Bicategories for TQFTs with Defects with Structure
aa r X i v : . [ m a t h . QA ] M a r BICATEGORIES FOR TQFTS WITH DEFECTS WITHSTRUCTURE
I. J. LEE AND D. N. YETTER
Abstract.
We provide a description of adequate categorical data togive a Turaev-Viro type state-sum construct of invariants of 3-manifoldswith a system of defects, generalizing the Dijkgraaf-Witten type invari-ants of our earlier work. We term the defects in our construction defects-with-structure because algebraic data associated to them is in generalricher than a module category over the spherical fusion category fromwhich the theory is constructed when no defect is present. Introduction
It is well known that spherical fusion categories provide the initial datafor a state-sum construction of 3-dimensional topological quantum field the-ories (TQFTs) generalizing that given by Turaev and Viro [16] using thefusion category constructed from the quantum group U q sl (2) for q a root ofunity. The generalization to arbitrary ribbon fusion categories (which arenecessarily spherical) was given by Yetter [19], with the general case for arbi-trary spherical fusion categories finally given given by Barrett and Westbury[1], who realized that a braiding and ribbon structure were superfluous andthe only the extra property of the coincidence of left and right traces wasneeded.In [12] the authors extended Wakui’s [17] state-sum construction of finite-gauge-group Dijkgraaf-Witten theory, which retrospectively is a special caseof the Barrett and Westbury general construction, to give invariants of 3-manifolds equipped with a link, a surface, or a link with a Seifert surface.The construction regarded the space equipped with subspaces as very simplestratified spaces and used the stratification to make the 3-manifold into adirected space in the sense of Grandis [10, 11] (see also [13]), replacing thefinite group with a finite category equipped with a conservative functor toa poset encoding the adjacencies of the (dimensions of) strata.It is the purpose of this paper to provide the generalization of this ear-lier work to a Turaev-Viro-style construction of invariants of 3-manifoldsequipped with links, surfaces or Seifert surfaces with their bounding link,again using a directed space structure, and using suitable bicategories suit-ably fibered over a small category encoding the adjacencies of (dimensionsof) strata. As [9, 12, 18] the data associated with a defect will be richer than simply a module category over the spherical category associated withthe bulk. We refer to this circumstance as “defects-with-structure.”2. Spherical Fusion Categories and Turaev-Viro Invariants
Let k be an algebraically closed field k , and A be a spherical fusioncategory over k . See [1, 5] for complete definitions. We recall explicitly onlyenough of the structure to fix our notation and inform our generalization.In particular A is semisimple with finitely many isomorphism classes ofsimple objects. We will choose as set S of simple objects representing theisomorphism clases, which without loss of generality contains the monoidalidentity object and necessarily contains dual objects for each of its elements.We let dim X ∈ k denotes the categorical dimension of the object X ,which is the scalar multiple of the identity map on the (simple) monoidalidentity object given by the right trace (or equivalently by sphericality theleft trace) of its identity arrow, represented in string diagrams by an orientedloop labeled with the object on its descending portion (and its dual on theascending portion).In any semisimple spherical category, there is an analogue of Schur’sLemma: namely any endomorphism f : k → k of a simple object k is theidentity morphism of that object multiplied by a scalar, namely tr( f ) / dim( k ).This has the consequence in manipulating string diagrams that if a string di-agram with an edge labeled k is multiplied by the trace of an endomorphismthe same object k , f : k → k , the result is the same as if f were inserted onthe string labeled k and the resulting diagram instead multiplied by dim( k )or a loop labeled k .We also choose bases for the hom-spaces A ( i ⊗ j, k ) for all i, j, k ∈ S ,whose union we denote by B . It is easy to see by simplicity that there aredual bases for A ( k, i ⊗ j ) in the sense that the endomorphisms of k resultingfrom composing elements of the chosen basis and the dual basis will be theidentity arrow on k or zero, and that the original bases give the projectionsof direct sum decompositions of the i ⊗ j , for which the dual bases are theinclusions. It is useful in conjunction with the analogue of Schur’s Lemma toreplace the dual bases with bases whose elements in A ( k, i ⊗ j ) are dim( i ) − times those in the dual basis. We denote the unions of these rescaled dualbases by ¯ B . In string diagrams, we will denote the scaled dual basis elementby the same symbol as that of the corresponding element in B , but labelinga node with one input and two outputs, rather than two inputs and oneoutput.The behavior of these bases when composed in the other order is thensummarized in string diagrams by ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 3 X k ∈ S α ∈ B dim( k ) i j k i j α ¯ α i j When we use this relation to manipulate sums of string diagrams in the direc-tion which results in the identity morphism on the tensor product depictedon the right, we will say we are “contracting the edge labeled k ”.3. Biparcels
Our goal in this section is to define the appropriate categorical structurewhich provides the appropriate common generalization of the spherical fu-sion categories of [1] and the parcels of [12, 13] to allow the description ofTuraev-Viro type TQFTs with defects-with-structure.The parcels of [12, 13] are both subsumed in the following definition:
Definition 3.1.
For a small gaunt category Γ, a Γ-parcel is a small category C equipped with a conservative functor γ : C →
Γ. A parcel is fiber-finite ifthe inverse image of any arrow in Γ is finite, and regular if it is injective onthe set of objects.In [12] fiber-finite regular P -parcels for P the totally ordered set withtwo or three elements were used as colors for a Dijkgraaf-Witten style con-struction of invariants of 3-manifolds with an embedded link and Seiftertsurface to the link, corresponding to a endowing the triple L ⊂ Σ ⊂ M as avery simple stratified space, with directed space structure [10, 11] in whichdirected paths may leave, but not enter, lower dimensional strata. In[13], parcels over the path category of the digraph with two vertices and twooppositely oriented edges joining them, were used as colors for a Dijkgraaf-Witten style construction of invariants of an oriented 3-manifold with anembedded oriented surface, corresponding to the directed space structure inwhich directed paths must enter or leave the surface in such a way that theorientation of the 3-manifold induced by appending a normal vector in theforward direction along the path to a basis of tangent vectors representingthe orientation on the surface agrees with the ambient orientation.It is, of course, possible to see Wakui’s original construction [17] of Dijkgraaf-Witten theory as an instance of Barrett and Westbury’s [1] generalizedTuraev-Viro construction, by replacing the finite group G with the spheri-cal fusion category of G -graded finite dimensional k -vector spaces, with its The authors recently learned that the idea used in [9, 12] of restricting admissiblepaths so that paths may not enter lower-dimensional strata had been used in 2009 byTreumann [T] in the context of constructible stacks on stratified spaces.
I. J. LEE AND D. N. YETTER associator given by the k × -valued 3-cocycle, α . Applying a similar trans-formation to the parcels of [12, 13], equipped with a cocycle (though nota partial cocycle) will result in a k -linear bicategory with finite semisim-ple hom-categories, fibered over the 2-category arising from the gaunt basecategory by simply adjoining identity 2-arrows to every arrow. Doing thissuggests a more general notion which we will employ to extend the gener-alized Turaev-Viro construction to various types of directed spaces arisingfrom manifolds stratified by submanifolds. Definition 3.2. A semisimple bicategory over k is a bicategory whose hom-categories are semisimple k -linear abelian categories, in which both 1- and2-dimensional composition of 2-arrows is k -bilinear. Recall that a semisimple k -linear category is called finite semisimple if there are only finitely manysimple objects. A semisimple bicategory is finite if each hom-category isfinite semisimple. Definition 3.3. A bicategory of fusion type over k is a finite semisimplebicategory over k in which the identity 1-arrows of every object are simpleas objects of the endomorphism category.Examples include fusion categories over k with their objects regardedas endo-1-arrows of a single object and their arrows regarded as 2-arrows.Indeed the point of the definition is to have a many-objects analogue of afusion category regarded as a one object bicategory. Definition 3.4.
A bicategory has pivotal duals if it is equipped with anassignment to each 1-arrow, a of a 1-arrow ¯ a satisfying ¯¯ a = a , s (¯ a ) = t ( a )and t (¯ a ) = s ( a ) for all 1-arrows a and ¯ X = X for all objects X , togetherwith 2-arrows ǫ a : a ¯ a ⇒ s ( a ) and η a : t ( a ) ⇒ ¯ aa. For which the composite 2-arrows a ρ − = ⇒ a t ( a ) aη a = ⇒ a (¯ aa ) α − a, ¯ a,a = ⇒ ( a ¯ a ) a ǫ a a = ⇒ a s ( a ) a λ = ⇒ a and ¯ a λ − = ⇒ t ( a ) ¯ a η a ¯ a = ⇒ (¯ aa )¯ a α ¯ a,a, ¯ a = ⇒ ¯ a ( a ¯ a ) ¯ aǫ a = ⇒ ¯ a s ( a ) ρ = ⇒ ¯ a are each identity 2-arrows.Observe that ǫ a and η a make ¯ a into a right dual 1-arrow for a , while ǫ ¯ a and η ¯ a make ¯ a into a left dual 1-arrow for a .Now observe that in any bicategory with pivotal duals, every 1-arrow hascanonically associated to it endo-2-arrows of the identity on its source andthe identity on its target, dim s ( a ) = η ¯ a ǫ a and dim t ( a ) = η a ǫ ¯ a , generalizingthe right and left dimensions in a pivotal category. And, as in the 1-objectcase, there are corresponding traces valued in the endo-2-arrows of the iden-tities on the source and target of the 1-arrow given for any endo-2-arrow f : a ⇒ a by tr s ( f ) = η ¯ a ¯ af ǫ a tr t ( f ) = η a f ¯ aǫ ¯ a ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 5
Now, at this level of generality, there is simply no analogue of the sphericalcondition that the right and left traces be equal: the source and target tracesare valued in different endo-hom-categories. However, if the bicategory is offusion type, or more generally has simple identity 1-arrows, each trace is anendo-2-arrow of a simple identity 1-arrow, and thus a scalar multiple of itsidentity 2-arrow, and we can thus make the following definition:
Definition 3.5. A bicategory with spherical duals is a bicategory with piv-otal duals, in which all identity 1-arrows are simple, and for all 1-arrows a ,for all endo-2-arrows f : a ⇒ a , if tr s ( f ) = c s ( f ) s ( a ) and tr t ( f ) = c t ( f ) t ( a ) ,the scalars c s ( f ) and c t ( f ) are equal. We refer to this common scalar as the scalar trace of the endo-2-arrow, denoting it by tr( f ), and the special casewhere f = 1 X by dim( X ).M¨uger [15] has called bicategories with spherical duals, “spherical bicate-gories”. We, however, eschew this terminology, since it more naturally wouldname a weak version of the spherical 2-categories of Mackaay [14].In what follows if C is any category, we will denote by C its trivial2-categorification, that is the 2-category created by adjoining a formal 2-identity arrow to each 1-arrow of the category, with the obvious 1- and2-dimensional compositions.It is also well established that string diagrams provide adequate compu-tational representation of 1-arrows (as strings) and 2-arrows (as nodes) ina bicategory, if the regions of the diagram are colored with the objects ofthe bicategory, with the source to the left of the right. In our calculationsbelow, we omit the colors of the regions, as these can be recovered from thesource/target date implied by the numbering on the vertices of the triangu-lation to which the string diagram has been associated. Definition 3.6.
For a small gaunt category Γ, a Γ -biparcel over k is asemisimple bicategory over k , equipped with a 2-functor β : B → Γ .A Γ-biparcel over k is finite if the preimage of each 1-arrow together withits identity 2-arrow is a finite semisimple additive category (of 1- and 2-arrows) and of fusion type if it is finite and, moreover, the preimage of everyidentity 1-arrow (together with its identity 2-arrow) is a fusion category(with 1-composition as tensor product and 2-composition as composition).A Γ-biparcel over k of fusion type is spherical if B has spherical duals, and quasi-spherical if B is a sub-bicategory of a bicategory with spherical duals.Note that for general Γ, the condition that a Γ-biparcel be of fusion typeis weaker than requring the bicategory B be of fusion type as a bicategory:the finiteness condition is imposed not on whole hom-categories in B , butonly on the pre-images of 1-arrows of Γ. For the finite posets used in [12],of course, the condition that a biparcel be of fusion type is equivalent to thebicategory B being of fusion type.In subsequent sections of the paper we will show that quasi-sphericalbiparcels of fusion type provide an appropriate notion of colorings for a I. J. LEE AND D. N. YETTER common generalization of our prior constructions [12, 13] and Barrett andWestbury’s generalized Turaev-Viro theory [1]. In the remainder of thissection, we give several ways of contructing examples of biparcels from cat-egorical structures more closely related to the usual toolkit for Turaev-Virotype TQFTs. These require a little bit of preparation:Many of our constructions will involve direct sums of abelian categories.It is convenient to introduce a notation for an object concentrated in onedirect summand. Let Γ [ δ ] denote the object of ⊕ d ∈ D C d , ( C d ) d ∈ D for which C δ = Γ and C d = 0 for all d = δ . Proposition 3.7. If ( C , I, ⊗ , . . . ) is a tensor (resp. semisimple tensor, piv-otal, spherical) category over k , and G is a groupoid, there is a k -linearbicategory (resp. semisimple bicategory, bicategory of with pivotal duals,bicategory with spherical duals) C ♯ G with the same objects as G and hom-categories given by C ♯ G ( X, Y ) = M f ∈G ( X,Y ) C with 1-dimensional composition given by ( C f ) f ∈G ( X,Y ) , ( D g ) g ∈G ( Y,Z ) ( ⊕ fg = h C f ⊗ D g ) h ∈G ( X.Z ) and 2-dimensional composition given coordinate-wise by the composition in C in each entry.If moreover, C is a fusion category and the hom-sets of G are all finite,then C ♯ G is of fusion type.Proof. The identity 1-arrow for an object X is I [1 X ] ∈ C ♯ G ( X, X ), where I is the monoidal identity of C .The associators and unitors for the bicategory structure are induced bythose of C together with the distributivity coherence maps that exist by theexactness of ⊗ in both variables in what by now is a fairly routine way.That the resulting bicategory is semisimple when C is immediate as adirect sum of semisimple abelian categories is semisimple.If C is pivotal, with the dual of C denoted C , then the dual of ( C f ) f ∈G ( X,Y ) is (Γ g ) g ∈G ( Y,X ) where Γ g = C g − . To describe the structure maps for theduality, it suffices to describe them for 1-arrows concentrated in a directsummand of some C ♯ G ( X, Y ), but here they are simply the structure mapsfor the duality in C . Thus if C is spherical, the necessary equality for thepivotal duals in C ♯ G to be spherical duals follows immediately from thecondition in C .Finally, the last statement of the proposition follows trivially, once it isnoted that the identity 1-arrow on an object X is I [ Id X ] . (cid:3) Recall from Etingoff et al. [5] the definition of a grading of a tensorcategory:
ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 7
Definition 3.8.
For a group G a G -grading of a tensor category C is anisomorphism of categories C ∼ = M g ∈ G C g where for each g ∈ G , C g is an abelian subcategory of G such that X ∈ C g and Y ∈ C h implies X ⊗ Y ∈ C gh .For our purposes it will be useful to think this in a slightly different way: Definition 3.9. A G -grading span for a tensor category C is a span ofbicategory functors G | | ←− C hom ι −→ C where C hom is a monoidal category with ι one functor of a monoidal equiv-alence of categories with its image in C , which image is a monoidal fullsubcategory of C , whose closure under direct sum is all of C , and the preim-age of each 1-arrow in G together with its identity 2-arrow is an abeliancategory.In what follows both C and C hom will be regarded as bicategories with asingle object.The spirit of the definition is that C hom should be the subcategory of C ofobjects homogeneous with respect to the grading and ι the inclusion functor,but the fussiness of the definition is needed since each fiber of the degreefunctor | | must have a zero object, and C need not have as many distinctzero objects as there are elements in G .It is easy to see that we have Proposition 3.10. A G -grading of a tensor category C is equivalent to a G -grading span for C .Proof. Given a G -grading define C hom to be a g ∈ G C g . The monoidal product on C induces a monoidal product on C hom given bythe restriction of the monoidal product to the grades followed by the inclu-sion of the grade indexed by the product of the degrees into the coproduct,and ι is the canonical functor induced by the inclusions of the grades into C ,which is trivially seen to be a monoidal functor. The image is precisely thehomogeneous objects of C with respect to the grading, and thus its closureunder direct sum is all of C .Plainly there is a bicategory functor from C hom to G sending the 1-arrowsto their degree and every 2-arrow to the identity 2-arrow of the commondegree of its source and target. That this assignment respects 1-dimensionalcomposition follows from the condition in a grading that X ∈ C g and Y ∈ C h implies X ⊗ Y ∈ C gh . I. J. LEE AND D. N. YETTER
Conversely given a grading span for a tensor category C , for each elementof G , define C g to be the image under ι of the preimage of g as a 1-arrowin G together with its identity 2-arrow. The generation condition on theimage of ι implies that C ∼ = ⊕ g ∈ G C g , while the functoriality of the degreefunctor implies that X ∈ C g and Y ∈ C h implies X ⊗ Y ∈ C gh . (cid:3) This last result gives us a construction for parcels.First, we consider the effect of pulling back the degree functor along the2-functor induced by a functor Φ :
G → G , for G a groupoid. Proposition 3.11.
For G a groupoid, given a grading span G | | ← C hom ι → C for a tensor (resp. pivotal, spherical) category over k , then for any functor Φ :
G → G , the pullback of | | : C hom → G along the induced 2-functor Φ : G → G is a bicategory over k (resp. bicategory with pivotal duals,bicategory with spherical duals). Moreover, if C is a fusion category, thepreimage of every 1-arrow in G together with its identity 2-arrow will befinite semisimple, and thus if every hom-set of G is finite, the bicategory willbe of fusion type.Proof. It is trivial that the pullback of C hom is a bicategory over k when C is a tensor category over k . Now suppose C is pivotal. It follows that C hom is pivotal as the dual of any object homogeneous of degree g is an objecthomogenenous of degree g − , and thus, C hom is closed under taking duals.Likewise the spherical condition on traces is preserved under restriction to C hom . Passing to the pullback, 1-arrows in the pullback are pairs ( X, γ )where X is an object of degree g and Φ( γ ) = g . It is easy to see that thestructure maps for X and X ǫ X and η X give rise to structure maps ( ǫ X , η X ,
1) making (
X, γ − ) into the dual 1-arrow of ( X, γ ).In the spherical case, since the identity 1-arrows of any object ( ∗ , S ) of thepullback are ( I, S ) the simplicity of I implies the simplicity of all identity1-arrows, the equality of the left and right trace in C implies the equality ofthe scalars when the source and target traces of ( f, γ ) are written as scalarmultiples of 1 ( I,s ( γ )) and 1 ( I,t ( γ )) , respectively.Finally, if C is fusion, note that the preimage of each γ in G is a copyof C Φ( g ) , and thus finite semisimple, from which the statement with thefiniteness condition on G follows immediately. (cid:3) Proposition 3.12. If Γ is a gaunt category which embeds in G (Γ) , thegroupoid it freely generates, then given a grading span G | | ← C hom ι → C fora tensor (resp. spherical, fusion) category, then for any functor Φ : Γ → G ,the pullback of | | : C hom → G along the induced 2-functor Φ : Γ → G isa Γ -biparcel (resp. quasi-spherical Γ -biparcel, Γ -biparcel of fusion type).Proof. This is immediate from the previous proposition and the definitionsof the various types of biparcels. (cid:3)
Similarly biparcels can be constructed by pulling back sectors of a multi-fusion category.
ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 9
Proposition 3.13.
A multifusion category ( C , I, ⊗ , . . . ) over k in which I has direct summands I j for j ∈ J for some finite indexing set J is equivalentto a bicategory of fusion type over k with objects set J . Moreover, C isspherical in the sense of Cui and Wang [3] if and only if the correspondingbicategory has spherical duals.Proof. Given a multifusion category C with I = ⊕ j ∈ J I j , define a bicategoryˆ C with object set J by letting ˆ C ( i, j ) = C i,j , the i, j -sector of C , and definingthe 1-dimensional composition to be the restriction of tensor product to thesectors.Conversely given a bicategory of fusion type B with object set J , thecorresponding multifusion category isˇ B = ⊕ i,j ∈ J B ( i, j )with tensor product defined by extending the 1-dimensional composition bythe universal property of the direct sum.It is easy to see that dual objects in the multifusion category inducedual 1-arrows in the corresponding bicategory and conversely, and that thespherical condition is preserved by the construction in both directions. (cid:3) Now observe that the bicategory ˆ C is naturally equiped with a functor tothe trivial 2-categorification of the chaotic preorder on J , Ch ( J ) .It is then easy to see that the following holds. Proposition 3.14.
For C a multifusion category with I = ⊕ j ∈ J I j , if Γ isa guant category and Φ : Γ → Ch ( J ) is a functor to the chaotic preorder on J , the pullback of ˆ C along Φ with its functor to Γ is a Γ -biparcel of fusiontype. If, moreover, Γ embeds in the groupoid it freely generates, and C is aspherical multifusion category, the pullback is a quasi-spherical Γ -biparcel. Manifolds with Defects, Flag-Like Triangulations andDirected Space Structures
In [12, 13] the authors considered various configurations of a 3-manifoldequipped with specified submanifolds, seeing them as very simple instancesof stratified spaces, and using the stratification to equip them in some waywith a directed space structure in the sense of Grandis [10, 11]. We hererecall from [10, 11, 7, 12, 13] for the present work.As in [12] we use the notions of Crane and Yetter [7]
Definition 4.1. A starkly stratified space is a PL space X equipped with afiltration X ⊂ X ⊂ . . . ⊂ X n − ⊂ X n = X satisfying(1) There is a triangulation T of X in which each X k is a subcomplex.(2) For each k = 1 , . . . n X k \ X k − is a(n open) k -manifold. (3) If C is a connected component of X k \ X k − , then T restricted to ¯ C gives ¯ C the structure of a combinatorial manifold with boundary.(4) For each combinatorial ball B k with ◦ B k ⊂ X k \ X k − , ◦ B k admits aclosed neighborhood N given inductively as a cell complex as follows(although we require B k to be a combinatorial ball, the triangulationis then ignored): N = N n , where N m for k ≤ m ≤ n is given inductively by N k = B k and N ℓ +1 = N ℓ ∪ [ v ∈ S ℓ L ( v ) ∗ v for S ℓ a finite set of points in X \ X ℓ , andl L ( · ) a function on S ℓ valued in { L | L is a combinatorial ball and B k ⊂ L ⊂ N ℓ } We will call such a neighborhood of the interior of a combinatorialball lying in a stratum of the same dimension a stark neighborhood .As we observed in [12] for a knot or link K with a Seifert surface Σ ina 3-manifold M , the filtered space ∅ ⊂ K ⊂ Σ ⊂ M is a starkly stratifiedspace. And, as in [12], even though K may, in fact, be a link, we will alwaysrefer to it as “the knot K ” to avoid confusion with the other meaning of theword “link” in PL topology.Recall also Definition 4.2. A simplicial flag is a finite sequence of simplexes σ ⊂ σ ⊂ . . . ⊂ σ n such that each σ i is a face of σ i +1 . A simplicial flag is complete if σ i is i -dimensional, and thus the sequence may be formed by starting with an n -simplex σ n and iteratively chosing a codimension 1 face until a vertex isreached. Definition 4.3.
A triangulation T of a stratified PL space X ⊂ X ⊂ . . . ⊂ X n − ⊂ X n = X is flag-like if each of the X i is a subcomplex, and moreover for each simplex σ of T , the restriction of the filtration to the simplex, that is the distinctnon-empty intersections in the sequence X ∩ σ ⊂ X ∩ σ ⊂ . . . ⊂ X n − ∩ σ ⊂ σ form a (possibly incomplete) simplicial flag. ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 11
As observed in [12], any triangulation of a stratified space for which theclosures of the strata are subcomplexes, gives rise to a flag-like triangulationupon barycentric subdivision.As was shown in [12], two flag-like triangulations of a starkly stratifiedspace are combinatorially equivalent if and only if they are related by asequence of extended Pachner moves (extended bistellar flips) each of whichis flag-like in the sense that both its initial and final state are flag-like.Moreover, in the case of a knot, Seifert surface, 3-manifold triple, a sufficientset of combinatorial moves remains if the 4-8 (resp. 8-4) extended bistellarmove in which and edge of the knot is subdivided (resp. its reverse in whichtwo edges on the knot are welded) is replaced with 3-6 (resp. 6-3) move inwhich an edge (resp. pair of edges) on the knot with a three-edge link, witha single vertex in the Seifert surface is subdivided (resp. welded).Building on the counting interpretation provided for some of the invariantsconstructed in [12] in terms of Grandis’s fundamental category [10], in [13]the authors considered various constructions of directed space structuresfrom stratifications. In particular we recall from Grandis [10]
Definition 4.4. A directed topological space or d -space X = ( X, dX ) is atopological space X , equipped with a set dX of continuous maps φ : I → X satisfying(1) Every constant map x : I → X for x ∈ X is in dX .(2) dX is closed under pre-composition with continuous weakly mono-tone functions from I to I .(3) dX is closed under concatenation of paths.Elements of dX are called directed paths in X . As was observed in [13]Grandis’s closure condition is equivalent to dX being closed under both(weakly monotone) reparametrizations of paths and arbitrary factorizationsof paths with respect to concatenation.A map of directed spaces f : ( X, dX ) → ( Y, dY ) is a continuous function f : X → Y such that p ∈ dX implies f ( p ) ∈ dY .In [12] a directed space structure was constructed from the knot, Seifertsurface, 3-manifold triple by defining a path to be directed if the functionassigning to each point the dimension of the stratum in which it lies was(weakly) monotone increasing on the path. While in [13] other ways ofproducing a directed space structure were considered, in particular thatdual to the one in [12] and one in which for a pair of an oriented manifoldwith an oriented submanifold of codimension one, paths were directed ifthey admitted a factorization into paths which either lay within a stratumor were factors of paths transverse to the submanifold with local intersectionnumber +1 at each intersection point.In each case the fundamental category ↑ Π( X ) admitted a conservativefunctor to a gaunt category with one object for each dimension of stratum, the poset or P (Γ ) the path category on the directed graph with a pair ofvertices and one edge in each direction between them.Finally, we need to describe the appropriate relationship between thedirected space structure induced from a stratification and flag-like triangu-lations of the stratified space. As in [13] the appropriate means of doing thisis to order the vertices within each stratum, then orient edges lying withina stratum from earlier vertex to later vertex, while orienting edges that gobetween strata so that the edge traversed in the direction of the orientationis a directed path. As in [13] we refer to a flag-like triangulation with suchan orientation on its edges as a directed triangulation .5. Biparcel Colorings and TQFTs withDefects-with-Structure
In the circumstances described in the last section, in which a starklystratified space has been equipped with a directed space structure for whichall paths lying entirely within a stratum are directed, and the fundamentalcategory of the directed space has been equipped with a conservative functorto a gaunt category with objects corresponding dimensions of the strata, wedescribe a procedure for producing state-sum invariants, which for reasonsanalogous to those discussed in [20] will be multiplicative under disjointunion and admit TQFT-type factorizations along codimension one subspacestransverse to the strata.
Definition 5.1.
For a quasi-spherical Γ-biparcel of fusion type Φ :
C → Γ ,with J : C → C ′ the inclusion of C into a bicategory with spherical duals, a system of Φ -coloring data consists of a choice of representatives for theisomorphism classes of simple 1-arrows in the preimage of each 1-arrow of Γ ,which we collect into a set of 1-arrows S with S chosen to be closed undertaking dual 1-arrows which lie in C , together with a choice of basis for eachof the spaces of 2-arrows C ( ab, c ) where a, b, c ∈ S , with a and b composableand c parallel to their composition. We denote the union of these bases by B , and the entire system of Φ-coloring data by (Φ , S, B ). Definition 5.2.
Let (
X, dX ) is the directed space associated to a 3-manifold X , filtered by the inclusion of a knot, a (closed) surface, or a knot and Seifertsurface to the knot, δ : ↑ Π( X, dX ) → Γ a functor from the fundamentalcategory to a gaunt category Γ, with objects the dimensions of the strata,which maps each object (point) to the dimension of the stratum in whichit lies. For Φ a quasi-spherical Γ-biparcel and Φ-coloring data (Φ , S, B ) a(Φ , S, B )-coloring λ of a directed triangulation T of ( X, dX ) is an assignmentto each vertex of T of the object naming the dimension of its stratum, toeach edge e of the triangulation, a 1-arrow λ ( e ) ∈ S which lies over the1-arrow of Γ to which the edge, as a directed path, maps under δ , and of a2-arrow λ ( uvw ) ∈ B with target λ ( uv ) λ ( vw ) ¯ λ ( uw ) to each 2-simplex uvw .We denote the set of (Φ , S, B )-colorings of T by Λ (Φ ,S,B ) ( T ) or simply Λ( T )if the coloring data are clear from contex. ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 13
As usual, the topological invariant will arise by summing over all coloringsa quantity which is a product of contributions from each simplex of the tri-angulation (here required to be flag-like and directed). The most importantfactor are those associated to colorings of 3-simplexes simplexes, which wedenote α ± ( λ ( σ )). There will be no factors associated to 2-simplexes. Thatassociated to a 1-simplex will be the scalar dimension of the 1-arrow withwhich it is colored, and that associated to vertices will be c − n where c n isthe sum over representatives of isomorphism classes of simple endo-1-arrowsof the object n of the squares of their dimensions, where n is the dimensionof the stratum in which the vertex lies.Here, α ǫ ( σ ) ( λ ( σ )) is the scalar trace of the endo-2-arrow given by thestring diagram shown next to the 3-simplex, where the sign ǫσ is positive ornegative according to whether the combinatorial orientation of the 3-simplexagrees with or is opposite that of the 3-manifold (taken in the illustrationto be the right-hand-rule orientation). α + ( λ ( σ )) λ (03) λ (13) λ (01) λ (12) λ (23) λ (02) λ (03) λ (013) λ (123) λ (012) λ (023) α − ( λ ( σ )) λ (03) λ (02) λ (23) λ (01) λ (12) λ (13) λ (03) λ (023) λ (012) λ (123) λ (013) Our main theorem is then
Theorem 5.3.
For any directed space ( X, dX ) as in 5.2, the quantity X λ ∈ Λ( T ) Y v ∈T c − λ ( v ) Y uv ∈T dim( λ ( uv )) Y σ ∈T α ǫ ( σ ) ( λ ( σ )) is independent of the triangulation T and thus an invariant of the directedspace ( X, dX ) . Proof.
The proof proceeds as usual, by verifying invariance under the ex-tended Pachner moves. The verification for the flag-like Pachner movesproper, regardless of how the boundary interacts with the stratification areessentially very similar to that provide below for the 2-6 and 3-6 moves, andentirely analogous to the proofs of invariance in [1, 12, 19] and are left to thereader. As usual for the reasons set forth in [12, 19] invariance under thesemoves, also implies invariance under reordering of the vertices within thestrata. For brevity we abuse notation and neither close the string diagramsrepresenting the instances of α ± to form their traces geometrically, nor writethem inside the symbols tr( ). All of the string diagrams in the calcula-tions below should be understood as the scalar traces of what is actuallywritten. We also, as noted above, omit the colorings of regions by objects,as these can be recovered from the numbering conventions of the verticeswhich recurs in the colors on edges and vertices of the string diagrams.For each of the moves we explicitly verify, we first indicate, with a depic-tion of the simplicial geometry, the orientations on the 3-simplexes, deter-mining whether an instance of α + or of α − is the appropriate coefficient, andthen give the algebraic local contribution of the simplexes depicted in thebefore and after states of the move. The calculation performed then showsthat the local contribution of both states is the same, thus establishing in-variance under the move, once it is observed that the full state-sum for anypair of triangulations related by an instance of the move can be rearrangedby distributivity and commutativity to have summands corresponding toeach coloring of the boundary of the region depicted, such summand hav-ing the quantity computed as a factor when the region is triangulated asindicated in the depictions of the two states.2-6 move. Here a subdivision of two 3-simplexes sharing a face which isa 2-simplex in a 2-dimensional stratum whose link is a pair of pointsis induced by applying an Alexander subdivision of the shared 2-simplex. There are several cases according to the way in which the2-dimensional stratum restricts directed paths.First, two tetrahedron states of 2-6 move are the following. ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 15 α − ( λ (0123)) α − ( λ (0134))For which the local contribution to the state-sum is given by X c − c − Y dim( λ ( ij )) λ (03) λ (02) λ (23) λ (01) λ (12) λ (13) λ (03) λ (023) λ (012) λ (123) λ (013) λ (04) λ (03) λ (34) λ (01) λ (13) λ (14) λ (04) λ (034) λ (013) λ (134) λ (014) , in which the sum ranges over all colorings of the depicted simplexesand the product over the edges.Applying the analogue of Schur’s lemma that holds by virtue ofthe simplicity of the edge labels gives X c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (34) λ (02) λ (23) λ (12) λ (01) λ (13) λ (03) λ (13) λ (01) λ (14) λ (04) λ (034) λ (023) λ (012) λ (123) λ (013) λ (013) λ (134) λ (014) dim( λ (03)) . Note that this string diagram has two edges labeled λ (03). Byvirtue of the fact that there is only the zero 2-arrow between anypair of parallel non-isomorphic 1-arrows, these two can be allowedto be given different labels ( λ (03) on the upper and λ (03) ′ on thelower) summing over both, without changing the value of the state-sum. Then contracting the edge of the string diagram labeled λ (03) ′ then gives X c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (34) λ (02) λ (23) λ (12) λ (01) λ (13) λ (14) λ (04) λ (034) λ (023) λ (012) λ (123) λ (134) λ (014) . Next, we have the following six tetrahedron states of 2-6 move.
ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 17 . α − ( λ ((012 )) α − ( λ (023 72 )) α + ( λ (123 )) α − ( λ α + ( λ (03 α − ( λ (13 The local state-sum for this configuration is then X c − c − Y dim( λ ( ij )) λ (0 ) λ (02) λ (2 ) λ (01) λ (12) λ (1 ) λ (0 ) λ (02 ) λ (012) λ (12 ) λ (01 ) λ (0 ) λ (03) λ (3 ) λ (02) λ (23) λ (2 ) λ (0 ) λ (03 ) λ (023) λ (23 ) λ (02 ) λ (1 ) λ (2 ) λ (12) λ (23) λ (3 ) λ (13) λ (1 ) λ (12 ) λ (23 ) λ (123) λ (13 ) × λ (04) λ (0 ) λ ( λ (01) λ (1 ) λ (14) λ (04) λ (0 λ (01 ) λ (1 λ (014) λ (04) λ (34) λ (03) λ (3 ) λ ( λ (0 ) λ (04) λ (034) λ (3 λ (03 ) λ (0 λ (14) λ (1 ) λ ( λ (13) λ (3 ) λ (34) λ (14) λ (1 λ (13 ) λ (3 λ (134) where again the sum is over all colorings of the local configurationand the product is over all of the edges occurring in the local con-figuration. Applying the analogue of Schur’s lemma rewrites thisas X c − c − Y dim( λ ( ij )) λ (0 ) λ (03) λ (3 ) λ (02) λ (23) λ (2 ) λ (0 ) λ (02) λ (2 ) λ (12) λ (1 ) λ (01) λ (12) λ (2 ) λ (23) λ (3 ) λ (13) λ (1 ) λ (0 ) λ (03 ) λ (023) λ (23 ) λ (02 ) λ (02 ) λ (012) λ (12 ) λ (12 ) λ (23 ) λ (123) λ (13 ) λ (01 ) λ (04) λ (03) λ (34) λ (3 ) λ (0 ) λ ( λ (04) λ (0 ) λ ( λ (1 ) λ (14) λ (01) λ (1 ) λ ( λ (3 ) λ (13) λ (34) λ (14) λ (04) λ (034) λ (3 λ (03 ) λ (0 λ (0 λ (01 ) λ (1 λ (1 λ (13 ) λ (3 λ (134) λ (014) ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 19 × dim( λ (0 72 )) dim( λ (1 72 )) dim( λ (04)) dim( λ (14)) . Repeated use of the same trick of allowing the two instances of thecolors occurring on two edges in the same connected string diagramseparated by a portion representing an endomorphism to vary inde-pendently, then contracting one reduces this to X c − c − Y dim( λ ( ij )) λ (0 ) λ (03) λ (02) λ (3 ) λ (12) λ (23) λ (13) λ (01) λ (1 ) λ (0 ) λ (03 ) λ (023) λ (012) λ (123) λ (13 ) λ (01 ) λ (04) λ (03) λ (0 ) λ (1 ) λ (13) λ (01) λ (3 ) λ (34) λ (14) λ (14) λ (034) λ (03 ) λ (03 ) λ (13 ) λ (134) λ (014) . Applying the analogue of Schur’s lemma once more to insert theleft diagram into the edge colored λ (0 ) and repeatedly contractingedges then gives. X c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (34) λ (02) λ (23) λ (12) λ (01) λ (13) λ (14) λ (04) λ (034) λ (023) λ (012) λ (123) λ (134) λ (014) λ (3 ) . Again, the sum is over colorings and the product is over the edgesmentioned in the string diagram. Note, however, that at this pointsomething new occurred: the loop labeled λ (3 ) has value dim( λ (3 ),so it, together with the factor of that same dimension, give becomea factor of c , canceling one factor of c and reducing the expres-sion to that computed from the 2-simplex configuration, thus givinginvariance under the 2-6 move.3-6 move. Here a subdivision of three 3-simplexes sharing an edge, lyingin a 1-dimensional stratum, whose link is the three edges in the 3-dimensional stratum, is induced by an Alexander subdivision of thecommon edge. There are again cases depending on how the stratainduce the directed space structure and whether, a common face ofa pair of the 3-simplexes lies in the 2-dimensional stratum (as in thecase we explicitly compute) or not.First, three tetrahedron state of 3-6 move is depicted below:
01 32 4
01 32 α − ( λ (0123))
01 3 4 α − ( λ (0134))
012 4 α + ( λ (0124))The corresponding local state sum is then ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 21 X c − c − c − Y dim( λ ( ij )) λ (03) λ (02) λ (23) λ (01) λ (12) λ (13) λ (03) λ (023) λ (012) λ (123) λ (013) λ (04) λ (03) λ (34) λ (01) λ (13) λ (14) λ (04) λ (034) λ (013) λ (214) λ (014) λ (04) λ (14) λ (01) λ (12) λ (24) λ (02) λ (14) λ (014) λ (124) λ (012) λ (024) . Applying the analogue of Schur’s lemma twice gives X c − c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (34) λ (02) λ (23) λ (01) λ (12) λ (13) λ (03) λ (13) λ (01) λ (14) λ (04) λ (14) λ (01) λ (12) λ (02) λ (24) λ (04) λ (034) λ (023) λ (012) λ (123) λ (013) λ (013) λ (134) λ (014) λ (014) λ (124) λ (012) λ (024) dim( λ (03)) dim( λ (04))The same trick of letting the labels on pairs of edged with thesame label vary independently, then contracting one of the edgesthen gives: X c − c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (34) λ (23) λ (02) λ (12) λ (13) λ (14) λ (24) λ (04) λ (034) λ (023) λ (123) λ (134) λ (124) λ (024) . Next, we have the following six tetrahedron states of 3-6 move.
ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 23
01 32 44 α − (0 , , , α + (0 , , , α − (0 , , , α − ( , , ,
12 4 α + ( , , , α − ( , , , Here the local state-sum is given by X c − c − c − Y dim( λ ( ij )) λ (03) λ (02) λ (23) λ (0 ) λ ( λ ( λ (03) λ (023) λ (0 λ ( λ (0 λ (04) λ ( λ (0 ) λ ( λ (24) λ (02) λ (04) λ (0 λ ( λ (0 λ (024) λ (04) λ (03) λ (34) λ (0 ) λ (23) λ ( λ (04) λ (034) λ (0 λ ( λ (0 × λ ( λ ( λ (23) λ ( λ (12) λ (13) λ ( λ ( λ ( λ (123) λ ( λ ( λ (14) λ ( λ (12) λ (24) λ ( λ ( λ ( λ (124) λ ( λ ( λ ( λ ( λ (34) λ ( λ (13) λ (14) λ ( λ ( λ ( λ (134) λ ( Applying the analogue of Schur’s lemma to insert the first (resp.second, third) diagram in the lower row on the edge labeled λ ( , λ ( , λ ( , λ ( ,
3) (resp. λ ( , λ ( , X c − c − c − Y dim( λ ( ij )) λ (03) λ (02) λ ( λ (0 ) λ (12) λ (23) λ ( λ (13) λ ( λ (03) λ (023) λ (0 λ ( λ (123) λ ( λ (0 λ (04) λ ( λ ( λ (14) λ (0 ) λ (12) λ (24) λ ( λ (02) λ (04) λ (0 λ ( λ (124) λ ( λ (0 λ (024) λ (04) λ (03) λ ( λ (0 ) λ (13) λ (34) λ ( λ (14) λ ( λ (04) λ (034) λ (0 λ ( λ (134) λ ( λ (0 . Another application of the analogue of Schur’s Lemma gives
ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 25 X c − c − c − Y dim( λ ( ij )) λ (04) λ (03) λ ( λ (0 ) λ (13) λ (34) λ ( λ (14) λ ( λ (04) λ (034) λ (0 λ ( λ (134) λ ( λ (0 λ ( λ ( λ (14) λ (0 ) λ (12) λ (24) λ ( λ (02) λ (04) λ (0 λ ( λ (124) λ ( λ (0 λ (024) λ (03) λ (02) λ ( λ (0 ) λ (12) λ (23) λ ( λ (13) λ ( λ (03) λ (023) λ (0 λ ( λ (123) λ ( λ (0 × dim( λ (04)) , Contracting the middle edge labeled λ (04) and then the resultingedge labeled λ (
4) (noting that this is the last edge so labeled so thedimension of the label no longer occurs in the product of dimensions)then gives X c − c − c − Y dim( λ ( ij )) λ (04) λ (03) λ ( λ (0 ) λ (34) λ (13) λ ( λ (14) λ (12) λ ( λ (24) λ (02) λ (04) λ (034) λ (0 λ ( λ (134) λ (124) λ ( λ (0 λ (024) λ (03) λ (02) λ ( λ (0 ) λ (12) λ (23) λ ( λ (13) λ ( λ (03) λ (023) λ (0 λ ( λ (123) λ ( λ (0 Applying Schur’s lemma again gives X c − c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (02) λ (23) λ ( λ (0 ) λ ( λ (12) λ (13) λ ( λ (03) λ ( λ (34) λ (0 ) λ ( λ (13) λ (14) λ (12) λ ( λ (24) λ (02) λ (04) λ (034) λ (023) λ (0 λ ( λ (123) λ ( λ (0 λ ( λ (0 λ (134) λ (124) λ ( λ (0 λ (024) × dim( λ (03)) . Contracting in sequence the vertical edge labeled λ (03), the re-sulting edge labeled λ (
3) and the resulting edge labeled λ (
1) (inthe latter two cases eliminating the last instance of the label andthe corresponding factor of its dimension from the coefficient), thengives
ICATEGORIES FOR TQFTS WITH DEFECTS WITH STRUCTURE 27 X c − c − c − Y dim( λ ( ij )) λ (04) λ (03) λ (34) λ (23) λ (02) λ (12) λ (13) λ (14) λ (24) λ (04) λ (034) λ (023) λ (123) λ (134) λ (124) λ (024) λ (0 ) . The loop, together with the corresponding factor of dim( λ (0 )in the coefficent, then sum to give an overall factor of c , whichwhen cancelled against one factor of c − leave the same result aswe obtained from the three-simplex configuration, thus establishinginvariance under the 3-6 move. (cid:3) Directions for Future Research
The constructions given in this paper, of course, call for the developmentof computational tools to actually compute examples. They also suggestother avenues of research into the construction of TQFTs with defects-with-structure. In no particular order: • Can they be extended to the study of general stratified spaces equippedwith the exit path directed space structure? • Can the assumption of quasi-sphericality be relaxed, either in generalor in particular circumstances? • What is the appropriate categorical structure to imitate the construc-tions in the present paper to give a Crane-Yetter [8] type theorywith defects-with-structure? Here, of course, there are many pos-sible types of defects, including knotted surfaces, with or withoutboundary, embedded 3-manifolds, again, with or without boundary,embedded curves, or even, subsuming them all, a stratification bysubmanifolds with boundary and corners.7.
Acknowledgements
The second author’s part in the completion of this work took place whilein residence at the Mathematical Sciences Research Institute, Berkeley, CA.
He wishes to thank MSRI for its hospitality and the National Science Foun-dation for financial support under grant DMS-0441170.
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I. J. Lee, Mathematics Department, Rowan University, Glass-boro, NJ 08028, U.S.A ([email protected])([email protected])