fi f admits a pictorial representation of the form: a aa ab bαf fi f
30e conclude that whenever ( f, ϕ ) is a morphism in V γC Φ then ( f, ϕ ) admitsa presentation as in condition 3 in the proposition. This concludes the proof. ∎ Proposition 6.2.
Let ( B ∗ , B ) be a decorated bicategory. Let Φ be a functorin F un ( B ∗ , Cat ⊗ ) B . In that case ℓC Φ = . Proof.
Let a, b be objects in C Φ . Let f ∶ a → b be a vertical morphism. Let α, α ′ be horizontal endomorphisms of a and let β, β ′ be horizontal endomor-phisms of b . Let ( f, ϕ ) and ( f, ϕ ′ ) be squares in V γC Φ , from α to β and from α ′ to β ′ respectively. By lemma 6.1 both ( f, α ) and ( f, α ′ ) admit a pictorialrepresentation as: a a a aa a a ab b b bb b b bαψf fi f βη α ′ ψ ′ f fi f β ′ η ′ The pictorial representation of the horizontal composition of the squaresabove is of the form: 31 aa aa ab bα ⊟ α ′ ψ ⊟ ψ ′ f fi f β ⊟ β ′ η ⊟ η ′ which is clearly a 2-morphism in V γC Φ . This proves that V nγC Φ = V γC Φ forevery positive integer n which proves that the category of morphisms γC Φ1 of C Φ is equal to V γC Φ . This proves that ℓγC Φ and thus that ℓC Φ = . Thisconcludes the proof of the proposition. ∎ Observe that the double categories described in theorem 3.1 are not in generalGG. To see this we use the computations provided by lemma 6.1 6.1.
Example 6.3.
Let ( B ∗ , B ) be the following decorated bicategory: We make B ∗ to be equal to the delooping category Ω N of the monoid N . We make B to be the single object bicategory Mat generated by the usual strictification
Mat of the category of complex vector spaces, i.e.
Mat is the monoidalcategory whose collection of objects is N and whose collection of morphisms isthe collection of complex matreces. Composition and tensor product in Mat are the usual composition and tensor product of matreces. Let Φ be an objectof the category F un ( Ω N , Cat ⊗ ) Mat be such that Φ ( ∗ ) = Mat and such thatfor every m ∈ N , Φ m acts on Mat by Φ m ( n ) = mn for every n ∈ N and by Φ m ( A ) = A ⊗ m for every complex matrix A . Thus defined Φ is a functor in F un ( Ω N , Cat ⊗ ) Mat . We show that the double category C Φ is not GG. Tothis end consider the square ( , id ) in C Φ . Observe that if ( , id ) were GGthen from lemma 6.1 ( , id ) = id ⊗ would admit a factorization as
40 0 ( , B ) ( , id )( , C ) ( , D ) But were this the case, since the rank rkB and rkD of both B and D is atmost 1 then the rank rkid ⊗ of id ⊗ would be at most 1, a contradiction. The2-morphism ( , id ) of C Φ is thus not GG and C Φ is thus not GG. In this section we study the GG condition on double categories constructedthrough the methods presented in theorem 3.1. We refer the reader to [16]for a treatment on GG double categories.We consider the following situation: Let G be a group. Let A be a com-mutative monoid. The pair ( Ω G, A ) is a decorated bicategory. Let Φ bea functor in F un ( Ω G, A ) . We consider situations in which the doublecategory C Φ associated to Φ in theorem 3.1 is GG. Compare this to example6.3. We prove the following proposition. Proposition 7.1.
Let
M, A be monoids. Suppose A is commutative. Let Φ be a functor in F un ( Ω M, Cat ⊗ ) A . Suppose that for every m ∈ M theendomorphism Φ m of A is surjective. In that case C Φ is GG. Proof.
Let
M, A be monoids. Suppose A is commutative. Let Φ be anobject of F un ( Ω M, Cat ⊗ ) A such that for every m ∈ M the endomorphism Φ m of A is surjective. We wish to prove in this case that the double category C Φ is GG.We wish to prove that every square in C Φ is GG. To do this it is enoughto prove that every morphism of ∫ Ω M Φ is GG. Let ( f, ϕ ) be a morphism in ∫ Ω M Φ . We represent ( f, ϕ ) pictorially as ∗ ∗∗ ∗ ( f, ϕ ) f f
33y lemma 6.1 and proposition 6.2 in order to prove that ( f, ϕ ) is GG it isenough to prove that ( f, ϕ ) admits a pictorial representaiton as: ∗ ∗∗ ∗∗ ∗∗ ∗ ψf f ( f, ϕ ) η for elements ψ, η ∈ A . We actually prove that ( f, ϕ ) admits a pictorialrepresentation as: ∗ ∗∗ ∗∗ ∗ ψf f ( f, ϕ ) for some element ψ ∈ M . The above pictorial equation is equivalent to theequation ( f, ϕ ) = ( f, A )( M , ψ ) which in turn is equivalent to the equation ( f, ϕ ) = ( f, Φ f ( ψ )) . A solution to this equation is thus found by solving theequation ϕ = Φ f ( ψ ) in A , but the condition that Φ f is surective guaranteesthe existence of such a solution. By lemma 6.1 we thus have that everymorphism in ∫ Ω M Φ is a morphism in V γC Φ . We conclude that C Φ is GG. ∎ Remark 7.2.
The arguments employed in the proof of proposition 7.1 workin the more general setting in which instead of considering bicategories of the34orm A we consider bicategories of the form C for a monoidal category C as long as the category C satisfies the condition that every morphism in C factors through the tensor unit 1 C of C . We avoid proving proposition 7.1in this generality for simplity. Corollary 7.3.
Let G be a group. Let A be a commutative monoid. Let Φ be a functor in F un ( Ω G, Cat ⊗ ) Ω A . In that case the double category C Φ isGG. In this final section we investigate a construction inverse to that presentedin theorem 3.1 in the case of bicategories decorated by groups. The resultsof this section should be regarded as an extension of corollary 4.1.We consider the following problem: Let G be a group. Let A be a com-mutative monoid. Let C be a double category satisfying the conditions ofproposition 7.1. We wish to construct a functor Φ in F un ( Ω G, Cat ⊗ ) A such that C and C Φ are somehow related. We begin ou investigation of thisproblem by proving the following proposition. Proposition 8.1.
Let G be a group. Let A be a commutative monoid. Let C be an object in dCat ( Ω G, A ) such that C is GG and such that ℓC = .In that case there exists a functor Φ in F un ( Ω G, Cat ⊗ ) A and a doublefunctor π C ∶ C Φ → C such that H ∗ π C = id ( Ω G, A ) and π C is full. Proof.
Let G be a group. Let A be a commutative monoid. Let C be aGG double internalization of ( Ω G, A ) such that ℓC = . We wish to provethat there exist a functor Φ in F un ( Ω G, Cat ⊗ ) A and a full double functor π C ∶ C Φ → C such that H ∗ π C = id ( Ω G, A ) .In order to obtain a functor Φ in F un ( Ω G, Cat ⊗ ) Ω A it is enough todefine a representation, which we will keep denoting by Φ , of G in End ( A ) .Given g ∈ G and a ∈ A we make Φ g ( a ) to be defined through the followingdiagram in C Φ : 35 ∗∗ ∗∗ ∗∗ ∗ f fi f af − f − i f − To see that this defines a representation of G in End ( A ) we first observethat by the equation HC = Ω A it follows that the above 2-morphism of C is globular and thus is an element of A . The fact that as defined abovethe function Φ defines a morphism from G to End ( A ) follows immediatlyfrom the structure equations defining C . We thus obtain a representation Φ ∶ G → End ( A ) which forms the morphism function of a functor Φ in F un ( Ω G, Cat ⊗ ) A .We now construct a full double functor π C ∶ C Φ → C satisfying the equa-tion H ∗ π C = id ( Ω G, A ) . By the fact that C is a length 1 GG internalizationof ( Ω G, A ) and by Lemma 4.11 of [15] we obtain a full double functor π C, ∶ V ( Ω G, A ) → C such that the restriction of π C, to the collections of 1- and 2-cells A and A of A is equal to id A and id A respectively. An easy computa-tion proves that V ( Ω G, A ) is equal to the delooping category Ω ( G ∗ A ) ofthe free product G ∗ A of G and A . From this and from the fact that forevery g ∈ G and a ∈ A the 2-morphisms i a and b satisfy the relation i − g ai g = Φ g ( a ) it follows that there exists an epic morphism π C ∶ G ⋊ Φ A → End C ( id ∗ ) making the following triangle commute:36 ∗ A End C ( id ∗ ) G ⋊ Φ Aπ C π ,C By the way π ,C is defined it follows that the pair π C = ( id G , π ) is a fulldouble functor from C Φ to C satisfyng the equation H ∗ π C = id ( Ω G, A ) .This concludes the proof. ∎ Let ( B ∗ , B ) be a decorated bicategory. We will write gCat ℓ = (B ∗ , B) for thecategory whose objects are GG double categories C satisfying the equations H ∗ C = ( B ∗ , B ) and ℓC = . Given objects C, C ′ in gCat ℓ = (B ∗ , B) we make thecollection of morphisms from C to C ′ to be the collection of double functors F from C to C ′ such that the object functor F of F is equal to the identityendofunctor of B ∗ . Observe that as defined gCat ℓ = (B ∗ , B) is not a subcategoryof dCat (B ∗ , B) . We extend proposition 8.1 through the following lemma. Lemma 8.2.
Let G be a group. Let A be a commuatitve monoid. Thefunction associating to every C in gCat ℓ = ( Ω G, A ) the functor Φ C extends toa functor Φ ● ∶ gCat ℓ = ( Ω G, A ) → F un ( Ω G, Cat ⊗ ) A Proof.
Let G be a group. Let A be a commutative monoid. We wishto prove that the operation of associating the functor Φ C to an object C in gCat ℓ = ( Ω G, A ) extends to a functor Φ ● ∶ gCat ℓ = ( Ω G, A ) → F un ( Ω G, Cat ⊗ ) A .Let C, C ′ be objects in gCat ℓ = ( Ω G, A ) . Let g ∈ G . We will write i g and i ′ g for the horizontal identities of g in C and in C ′ respectively. The horizontalidentities i g , i ′ g are related by the equation F ( i g ) = i ′ g . From this equationwe obtain: F ( i g − ai g ) = i ′− g F ( a ) i ′ g for every a ∈ A . The equation: F Φ Cg ( a ) = Φ C ′ g ( F ( a )) follows for every g ∈ G and for every a ∈ A . Observe that the fact that F is adouble functor implies that F ↾ A is an endomorphism of A . From this and37rom the equations above it is immediate that the following square commutesfor every g ∈ G : Ω A Ω A Ω A Ω A Φ Cg F ↾ A F ↾ A Φ C ′ g This proves that F ↾ A defines a natural transformation from Φ C to Φ C ′ . It isimmediate that the function associating to every morphism F in gCat ℓ = ( Ω G, A ) the natural transformation F ↾ A extends the function associating to ev-ery double category C in gCat ℓ = ( Ω G, A ) to a functor from gCat ℓ = ( Ω G, A ) to F un ( Ω G, Cat ⊗ ) A . This concludes the proof of the lemma. ∎ Proposition 8.3.
Let G be a group. Let A be a commutative monoid. Inthat case the following relation holds: Φ ● ⊢ C ● Proof.
Let G be a group. Let A be a commutative monoid. We wish toprove in this case that Φ ● ⊢ C ● .We provide a counit-unit pair ( ǫ, η ) for the adjunction Φ ● ⊢ C ● . Webegin by defining ǫ . Let C be an object in gCat ℓ = ( Ω G, A ) . We make ǫ C to be the functor π C deifned in lemma 8.2. We prove that the collection ǫ of double functors π C with C running through gCat ℓ = ( Ω G, A ) is a naturaltransformation from C ● Φ ● to the identity endofunctor of gCat ℓ = ( Ω G, A ) . Tosee this let C, C ′ be objects in gCat ℓ = ( Ω G, A ) and let F ∶ C → C ′ be adouble functor such that F = id Ω G . In that case the restriction F ↾ A of themorphism functor F of F is a natural transformation from Φ C to Φ C ′ bylemma 8.2. From this and from the equation H ∗ π C = id ( Ω G, A ) it followsthat the following diagram is commutative: C Φ C C ′ Φ C ′ C C ′ π C FF π C ′ ǫ is indeed a natural transformation from C ● Φ ● to the identity endofunctor of gCat ℓ = ( Ω G, A ) .We now define η . Observe that for every Φ in F un ( Ω G, Cat ⊗ ) ( Ω G, A ) the monoid of endomorphisms of the horizontal identity i ∗ of the only objectin C Φ is equal to G ⋊ Φ A and thus π C Φ is equal to Φ . We make η to be thecollection of identity natural transformations id Φ with Φ running throughthe collection of all functors in F un ( Ω G, Cat ⊗ ) ( Ω G, A ) .We prove that the pair ( ǫ, η ) satisfies the triangle identities. That is, weprove that the following triangles commute: C ● C ● Φ ● Φ ● Φ ● C ● Φ ● C ● Φ ● C ● Φ ● η id Φ ● ǫ Φ ● ηC ● id C ● C ● ǫ The commutativity of the triangle on the left hand side reduces, in our set-ting, to proving that for every functor Φ in F un ( Ω G, Cat ⊗ ) ( Ω G, A ) theequation π C Φ = id C Φ holds. To prove this observe that the monoid of en-domorphisms End C Φ ( i ∗ ) of the horizontal identity i ∗ of the only object ∗ in C Φ is equal to G ⋊ Φ A . From this and from the way π C was defined weconclude that π Φ = id C Φ . The commutativity of the second triangle abovereduces to proving that the equation id Φ C = Φ π C holds for every object C in gCat ℓ = ( Ω G, A ) . This follows from arguments analogous as the ones describedin proving the commutativity of the triangle on the left hand side above.This concludes the proof. ∎ References [1] Andre Henriques Arthur Bartels, Christopher L. Douglas. Dualizabilityand index of subfactors.
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