Homotopy Coherent Mapping Class Group Actions and Excision for Hochschild Complexes of Modular Categories
aa r X i v : . [ m a t h . QA ] A p r [ZMP-HH/20-12]Hamburger Beitr¨age zur Mathematik Nr. 835April 2020 Homotopy Coherent Mapping Class Group Actions and Excisionfor Hochschild Complexes of Modular Categories
Christoph Schweigert and Lukas Woike
Fachbereich MathematikUniversit¨at HamburgBereich Algebra und ZahlentheorieBundesstraße 55D – 20 146 Hamburg
Given any modular category C over an algebraically closed field k , we extract a sequence ( M g ) g ≥ of C -bimodules. We show that the Hochschild chain complex CH ( C ; M g ) of C with coefficients in M g carries a canonical homotopy coherent projective action of the mapping class group of the surfaceof genus g + 1. The ordinary Hochschild complex of C corresponds to CH ( C ; M ).This result is obtained as part of the following more comprehensive topological structure: Weconstruct a symmetric monoidal functor F C : C - Surf c −→ Ch k with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled withprojective objects in C . The functor F C satisfies an excision property which is formulated in termsof homotopy coends. In this sense, any modular category gives naturally rise to a modular functorwith values in chain complexes. In zeroth homology, it recovers Lyubashenko’s mapping class grouprepresentations.The chain complexes in our construction are explicitly computable by choosing a marking on thesurface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connectedand simply connected groupoid of cut systems that appears in the Lego-Teichm¨uller game by acontractible Kan complex. Contents
Introduction and summary
It is an important insight of the last decades that the deep connections between low-dimensional topology andrepresentation theory can be profitably used in both directions.Given a certain type of representation category (typically a monoidal category with plenty of additionalstructure and often subject to finiteness conditions), one can construct topological invariants: Via a surgeryconstruction, a semisimple modular category gives rise to the Reshetikhin-Turaev invariants [RT90, RT91,Tur10]. These include the Turaev-Viro invariants [TV92] that can be obtained from a spherical fusion categoryvia a state sum construction. These constructions actually extend to three-dimensional topological field theoriesin the sense of [At88].By a change of perspective, such constructions can be read backwards in the sense that low-dimensionaltopology can be used to construct meaningful algebraic quantities from a representation category (or a relatedalgebraic object). As a second and often even more important step, one will then use topology to establishproperties of these algebraic quantities. In a lot of cases, such topological manipulations are not only moreconceptual, but also turn out to be easier than purely algebraic manipulations.In this article, we present such a topological perspective on modular categories — with a special emphasis onthe non-semisimple case. On the one hand, it has been known for more than 25 years that one can constructfrom a not necessarily semisimple modular category a system of projective mapping class group representations[Lyu95a, Lyu95b, Lyu96] that can be used to build a modular functor with values in vector spaces. On theother hand, it is clear that non-semisimplicity will result in a non-trivial homological algebra, an importantaspect that from a Hopf algebraic perspective appears e.g. in [GK93, MPSW09, Bic13]. In this paper, we un-ravel within a homotopy coherent framework the interplay of the homological algebra of a modular category andlow-dimensional topology. This leads to homotopy coherent projective mapping class group actions and excisionresults for certain Hochschild complexes of a modular category. Our methods will allow us to systematicallytrace back this structure to a clear topological origin.Let us first recall the notion of a modular category: A finite category is a linear category (over a fixed field k that we will assume to be algebraically closed throughout this article) with finite-dimensional morphism spaces,enough projectives, finitely many isomorphism classes of simple objects such that every object has finite length.A finite tensor category [EO04] is a tensor category (linear Abelian rigid monoidal category with simple unit)whose underlying linear category is a finite category. A finite tensor category that is also equipped with abraiding and a ribbon structure, is called a finite ribbon category . For a braided finite tensor category C withbraiding c , one defines the M¨uger center as the full subcategory of C spanned by all transparent objects, i.e. allobjects X ∈ C satisfying c Y,X c X,Y = id X ⊗ Y for every Y ∈ C . The braiding (and then also the braided finitetensor category) is called non-degenerate if its M¨uger center is trivial, i.e. spanned by the monoidal unit underfinite direct sums. A modular category is a non-degenerate finite ribbon category. Modular categories appearas categories of modules over certain Hopf algebras [Tur10, EGNO17], vertex operator algebras [Hua08] or netsof observable algebras [KLM01], see in particular [LO17, GLO18, CGR20] for the non-semisimple case. Recallthat a modular category (more generally a finite tensor category) is semisimple if and only if all of its objectsare projective.For a treatment of modular categories in topological terms, non-semisimplicity is a major challenge: From asemisimple modular category, a once-extended three-dimensional oriented topological field theory can be builtvia the Reshetikhin-Turaev construction [RT90, RT91, Tur10]. In the non-semisimple case, such a constructionis not available. In fact, once-extended three-dimensional oriented topological field theories are equivalent tosemisimple modular categories [BDSPV15] by evaluation on the circle (interestingly enough, if one changes thebordism category to the extent that it loses rigidity, some constructions are still possible [DRGGPMR19]).For this reason, we will work throughout this article with a different kind of topological structure thatcomprises slightly less than the notion of a once-extended oriented three-dimensional topological field theory:the notion of modular functor [Til98, BK01], or rather a suitable version thereof. Roughly, a modular functoris a consistent system of (projective) mapping class group actions. These are classically valued in vector spaces,but in order to capture the homological algebra of a modular category, we will consider a differential gradedversion.Let us discuss the definition of a modular functor in more detail: An extended surface Σ is a compactoriented two-dimensional smooth manifold (possibly with boundary) with the choice of a point on each boundarycomponent and an orientation on each boundary component (which may either agree or disagree with theorientation induced by the surface making this boundary component either outgoing or incoming ). For a set X (to be thought of as label set), we define the category X - Surf c whose objects are extended surfaces withan element in X for each boundary component and whose morphisms are generated by mapping classes andsewings; the superscript c indicates that some relations between mapping classes will just be satisfied up to an2dditional central generator (this is to allow for (a certain type of) projective actions). Disjoint union endows X - Surf c with a symmetric monoidal structure. We refer to Section 3.1 for the detailed definition of this surfacecategory.A modular functor (with values in chain complexes over k ) is defined as a symmetric monoidal functor X - Surf c −→ Ch k from X - Surf c to the category of chain complexes over k satisfying an excision property formu-lated in terms of homotopy coends (Definition 3.2). We refer to the values of a modular functor as conformalblocks . The notion of symmetric monoidal functor may of course be relaxed from a strict version to a homotopycoherent version by considering instead of X - Surf c a suitable resolution.While considering modular functors with values in chain complexes as a generalization of the classical notionis certainly natural, it is not clear that a non-trivial class of examples exists. The goal of this article is toprove that modular categories produce such a non-trivial class of examples of modular functors. This class ofexamples will lead to concrete applications to Hochschild complexes of modular categories.Let us state the main topological result: To this end, we fix a modular category C and consider the surfacecategory (as defined above) for the label set ( Proj C ) , the set of projective objects of C . We denote this surfacecategory by C - Surf c . The projectivity assumption for boundary labels is not essential and is used here to simplifythe presentation, see Remark 3.9. Theorem 3.6 (Main Theorem, Part I) . Any modular category C gives rise in a canonical way to a modularfunctor F C : C - Surf c −→ Ch k (1.1) with values in chain complexes. The specific model for F C that we provide will actually be strictly functorial in C - Surf c ; in particular, theresulting projective mapping class group actions are strict. However, below we will transfer these actions alongequivalences to certain Hochschild complexes leading to non-strict actions.The category C (or rather its subcategory Proj C ) enters the definition of C - Surf c just through its object set ,but it is actually recovered as a linear category by evaluation of (1.1) on the cylinder, for more details we referto Section 3.2.As an example, we explicitly describe the modular functor for modules over the Drinfeld double of a finitegroup G in finite characteristic as chains on groupoids of G -bundles over surfaces (Example 3.13).The second part of the main result is concerned with the concrete computation of the modular functor F C on a given extended surface Σ with projective boundary label X (of course, Σ can be closed and hence X the empty collection): We choose an auxiliary datum, namely a marking Γ on Σ (roughly: a cut system andan embedded graph). By a prescription using the combinatorial data provided by the marking, the morphismspaces of C and homotopy coends we define a chain complex B Σ,Γ C ( X ), the so-called marked block for ( Σ, Γ, X );see Section 2.2 for details. For the example of the closed torus and a sufficiently simple marking, this complexis given by the (normalized) chains on the simplicial vector space . . . M X ,X ,X ∈ Proj C C ( X , X ) ⊗ C ( X , X ) ⊗ C ( X , X ) M X ,X ∈ Proj C C ( X , X ) ⊗ C ( X , X ) M X ∈ Proj C C ( X , X ) , (1.2) where C ( X, Y ) is the space of morphisms from X to Y , and the face and degeneracy maps are given bycomposition in C and insertion of identities, respectively, see Example 2.2. Hence, it is given by the Hochschildcomplex [MCar94, Kel99] for the category of projective objects in C . Theorem 3.6 (Main Theorem, Part II) . After any choice of marking Γ for an extended surface Σ with projectiveboundary label X , there is a canonical equivalence B Σ,Γ C ( X ) ≃ −−→ F C ( Σ, X ) . (1.3)Note that this equivalence is canonical after the choice of the marking; the marking itself is not canonical.The modular functor F C : C - Surf c −→ Ch k relates to classical constructions of modular functors with valuesin vector spaces: 3 Reshetikhin-Turaev construction . As mentioned above, from a semisimple modular category, one canbuild a once-extended three-dimensional oriented topological field theory via the Reshetikhin-Turaev con-struction [RT90, RT91, Tur10]. Building the modular functor F C : C - Surf c −→ Ch k for a semisimplemodular category does not add anything to the picture: It has non-trivial homology only in degree zeroand recovers in zeroth homology the modular functor obtained by restriction of the Reshetikhin-Turaevtopological field theory to surfaces. • Lyubashenko construction.
The classification result from [BDSPV15] tells us that from a non-semisimplemodular category, we cannot obtain a once-extended oriented three-dimensional topological field theory.However, by a remarkable result of Lyubashenko [Lyu95a, Lyu95b, Lyu96] any modular category (notnecessarily semisimple) still gives rise to a mapping class group representations (in the semisimple case,they agree with the ones obtained from the Reshetikhin-Turaev construction). A key ingredient for theconstruction of these representations is the canonical coend F = R X ∈C X ⊗ X ∨ ∈ C that is also referredto as Lyubashenko coend . The modular functor F C : C - Surf c −→ Ch k will recover in zeroth homology thelinear dual of Lyubashenko’s mapping class group representations. However, the modular functor F C willgenerally have non-trivial higher homologies.We can now provide a topological perspective on Hochschild complexes of a modular category C withcoefficients in specific bimodules: For a modular category C and g ≥
0, the evaluation of the modularfunctor F C on a surface of genus g and with two oppositely oriented boundary components yields a bi-module, i.e. a functor M g : C op ⊗ C −→ Ch k . Up to equivalence, M g is concentrated in degree zero andgiven by M g ( X, Y ) = C ( X, Y ⊗ F ⊗ g ) for X, Y ∈ C , where C ( − , − ) denotes the morphism spaces of C and F = R X ∈C X ⊗ X ∨ ∈ C the canonical coend. We recall in Section 3.4 the definition of the Hochschild chains CH ( C ; M g ) of C with coefficients in M g and prove: Theorem 3.10.
For any modular category C and g ≥ , the Hochschild chains CH ( C ; M g ) with coefficients inthe bimodule M g carry a canonical homotopy coherent projective action of the mapping class group Map ( Σ g +1 ) of the closed surface of genus g + 1 . The complex CH ( C ; M ) is the ‘ordinary’ Hochschild complex (1.2), and the homotopy coherent projectiveaction of Map ( Σ ) = SL(2 , Z ) was already established in [SW19], see [LMSS18] for a Hopf algebraic analogueof this result on (co)homology level. The projective mapping class group actions induced on the homologies H ∗ ( CH ( C ; M g )) can be related to the projective mapping class group actions on certain Ext groups in [LMSS20](Remark 3.12).Our main result provides the following topological proof for Theorem 3.10: Using the excision property formarked blocks, we observe that CH ( C ; M g ) can be seen as the marked block for Σ g +1 and a specific mark-ing . This makes the Hochschild complexes CH ( C ; M g ) canonically equivalent to the conformal block F C ( Σ g +1 )thanks to (1.3). The conformal block F C ( Σ g +1 ) carries even a strict projective action of Map ( Σ g +1 ). As aconsequence, CH ( C ; M g ) carries also a projective Map ( Σ g +1 )-action through transfer which, in general, willjust be homotopy coherent. Note that constructing directly a homotopy coherent action on the Hochschildcomplex CH ( C ; M g ), i.e. without using the relation to F C ( Σ g +1 ), would be rather involved (as the treatment of CH ( C ; M ) in [SW19] shows). The reason for this difficulty is clear: From a topological perspective, the complex CH ( C ; M g ) corresponds to a specific marking, and the action of the mapping class group will not preserve thismarking! Therefore, it is easier to obtain the mapping class group action through the complex F C ( Σ g +1 ), whichis a genuinely topological quantity.Theorem 3.10 implies a Hopf algebraic statement: Let A be a ribbon factorizable Hopf algebra and denote by A ∗ coadj the dual of A equipped with the coadjoint action. Consider now for g ≥ A -module A ⊗ (cid:16) A ∗ coadj (cid:17) ⊗ g (tensor product in the monoidal category of A -modules). By multiplication from the right on the A -factor, thisbecomes an A -bimodule. Corollary 3.11.
Let A be a ribbon factorizable Hopf algebra and g ≥ . Then the Hochschild chains of A with coefficients in the A -bimodule A ⊗ (cid:16) A ∗ coadj (cid:17) ⊗ g carry a canonical homotopy coherent projective action ofthe mapping class group Map ( Σ g +1 ) of the closed surface of genus g + 1 . While the proof of our Main Theorem 3.6 uses Lyubashenko’s work on the canonical coend F = R X ∈C X ⊗ X ∨ of a modular category C and the S-transformation, it does not directly build on Lyubashenko’s construction of theprojective mapping class group representations in [Lyu95a]. These are based on a presentation of mapping classgroups in terms of generators and relations and seem hard to adapt to a differential graded framework. Instead,we adapt the Lego Teichm¨uller game developed by Bakalov and Kirillov in [BK00] based on [HT80, Har83, Gro84]to our purposes by replacing their connected and simply connected groupoid of markings on an extended surface4y a contractible ∞ -groupoid: First we define a category b M ( Σ ) of colored markings on an extended surface Σ formed by markings on Σ with the additional datum of a subset of distinguished cuts which we call colored cuts .We require that there is at least one such colored cut per closed connected component. We also add uncolorings ,new non-invertible morphisms that reduce the number of colored cuts. We then prove the crucial result thatthe category of colored markings b M ( Σ ) (Theorem 4.11) is contractible. The reason for the significance of thecategory of colored markings is Theorem 5.4 which states that marked blocks can actually be naturally extendedto functors b M ( Σ ) −→ Ch k out of the category b M ( Σ ) of colored markings on Σ . The idea is to send a coloredmarking to a version of marked blocks which uses homotopy coends for gluing at all colored cuts and ordinarycoends at uncolored cuts. This is motivated by the key observation that the marked blocks do not change upto equivalence if we replace the homotopy coends used for the gluing by ordinary coends at all but one cut perclosed connected component (Corollary 5.2). As a consequence, the functor b M ( Σ ) −→ Ch k sends all uncoloringsto equivalences and hence descends to the ∞ -groupoid obtained by localizing b M ( Σ ) at all uncolorings. Thisconstruction allows us to reduce some statements about our differential graded marked blocks to statementsabout marked blocks with values in vector spaces.The functors b M ( Σ ) −→ Ch k descend to the category obtained by gluing colored markings for different surfacestogether (the gluing is accomplished via the Grothendieck construction). By a homotopy left Kan extension,we obtain a symmetric monoidal functor defined on labeled surfaces — this will be our modular functor. Theproof of the equivalence B Σ,Γ C ( X ) ≃ −−→ F C ( Σ, X ) from (1.3) relies on the contractibility result Theorem 4.11.Finally, we use (1.3) to conclude excision from a marked version of excision (Proposition 2.3) which is easier toprove.It should be mentioned that the methods developed in this article could also be helpful to study modularfunctors with values in vector spaces . We consistently use the Lego Teichm¨uller game from [BK00] and con-struct the modular functor by gluing together (in a categorical sense) markings for different surfaces via theGrothendieck construction and a left Kan extension (note that this is different from the strategy in [FS17], seeRemark 5.7). The important, but subtle concept of coends in categories of left exact functors between finitecategories from [Lyu96] is avoided and replaced by techniques which are easier to adapt to a differential gradedframework. One key simplification in comparison to [Lyu95a, LMSS18, SW19, LMSS20] (regardless of whetherthese works cover the vector space valued case or work at chain level or in (co)homology) is that our constructiondoes not rely on a concrete presentation of mapping class groups in terms of generators and relations, but isgenuinely topological.
Conventions.
Throughout this text, we will work over an algebraically closed field k which is not assumed tohave characteristic zero. By Ch k we denote the symmetric monoidal category of chain complexes over k equippedwith its projective model structure in which weak equivalences (for short: equivalences) are quasi-isomorphismsand fibrations are degree-wise surjections. A (small) category enriched over Vect k or Ch k will be called a linearor differential graded category, respectively. Unless otherwise stated, functors between linear and differentialgraded categories will automatically be assumed to be enriched. By a (canonical) equivalence between chaincomplexes we do not necessarily mean a map in either direction, but also allow a (canonical) zigzag. Acknowledgments.
We would like to thank Adrien Brochier, Damien Calaque, J¨urgen Fuchs, David Jordan,Andr´e Henriques, Simon Lentner, Svea Nora Mierach, Lukas M¨uller, Claudia Scheimbauer, Yorck Sommerh¨auserand Nathalie Wahl for helpful discussions.CS and LW are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)within the framework of the RTG 1670 “Mathematics Inspired by String Theory and QFT” and under Germany’sExcellence Strategy – EXC 2121 “Quantum Universe” – 390833306.
We start by giving the definition of marked blocks which should be seen as auxiliary objects needed for theconstruction of modular functors. They will later enable us to perform concrete computations. Moreover, wewill already establish a version of excision for marked surfaces that may be seen as a preparation for the excisionproperty that will be a part of the modular functor.
Before defining marked blocks, we recall some terminology and conventions on surfaces, cut systems and mark-ings from [BK00, BK01, FS17]: In the sequel, a surface will be an abbreviation for compact oriented two-dimensional smooth manifold with boundary. A surface with oriented boundary is a surface such that everyboundary component is endowed with an orientation. If this orientation of a specific boundary component5oincides with the orientation inherited from the surface, we refer to this component as outgoing , otherwise as incoming . An extended surface is a surface with oriented boundary and the choice of a marked point on everyboundary component.
Cut systems.
Every surface can be non-uniquely cut into spheres with several open disks removed. Thisis formalized as follows: We define a cut on an extended surface Σ as an oriented simple closed curve in theinterior of Σ with the choice of a point on this curve. An isotopy class of a finite family C of disjoint cuts on Σ iscalled a cut system if every component of Σ \ C has genus zero (here the isotopy has to preserve the disjointnessof the cuts). The number of cuts in a cut system C will be denoted by | C | . The manifold with boundaryobtained by cutting Σ along C will be denoted by cut C Σ (we must keep in mind that, strictly speaking, cut C Σ is only well-defined up to diffeomorphism because cut systems are defined as isotopy classes). If C consists of | C | cuts, then cut C Σ has 2 | C | boundary components more than Σ . The orientation and the marked point onthese additional boundary components are inherited from the cutting curve. The relations between differentcut systems will be covered in Section 4.1. Standard spheres.
For n ≥ ε = ( ε , . . . , ε n ) ∈ {± } n of signs, we define a particular extendedsurface, namely the standard sphere S ◦ n,ε . The underlying surface is the Riemann sphere C ∪ {∞} with n opendisks D , . . . , D n with radius 1 / , . . . , n removed. The orientation of this surface is the standardone, and the boundary circle with center j , where 1 ≤ j ≤ n , is endowed with the inherited orientation if ε j = +1 (making this component outgoing), and endowed with the opposite orientation if ε j = − incoming ). The marked points lie at j − i / ≤ j ≤ n . This extended surface is decoratedwith a graph Γ ◦ n called the standard marking whose vertices are the marked points and the so-called internalvertex at − n straight lines between the internal vertex and each marked point. Thepoint 1 − i / distinguished vertex , and the edge connecting the internal vertex to the distinguishedvertex is called the distinguished edge . The standard marking for a spheres with three holes and some boundaryorientation is depicted in Figure 1. Figure Standard marking on a sphere with three holes and sign tuple ε = ( − , +1 , +1) . The long straight arrows arethe coordinate axes of the complex plane. On the three boundary components, the marked points are drawnas white dots and the orientation is indicated by an arrow. The distinguished vertex of the marking is a bluedot, and the edges of the marking are drawn in blue. The distinguished edge is drawn as a double line. Markings.
Let Σ be a connected extended surface of genus zero. A chart for Σ is a diffeomorphism Φ : Σ −→ S ◦ n,ε for some n ≥ ε ∈ {± } n which preserves the orientation, the orientation of the boundaryand sends marked points to marked points. We call a graph Γ embedded in Σ a Φ -compatible graph for achart Φ : Σ −→ S ◦ n,ε if Φ sends Γ to the standard graph Γ ◦ n on S ◦ n,ε . Consider two pairs ( Φ j , Γ j ) for j = 0 , Φ j : Σ −→ S ◦ n,ε is a chart (both for the same n and ε ) and Γ j a Φ j -compatible graph on Σ . An isotopy( Φ , Γ ) −→ ( Φ , Γ ) is an isotopy Φ t from Φ to Φ through charts. An isotopy class of a pair formed by achart for Σ and a compatible graph is called a marking without cuts on Σ . A marking on an arbitrary extendedsurface Σ is a cut system C together with a marking without cuts on every connected component of cut C Σ .Note that this equips Σ in particular with an isotopy class of graphs Γ , see Figure 2 for an example. Often,we will use the symbol Γ for this graph to denote the entire marking, thereby suppressing the underlying cutsystem and the charts in the notation. 6 igure A marking on a torus T with two boundary components. The cut system has one cut such that cut C T is asphere with four holes. We use the same drawing convention for the marked points, the orientation and themarking as in Figure 1. After these preparations, we may define the marked blocks for a pivotal k -linear monoidal category C . Themonoidal product and the monoidal unit for the underlying k -linear monoidal category will be denoted by ⊗ and I , respectively. Pivotality means that C is rigid (we denote by − ∨ the duality functor; our conventionsfor the duality are the ones from [EGNO17]) and equipped with a monoidal natural isomorphism id C ∼ = − ∨∨ from the identity functor on C to the double dual functor. Thanks to the pivotal structure, left and right dualscoincide. We introduce the notation X ε for ε ∈ {± } by X := X and X − := X ∨ .In order to establish later the crucial results for these marked blocks, we will need C to be modular (a notionrecalled in Section 3.3), but the mere definition of marked blocks makes sense in greater generality.Denote by Σ p | n − p a surface of genus zero with n ≥ p of which are incoming. Let Σ p | n − p be endowedwith a marking without cuts, i.e. an isotopy class of a chart Φ : Σ p | n − p −→ S ◦ n,ε and a graph Γ that is mappedby Φ to the standard marking Γ ◦ n on the standard sphere S ◦ n,ε . Now let X be a labeling of the boundarycomponents of Σ p | n − p with projective objects in C , i.e. a function from π ( ∂Σ p | n − p ) to the projective objectsof C . The map π ( ∂ S ◦ n,ε ) −→ π ( ∂Σ p | n − p ) induced by Φ provides a numbering ( X , . . . , X n ) of the objects thatare part of the labeling. We define the vector space B Σ p | n − p ,Γ C ( X ) := C ( I, X ε ⊗ · · · ⊗ X ε n n ) , (2.1)where C ( − , − ) is our notation for the morphism spaces of C , and observe that this is well-defined, i.e. it doesnot depend on the representative of our marking (recall that in our language a marking is always an isotopyclass of certain data as precisely defined above).In the next step, let ( Σ, Γ ) be a connected marked surface. The surface cut C Σ obtained by cutting Σ alongthe cut system underlying the marking yields components ( Σ , Γ ) , . . . , ( Σ ℓ , Γ ℓ ) with each Σ j being a spherewith m j ≥ Σ j and we will do the same for thecuts in a moment; this is done mainly for the readability of this rather technical definition — we will explainafterwards how the dependence on this order is dealt with. If n j is the number of boundary components of Σ j that do not arise from the cutting, then we can label these boundary components by a family X j of length n j ofprojective objects in C . These boundary labels form a boundary label X for Σ (and of course, every boundarylabel for Σ arises this way). Additionally, we label the two oppositely oriented boundaries arising from a cut c i by a projective object Y i , where 1 ≤ i ≤ | C | . Denote by Y j the boundary labels of Σ j arising from cuts. Nowit makes sense to consider the vector spaces B Σ j ,Γ j C ( X j , Y j ) for 1 ≤ j ≤ ℓ . As these vector spaces run over j , each Y i appears precisely twice, for the two boundary components resultingfrom the cut — once as covariant and once as contravariant argument. Therefore, we may define the chaincomplex B Σ,Γ C ( X ) := Z Y ,...,Y | C | ∈ Proj C L ℓ O j =1 B Σ j ,Γ j C ( X j , Y j ) (2.2)7ia an iterated homotopy coend [SW19, Section 2]; we spell this definition out for the torus in Example 2.2.In (2.2) we have used a numbering for the iterated homotopy coends and the multiple tensor products, which,strictly speaking, is problematic because those numberings are not part of the data of a marking (for instance,the set of cuts was not assumed to be ordered). Let us discuss the remedy in detail for the iterated homotopycoends (we comment on the multiple tensor products afterwards): Denote by O C the action groupoid of thefree and transitive action of the permutation group on | C | letters on the bijections from the set of cuts of C to { , . . . , | C |} . Then the right hand side of (2.2), when evaluated for all possible orderings of cuts, will actuallygive us a functor O C −→ Ch k . To this end, a permutation of such an ordering must be sent to an isomorphismcompatibly with the composition of permutations. The latter can be done using the Fubini Theorem from[SW19, Proposition 2.7]. The marked block B Σ,Γ C ( X ) can then be defined as the colimit of that functor (whichis also the homotopy colimit). This gives us the notion of an unordered iterated homotopy coend. Of course, ifwe pick an ordering for the set of cuts, the complex computed for that ordering will be canonically isomorphicto colimit over all orderings. In the same way, we can use an unordered tensor product (defined using thesymmetric braiding of Ch k ) to get rid of the numbering of the surfaces that Σ is cut into. In the sequel,however, we will suppress such subtleties in the notation and will understand an expression like the right handside of (2.2) always as an unordered homotopy coend and/or tensor product.As for (2.1), we can observe that (2.2) does not depend on any representatives chosen for the marking (inparticular, it is not a problem that cut C Σ is only well-defined up to diffeomorphism). The reason for this is thatthe definitions (2.1) and (2.2) just depend on the combinatorics coming from the incidences of cuts and graphs,i.e. their relative location to each other. This important point is already implicit in related constructions in[BK01, FS17].The definition (2.2) is extended to non-connected surfaces by sending the disjoint union of connected markedsurfaces with boundary components labeled by projective objects to the tensor product of the chain complexesdefined for the connected case. Definition 2.1.
We refer to B Σ,Γ C ( X ) as the marked block for the marked surface ( Σ, Γ ), the pivotal linearmonoidal category C and the projective boundary label X .Of course, the marked blocks are functorial in the boundary label. This will play a role later, see Section 3.2and in particular Remark 3.1. Example 2.2 (Relation to Hochschild chains) . For the closed torus T with the marking Γ with one cut shownin Figure 3, the marked block B T ,Γ C is given by the homotopy coend B T ,Γ C = Z X ∈ Proj C L C ( X, X ) , which by the definition is given by the normalized chains on the simplicial vector space . . . M X ,X ,X ∈ Proj C C ( X , X ) ⊗ C ( X , X ) ⊗ C ( X , X ) M X ,X ∈ Proj C C ( X , X ) ⊗ C ( X , X ) M X ∈ Proj C C ( X , X ) , where the face maps are defined using the composition in C and the degeneracies insert identities. This is theHochschild complex of the linear category Proj C . It is called the Hochschild complex of C (instead of Proj C )in [SW19] because it is implicitly assumed that only the projective objects are used to construct the complex.The latter is motivated by the Agreement Principle [MCar94, Kel99] that says that the Hochschild complexbuilt from the projective objects of a linear category C is equivalent to the usual Hochschild complex of afinite-dimensional algebra A if C is the category of finite-dimensional modules over A (see also Section 3.4). Figure Marking for the closed torus (same drawing conventions as in Figure 1). .3 Excision properties of marked blocks The locality properties of modular functors are usually phrased in terms of factorization and self-sewing , see[BK01, FS17]. For the marked blocks, we will already formulate here a marked version of such a localityproperty that will follow in a straightforward way from the definitions. Factorization and self-sewing can evenbe packaged into one property that we call excision .In order to formalize the excision property, let (
Σ, Γ ) be a marked surface and denote by Σ ′ the result of sewinga specific incoming boundary component to a specific outgoing boundary component (more complicated sewingoperations can be considered as well, but they can the decomposed into sewings of the form just described).We write this sewing operation symbolically as an arrow s : Σ −→ Σ ′ (in Section 3.1 we will discuss a categoryof surfaces in which these sewings are promoted to actual morphisms, but this is not needed for the moment).The marking Γ on Σ induces a marking Γ ′ on Σ ′ .Consider the marked block B Σ,Γ C ( X, P, Q ) for (
Σ, Γ ) and boundary label (
X, P, Q ), where P and Q arethe labels for the incoming and outgoing boundary component along which we glue, and X is a label for theremaining boundary components. Since C ( − , − ) : C op ⊗ C −→ Vect is a functor, it follows from the definition ofmarked blocks in (2.2) that the assignment P ⊗ Q B Σ,Γ C ( X, P, Q ) yields a functor (
Proj C ) op ⊗ Proj
C −→ Ch k ,i.e. a differential graded bimodule over Proj C . We now obtain a canonical isomorphism B Σ ′ ,Γ ′ C ( X ) ∼ = Z P ∈ Proj C L B Σ,Γ C ( X, P, P ) (2.3)because both complexes describe the colimit by which the unordered homotopy coend is defined. In particular,we obtain a sewing map s P : B Σ,Γ C ( X, P, P ) −→ Z P ∈ Proj C L B Σ,Γ C ( X, P, P ) ∼ = B Σ ′ ,Γ ′ C ( X ) , (2.4)which is just the structure map B Σ,Γ C ( X, P, P ) −→ R P ∈ Proj C L B Σ,Γ C ( X, P, P ) composed with the isomorphism(2.3). By definition of the homotopy coend we now obtain the following statement:
Proposition 2.3 (Excision with marking) . For every pivotal linear monoidal category C , any marked surface ( Σ, Γ ) and a sewing s : ( Σ, Γ ) −→ ( Σ ′ , Γ ′ ) that glues one incoming boundary component to an outgoing one,the sewing maps (2.4) induces an equivalence Z P ∈ Proj C L B Σ,Γ C ( X, P, P ) ≃ −−→ B Σ ′ ,Γ ′ C ( X ) of chain complexes from the homotopy coend over the Proj C -bimodule B Σ,Γ C ( X, − , − ) to B Σ ′ ,Γ ′ C ( X ) . In fact, this map is actually an isomorphism with the concrete models for the homotopy coend that we use.Of course, the proof of the marked version of excision is immediate. The more non-trivial task of formulatingsuch statements independently of the marking will be addressed later.
In [SW19, Section 3.2], we have proven a relation between homotopy coends over projective objects and theLyubashenko coend whose definition we recall in a moment (we have also argued there that this is an instance ofthe general principle to express traces via class functions, but this background is not needed to understand thestatement). After recalling some terminology, we will prove a generalization of the statements made in [SW19,Section 3.2] that we will need in the sequel.
Finite tensor categories. A finite category is a linear category with finite-dimensional morphism spaces,enough projectives, finitely many isomorphism classes of simple objects such that every object has finite length.A linear category is finite if and only if it is linearly equivalent to the category of finite-dimensional modules overa finite-dimensional algebra (which does not mean that choosing such an equivalence will be necessarily helpful).A tensor category is a linear Abelian rigid monoidal category with simple unit. A finite tensor category [EO04]is a tensor category whose underlying linear category is a finite category. From these definitions, one concludesthe following: In a finite tensor category C , any tensor product X ⊗ Y is projective if X or Y is projective.Moreover, the tensor product is exact in both arguments. Finally, any finite tensor category is self-injective,i.e. the projective objects are precisely the injective ones.9 he Lyubashenko coend. For any finite tensor category C , one may define the coend F := Z X ∈C X ⊗ X ∨ (2.5)which is called the canonical coend of C or also the Lyubashenko coend due to its appearance in [Lyu95a,Lyu95b, Lyu96]. This object is the key to the construction of the mapping class group actions in [Lyu95a].In [SW19, Section 3.2], the coend (2.5) is replaced by a (finite version of a) homotopy coend leading to adifferential graded object R P ∈ Proj C f L P ⊗ P ∨ in C which is proven to be a projective resolution of F . This projectiveresolution appears in the following important homological algebra result that is the key to understanding markedblocks because it gives us the possibility to express iterated homotopy coends over morphism spaces in a differentway: Proposition 2.4.
For any pivotal finite tensor category C , there is a canonical equivalence of chain complexes Z P ,...,P g ∈ Proj C L C ( X, P ⊗ P ∨ ⊗ · · · ⊗ P g ⊗ P ∨ g ) ≃ C X, Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g for any X ∈ C and g ≥ . Recall from our conventions that by (canonical) equivalence we do not mean necessarily a map in eitherdirection, but also allow a (canonical) zigzag.
Proof.
For g = 0, the statement is true by the convention that a tensor product over an empty index set is themonoidal unit. We now prove the statement by induction on g ≥
1. For g = 1, we find by duality in C and[SW19, Theorem 3.5] Z P ∈ Proj C L C ( X, P ⊗ P ∨ ) ∼ = Z P ∈ Proj C L C ( X ⊗ P, P ) (by duality) ≃ C I, Z P ∈ Proj C f L P ⊗ ( X ⊗ P ) ∨ ! (by [SW19, Theorem 3.5]) ∼ = C I, Z P ∈ Proj C f L P ⊗ P ∨ ⊗ X ∨ ! (since ( X ⊗ P ) ∨ ∼ = P ∨ ⊗ X ∨ ) ∼ = C X, Z P ∈ Proj C f L P ⊗ P ∨ ! (by duality) . This proves the statement for g = 1.In order to complete the induction step g −→ g + 1, we observe Z P ,...,P g +1 ∈ Proj C L C ( X, P ⊗ P ∨ ⊗ · · · ⊗ P g +1 ⊗ P ∨ g +1 ) ∼ = Z P ,...,P g +1 ∈ Proj C L C ( X ⊗ P g +1 ⊗ P ∨ g +1 , P ⊗ P ∨ ⊗ · · · ⊗ P g ⊗ P ∨ g ) (by duality) ≃ Z P g +1 ∈ Proj C L C X ⊗ P g +1 ⊗ P ∨ g +1 , Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g (by the induction hypothesis) ≃ Z P g +1 ∈ Proj C L C (cid:0) X ⊗ P g +1 ⊗ P ∨ g +1 , F ⊗ g (cid:1) (by [SW19, Corollary 3.7]and since X ⊗ P g +1 ⊗ P ∨ g +1 is projective) ∼ = Z P g +1 ∈ Proj C L C (cid:16)(cid:0) F ⊗ g (cid:1) ∨ ⊗ X, P g +1 ⊗ P ∨ g +1 (cid:17) (by duality) ≃ C (cid:0) F ⊗ g (cid:1) ∨ ⊗ X, Z P ∈ Proj C f L P ⊗ P ∨ ! (by the induction start) ∼ = C X, F ⊗ g ⊗ Z P ∈ Proj C f L P ⊗ P ∨ ! (by duality) ≃ C X, Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ ( g +1) (by [SW19, Lemma 3.8]).10 nimodularity. Let C be a pivotal finite tensor category. Since C is assumed to be over an algebraicallyclosed field, we have by [GKP18, Section 5.3] a so-called modified trace on the tensor ideal of projective objectsthat is unique up to invertible scalar. It gives us canonical and natural isomorphisms C ( X, P ) ∼ = C ( α ⊗ P, X ) ∗ for any X ∈ C as long as P is projective, where α is the socle of the projective cover of the monoidal unit. If C is unimodular, α is isomorphic to the monoidal unit. In this case, we even find C ( X, P ) ∗ ∼ = C ( P, X ) , X ∈ C , P ∈ Proj C . (2.6)Therefore, we may conclude from Proposition 2.4: Corollary 2.5.
For any unimodular pivotal finite tensor category C and X ∈ C , there is a canonical equivalenceof chain complexes Z P ,...,P g ∈ Proj C L C ( X, P ⊗ P ∨ ⊗ · · · ⊗ P g ⊗ P ∨ g ) ≃ C Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g , X ∗ . Example 2.6.
For p, q ≥ g ≥
1, consider the standard marked sphere S ◦ p + q +2 g,ε such that the first p holes are incoming, the next q holes are outgoing, and the last 2 g holes are pairs of an outgoing followedby an incoming hole. By gluing together these g pairs, we obtain a marking Γ ◦ g,p,q on the surface Σ p | qg ofgenus g with p incoming and q outgoing boundary components. If C is a pivotal finite tensor category and X = ( X ′ , . . . , X ′ p , X ′′ , . . . , X ′′ q ) a projective boundary label for Σ p | qg (the first p labels are for the incomingboundary components, the last q labels for the outgoing ones), then by the definition of marked blocks andduality (recall that left and right duals coincide thanks to pivotality), we have B Σ p | qg ,Γ ◦ g,p,q C ( X ) ∼ = Z P ,...,P g ∈ Proj C L C ( X ′ p ⊗ · · · ⊗ X ′ , X ′′ ⊗ · · · ⊗ X ′′ q ⊗ P ⊗ P ∨ ⊗ · · · ⊗ P g ⊗ P ∨ g ) . From Proposition 2.4, we conclude B Σ p | qg ,Γ ◦ g,p,q C ( X ) ≃ C X ′ p ⊗ · · · ⊗ X ′ , X ′′ ⊗ · · · ⊗ X ′′ q ⊗ Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g . (2.7)If C is unimodular, we arrive at B Σ p | qg ,Γ ◦ g,p,q C ( X ) ≃ C X ′′ ⊗ · · · ⊗ X ′′ q ⊗ Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g , X ′ p ⊗ · · · ⊗ X ′ ∗ (2.8)thanks to Corollary 2.5. We will now relate marked blocks to the ‘classical’ marked blocks with values in vector spaces from [FS17]which are based on [Lyu95a]. First we observe that the marked block B Σ,Γ C ( X ) for a marked surface ( Σ, Γ )with projective boundary label X comes with a canonical surjection B Σ,Γ C ( X ) −→ H B Σ,Γ C ( X ) (2.9)to its zeroth homology that we refer to as the augmentation fibration . The zeroth homology of the markedblock B Σ,Γ C ( X ) is given the same expression as in (2.2), but with the homotopy coend replaced by an ordinarycoend. It is important that both types of coends run over the subcategory of projective objects. Marked blocks with values in vector spaces.
In [FS17, Section 2.4] marked block functors are defined basedon [Lyu95a, Lyu96] as coends in categories of left-exact functors. This definition can be made for any pivotalfinite tensor category C (for the construction of mapping class group actions, more structure is needed). Thevalue of the these block functors on a given boundary label is a vector space and will denoted by b Σ,Γ C ( X ); in[FS17] the notation e Bl Σ,Γ is used for the block functors.
Proposition 2.7 (Augmentation fibration in the case of non-empty boundary) . Let C be a pivotal finite tensorcategory. If ( Σ, Γ ) is a marked surface with at least one boundary component on each connected component,then the augmentation fibration (2.9) is a trivial fibration for every projective boundary label X , i.e. in this case,the marked block has non-trivial homology only in degree zero. Moreover, this zeroth homology is canonicallyisomorphic to the vector spaces valued marked block b Σ,Γ C ( X ) such that the augmentation fibration takes theform of a trivial fibration B Σ,Γ C ( X ) ≃ −−→ b Σ,Γ C ( X ) . (2.10)11 roof. Without loss of generality, we may assume that Σ is a connected surface of genus g with n ≥ B Σ,Γ C ( X ) C (cid:18) X ⊗ , (cid:16)R P ∈ Proj C f L P ⊗ P ∨ (cid:17) ⊗ g (cid:19) b Σ,Γ C ( X ) C ( X ⊗ , F ⊗ g ) induced by R P ∈ Proj C f L P ⊗ P ∨ ≃ −−−→ F [FS17, (2.12)] H −−−→ H B Σ,Γ C ( X ) H C (cid:18) X ⊗ , (cid:16)R P ∈ Proj C f L P ⊗ P ∨ (cid:17) ⊗ g (cid:19) b Σ,Γ C ( X ) C ( X ⊗ , F ⊗ g ) ∼ = ∼ = ∼ = We first describe the left diagram: The lower horizontal map is the canonical isomorphism [FS17, (2.12)] fromthe vector spaces valued marked block to the morphism space C ( X ⊗ , F ⊗ g ), where X ⊗ = X σ (1) ⊗ · · · ⊗ X σ ( n ) for some permutation σ on n letters (that is fixed by our definition of marked blocks, but not relevant for theargument because it is the same regardless of whether we consider blocks with values in vector spaces or chaincomplexes). This isomorphism is a specific sequence coming from the application of the Yoneda Lemma in theform [FS17, (2.2)] and the relation [FS17, (2.4)] between the coend in left-exact functors and the Lyubashenkocoend. The upper horizontal double-headed arrow in the left diagram is a zigzag that is obtained by performingthe same sequence of operations on marked blocks, but with the following replacements for [FS17, (2.2)] and[FS17, (2.4)]: • Instead of [FS17, (2.2)], we use the canonical equivalence Z P ∈ Proj C L C ( X, P ) ⊗ C ( P, Y ) ≃ C ( X, Y )from [SW19, Lemma 4.11] that holds for objects X and Y in C if either X or Y is projective (in order toapply this, it is crucial that X ⊗ is projective, which uses n ≥
1; for n = 0, the object X ⊗ = I would notbe necessarily projective — in fact, it is projective if and only if C is semisimple). • Instead of [FS17, (2.4)], we use Proposition 2.4.Therefore, the upper horizontal arrow in the left diagram is a zigzag of equivalences. Moreover, since X ⊗ isprojective, C ( X ⊗ , − ) is exact, which means that the right vertical map in the left diagram is an equivalence(again, this only holds because of n ≥ B Σ,Γ C ( X ) −→ H B Σ,Γ C ( X )is an equivalence.It remains to exhibit a canonical isomorphism H B Σ,Γ C ( X ) ∼ = b Σ,Γ C ( X ). To this end, we take the zerothhomology of the left diagram which gives us the right diagram. Note that we invert the lower horizontal map.Now we define the dashed isomorphism such that the right square commutes. Remark 2.8.
The triviality of higher homologies for marked blocks for surfaces with non-empty boundarycomes here from the fact that we only allow projective boundary labels (which we do to give a technicallymore uniform treatment). Non-projective boundary labels can be considered in a straightforward way once themodular functor is constructed as we will explain in Remark 3.9.
In this section we state our main result that a (not necessarily semisimple) modular category gives rise to a modular functor with values in chain complexes (we will give the precise notion in Definition 3.2 and 3.4 below)and that we can compute this modular functor in terms of marked blocks. The proof of this result will occupythe rest of the article.
The notion of a modular functor uses the category of extended surfaces
Surf defined in [FS17, Section 3.1]based on [BK00, HLS00, BK01]. Objects are extended surfaces as defined in Section 2.1. Morphisms aregenerated by mapping classes (isotopy classes of orientation-preserving diffeomorphisms mapping marked pointsto marked points) and sewings of surfaces that glue one or several pairs of incoming and outgoing boundarycomponents together. These generators are subject to the obvious relations for the composition of mappingclasses and sewings separately and the following mixed relation involving both mapping classes and sewings: If φ : Σ −→ Σ is a mapping class and s : Σ −→ Σ ′ a sewing, then φ induces a sewing s ′ : Σ −→ Σ ′ of Σ ,and s induces a mapping class φ ′ : Σ ′ −→ Σ ′ . The morphisms are subject to s ′ φ = φ ′ s . Disjoint union makes
Surf into a symmetric monoidal category.12 entral extensions.
The group
Map ( Σ ) := Aut Surf ( Σ ) is the mapping class group of the extended surface Σ .Hence, functors out of the category Surf of surfaces describe in particular mapping class group representations.In order to describe certain projective mapping class group actions relevant in the theory of modular functors,one may use the central extension
Surf c of Surf discussed in [FS17, Section 3.2] based on [Seg04]. Here
Surf c isa certain category with the same objects as Surf c which comes with a functor P : Surf c −→ Surf inducing forevery extended surface Σ a short exact sequence0 −→ Z −→ Aut
Surf c ( Σ ) −→ Aut
Surf ( Σ ) −→ Surf c from Surf by introducing for each extended surface Σ an additionalgenerator C Σ which commutes with all mapping classes and which behaves multiplicatively under sewing, i.e.for any sewing s : Σ ⊔ Σ ′ −→ Σ ′′ , the relation s ( C pΣ ⊔ C qΣ ′ ) = C p + qΣ ′′ s for p, q ∈ Z holds. For details, we refer to [FS17, Section 3.2] and also Remark 3.8 and 5.12, where we discuss the origin ofthe projectiveness and give a concrete model for the central extension in (5.15) (that one could also take as adefinition of Surf c ). The functor P : Surf c −→ Surf sends the central generators of
Surf c to identities. Labeled surfaces.
For any set X (to be thought of as label set ), we can define the category X - Surf whoseobjects are extended surfaces with boundary components labeled by elements of X . Morphisms are again givenby mapping classes (acting in the obvious way on the labels) and sewings with the restriction that a sewing of anincoming to an outgoing boundary component is only allowed if the labeling objects coincide. The label for thesewn surface is obtained by omitting the labels of the boundary components that are sewn together. Disjointunion provides again a symmetric monoidal structure. Of course, we also obtain a labeled version X - Surf c ofthe central extension Surf c . All definitions for the notion of a modular functor that abound in the literature have in common that a modularfunctor should provide a consistent system of (projective) mapping class group representations in vector spacescompatible with the gluing of surfaces.We will now lift this structure to a differential graded framework: Vector spaces will be replaced with chaincomplexes, the projective mapping class group actions will not necessarily be strict, but possibly up to coherenthomotopy, also the gluing properties have to be formulated homotopy coherently.Let X be a label set and M : X - Surf c −→ Ch k a symmetric monoidal functor whose value on an extendedsurface Σ with label X = ( X , . . . , X n ) for the n boundary components we denote by M ( Σ, X ). For themoment, this functor is assumed to be strict, but see Remark 3.3.
Remark 3.1 (Cylinder category and functorial dependence on boundary label) . To the cylinder with incomingboundary label X ∈ X and outgoing boundary label Y ∈ X , a symmetric monoidal functor M : X - Surf c −→ Ch k assigns a chain complex M ( S × [0 , , ( X, Y )). Since the sewing of cylinders is associative, X becomes the setof objects for a (not necessarily unital) differential graded category that we denote by Z M and refer to as the cylinder category of M . We can now deduce a functorial dependence of the values of M on cylinder category:Let Σ be an extended surface with one outgoing boundary component (more boundary components can betreated in the same way). Then M ( Σ, − ) : Z M −→ Ch k is a functor on the cylinder category. In fact, for X, Y ∈ X , sewing a cylinder to Σ yields a map Z M ( X, Y ) ⊗ M ( Σ, X ) = M ( S × [0 , , ( X, Y )) ⊗ M ( Σ, X ) −→ M ( Σ ∪ S ( S × [0 , , Y ) ∼ = M ( Σ, Y ) . Let s : Σ −→ Σ ′ be a sewing that glues an incoming to an outgoing boundary component. Then for anylabel X of the remaining boundary components, M ( Σ, ( X, − , − )) provides a functor Z op M ⊗ Z M −→ Ch k (bythe explanations in Remark 3.1), i.e. an Z M -bimodule. Evaluation of M on s induces a map Z P ∈Z M L M ( Σ, ( X, P, P )) −→ M ( Σ ′ , X ) . (3.1)(Note that in [SW19] homotopy coends over differential graded categories are defined, but the above homotopycoend runs over a not necessarily unital differential graded category. In that case, the homotopy coend can stillbe defined as the realization of a semisimplicial object instead of a simplicial one.)We say that the symmetric monoidal functor M : X - Surf c −→ Ch k satisfies excision if (3.1) is an equivalence.13 efinition 3.2. For a set X , we call a symmetric monoidal functor M : X - Surf c −→ Ch k a modular functor(with values in chain complexes) if M satisfies excision. Remark 3.3.
As mentioned above, one can relax the definition in the sense that one requires this functor notto be strict, but just homotopy coherent. Technically, one would accomplish this by passing from C - Surf c toa resolution, see [Rie18] for the necessary techniques. For the concrete constructions in this article, however,this will not be necessary because the modular functor that we will build will be strict. Nonetheless, it willlead to non-strict actions on certain Hochschild complexes because these will be obtained by transfer along anequivalence.In practice, a modular functor is constructed from a certain linear category (with a lot more structure andproperties) that is recovered by evaluation of the modular functor on the cylinder leading to a notion of amodular functor for a given linear category . We will take this into account in our definitions as follows: Fora linear category A , we set A - Surf c := ( Proj A ) - Surf c , where ( Proj A ) is the set of objects of the subcategory Proj
A ⊂ A of projective objects.
Definition 3.4.
Let A be a linear category. We call a modular functor M : A - Surf c −→ Ch k a modular functorfor A if the cylinder category Z M of M is equivalent to Proj A . Remark 3.5 (Modular functors with values in vector spaces) . Replacing in the above Definitions 3.2 and 3.4the category of chain complexes with the category of vector spaces, we obtain the definition of an modularfunctor with values in vector spaces (we should say ‘a’ definition because other definitions might be used indifferent contexts; our definition is close to [Til98]). It is then clear that the zeroth homology of a modularfunctor with values in chain complexes is an modular functor with values in vector spaces.
Before stating our main result that a (not necessarily semisimple) modular category gives rise to a modularfunctor with values in chain complexes, we recall some terminology.
Ribbon categories, non-degeneracy and modularity. A ribbon structure (also called ribbon twist ) on a finitetensor category C with braiding c is a natural automorphism of the identity whose components θ X : X −→ X satisfy the conditions θ X ⊗ Y = c Y,X c X,Y ( θ X ⊗ θ Y ) ,θ I = id I ,θ X ∨ = θ ∨ X for all X, Y ∈ C . The first two conditions say precisely that the braided monoidal structure and the naturalautomorphism θ form an algebra over the framed little disk operad [SW03]; the last condition requires an ad-ditional compatibility with duality. A finite ribbon category is a braided finite tensor category with the choiceof a ribbon structure. Recall that any ribbon structure induces a pivotal structure. If we are given a braidedfinite tensor category C , we can define the M¨uger center as the full subcategory of C spanned by the transparentobjects, i.e. those objects X ∈ C such that c Y,X c X,Y = id X ⊗ Y for all Y ∈ C . We call the braiding (and then alsothe braided finite tensor category C ) non-degenerate if the M¨uger center is trivial, i.e. generated by the monoidalunit under finite direct sums. A modular category is a finite ribbon category with non-degenerate braiding.Note that, in our terminology, modularity does not include semisimplicity (but still finiteness assumptions).Semisimple modular categories are a standard object in quantum topology and known as the input datum forthe Reshetikhin-Turaev construction [RT90, RT91, Tur10]. For non-semisimple modular categories, there werefor a long time different definitions of the non-degeneracy of the braiding which were proven to be equivalent byShimizu [Shi19] only recently — the one given above in terms of the M¨uger center is one of possible definitionsof non-degeneracy.We may now state our main result: Theorem 3.6 (Main Theorem) . Any modular category C gives rise in a canonical way to a modular functor F C : C - Surf c −→ Ch k with values in chain complexes. This modular functor can be computed by the marked blocks from Section 2.2as follows: After the choice of a marking Γ on Σ , there is a canonical equivalence B Σ,Γ C ( X ) ≃ −−→ F C ( Σ, X ) (3.2) of chain complexes. H F C of F C to classical constructions will be explained in Remark 5.13. The remaining Sections 4and 5 of this article are devoted to the proof of Theorem 3.6. In the remainder of this section, we will discussimplications and additions to Theorem 3.6 and also examples.Since homotopy coherent actions transfer along equivalences, we obtain the following immediate consequenceof (3.2): Corollary 3.7.
Let C be a modular category and Σ an extended surface with marking Γ and projectiveboundary label X . Then the marked block B Σ,Γ C ( X ) comes canonically with a homotopy coherent projectiveaction of the mapping class group of Σ . Remark 3.8 (Framing anomaly) . For every modular category C , one can define an invertible scalar ζ ∈ k × ,the framing anomaly (or central charge ), see e.g. [FS17, Section 3.2]. A modular category whose central chargeis 1 ∈ k will be called anomaly-free . The framing anomaly controls the projectivity of the mapping class groupactions that are part of the vector space valued modular functor for C in the sense that the central generatorof a surface with genus g is sent to multiplication with ζ g . Hence, in the anomaly-free case, these mappingclass group actions will be linear and not only projective. The projectivity of the mapping class group actionsobtained from the modular functor with values in chain complexes from Theorem 3.6 is precisely the same. Thiswill explained in detail in Remark 5.12. Remark 3.9 (Non-projective boundary labels) . In the definition of C - Surf c , we only allow projective boundarylabels. The choice leads to simplifications in the presentation (it matches better with excision argumentsbecause our homotopy coends always run over subcategories of projectives), but we can define conformal blocksfor non-projective labels although this does not amount to a substantial addition (because our modular functorwith its present definition already contains all the necessary information): Let Σ be an extended surface. For any boundary label X , we denote by Q X the boundary label for Σ with differential graded objects in C thatresolves every outgoing label projectively and every ingoing label injectively (this replacement can be chosenfunctorially). Since C is self-injective (i.e. the projective objects are precisely the injective ones), Q X is degree-wise projective. Therefore, we can apply F C ( Σ, − ) degreewise and obtain the | X | + 1-fold complex F C ( Σ, Q X )whose totalization we define to be F C ( Σ, X ). In order to see that this does not depend on the choice of functorialresolution, let Σ | g be a surface of genus g ≥ g = 0 follows directly from [SW19, Lemma 3.8]) andwith one boundary component that is outgoing (more boundary components, some of them possibly incoming,can be treated analogously). Denote by Γ ◦ g, , the marking on Σ | g discussed in Example 2.6. There we foundin (2.8) a canonical equivalence B Σ | g ,Γ ◦ g, , C ( Y ) ≃ C I, Y ⊗ Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g , Y ∈ Proj C . (3.3)In order to prove independence of F C ( Σ | g , Q X ) of the chosen resolution, it suffices by naturality of the maps(3.2) to observe that the chain complex valued functor C (cid:18) I, − ⊗ (cid:16)R P ∈ Proj C f L P ⊗ P ∨ (cid:17) ⊗ g (cid:19) preserves equivalencesbetween non-negatively graded complexes of projective objects in C . But this follows from the exactness of themonoidal product and [SW19, Lemma 3.8]. By functoriality of F C in boundary labels (Remark 3.1) the complexeswill again carry action of the mapping class group of Σ . It should also be noted that from (3.3), (3.2) and[SW19, Lemma 3.8] it follows that F C ( Σ | g , I ) ≃ F C ( Σ g ) , where Σ g = Σ | g is the closed surface of genus g . The marked block for the closed torus and a specific marking is given by the Hochschild complex (Example 2.2),and it was shown already in [SW19] that the Hochschild complex of a modular category carries a homotopycoherent SL(2 , Z )-action, see [LMSS18] for a related Hopf algebraic result on (co)homology level. The fullyfledged modular functor of a modular category provides now for us complexes that we may interpret as highergenus analoga of the Hochschild complex together with (homotopy coherent) projective mapping class groupactions on them. 15 imodules and the Agreement Principle. For a linear category C , a C -bimodule is a functor M : C op ⊗ C −→ Vect k . We define the Hochschild complex of C with coefficients in this bimodule as the homotopy coend CH ( C ; M ) := R X ∈ Proj C L M ( X, X ). This definition is made in a way such that the following holds: If C = Mod k A is given as the category of finite-dimensional modules over a finite-dimensional algebra A , then by the AgreementPrinciple , that goes back to McCarthy and Keller [MCar94, Kel99] and is stated in a modified form in [SW19,Theorem 2.9], the canonical embedding ι A : ⋆//A op −→ Proj k A = Proj C induces an equivalence CH ( A ; M ( A, A )) ≃ −−→ CH ( C ; M ) (3.4)of chain complexes, where CH ( A ; M ( A, A )) are the ‘ordinary’ Hochschild chains of A with coefficients in thebimodule M ( A, A ).For any finite tensor category C , there is a canonical family ( M g ) g ≥ of C -bimodules defined by M g ( X, Y ) = C ( X, Y ⊗ F ⊗ g ) , X, Y ∈ C where F = R X ∈C X ⊗ X ∨ is the canonical coend. Theorem 3.10.
For any modular category C and g ≥ , the Hochschild chains CH ( C ; M g ) with coefficients inthe bimodule M g carry a canonical homotopy coherent projective action of the mapping class group Map ( Σ g +1 ) of the closed surface of genus g + 1 . The proof strategy was already outlined in the introduction: We realize that CH ( C ; M g ) is the marked blockfor a Σ g +1 and a specific marking and then apply our main result. Proof.
Denote by Γ ◦ g, , the marking from Example 2.6 on the surface Σ | g with genus g and one incomingboundary component labeled by X ∈ Proj C and one outgoing boundary component labeled by Y ∈ Proj C . By(2.7) we have a canonical equivalence B Σ | g ,Γ ◦ g, , C ( X, Y ) ≃ C X, Y ⊗ Z P ∈ Proj C f L P ⊗ P ∨ ! ⊗ g . Since X is projective, we arrive at B Σ | g ,Γ ◦ g, , C ( X, Y ) ≃ C ( X, Y ⊗ F ⊗ g ) = M g ( X, Y ). It is now a direct consequenceof excision for marked surfaces (Proposition 2.3) that B Σ | g +1 ,Γ ◦ g +1 , , C ≃ CH ( C ; M g ). Now the statement followsfrom Corollary 3.7.Theorem 3.10 can be used to obtain homotopy coherent projective mapping class group actions on theHochschild complexes of ribbon factorizable Hopf algebras with coefficients in specific bimodules: For a rib-bon factorizable Hopf algebra A , the category Mod k A is modular, see e.g. [LMSS20, Section 2.3], and theLyubashenko coend F = R X ∈ Mod k A X ⊗ X ∨ in Mod k A is isomorphic to the A -module A ∗ coadj given by the dual A ∗ of A and the so-called coadjoint action A ⊗ A ∗ −→ A ∗ , a ⊗ α (cid:0) b α (cid:0) S ( a (1) ) ba (2) (cid:1)(cid:1) , see [KL01, Theorem 7.4.13], where S is the antipode of A , and we have used the Sweedler notation ∆ a = a (1) ⊗ a (2) for the coproduct ∆ of A . Now for g ≥
0, we can consider the A -module A ⊗ (cid:16) A ∗ coadj (cid:17) ⊗ g thatis defined using the monoidal structure on Mod k A . By right multiplication on the A -factor this becomes an A -bimodule. We obtain M g ( A, A ) = Hom A (cid:16) A, A ⊗ (cid:0) A ∗ coadj (cid:1) ⊗ g (cid:17) = A ⊗ (cid:0) A ∗ coadj (cid:1) ⊗ g , where A ⊗ (cid:16) A ∗ coadj (cid:17) ⊗ g has to be seen as A -module in the middle of the equation and as A -bimodule on the righthand side. The Agreement Principle (3.4) provides us with a canonical equivalence CH (cid:16) A ; A ⊗ (cid:0) A ∗ coadj (cid:1) ⊗ g (cid:17) ≃ −−→ CH ( Mod k A ; M g ) . This leads to the following Hopf algebraic version of Theorem 3.10:
Corollary 3.11.
For any ribbon factorizable Hopf algebra A and g ≥ , the Hochschild chains of A withcoefficients in the A -bimodule A ⊗ (cid:16) A ∗ coadj (cid:17) ⊗ g come with a canonical homotopy coherent projective action ofthe mapping class group of the closed surface of genus g + 1 . emark 3.12. For a specific marking, the homology of the marked block of a modular category C can becomputed rather explicitly. We will focus on a closed surface Σ g of genus g ≥
0. In the presence of boundaries,we may use Proposition 2.7. First assume g ≥ g = 0 in a moment). Since anymodular category is unimodular [ENO04, Proposition 4.5], the equivalence (2.8) gives us a canonical isomorphism H ∗ B Σ g ,Γ ◦ g C ∼ = Ext (cid:0) F ⊗ g , I (cid:1) ∗ ∼ = Ext (cid:0) I, F ⊗ g (cid:1) ∗ . (3.5)The second isomorphism uses the self-duality F ∨ ∼ = F of the canonical coend of a modular category comingfrom the non-degenerate Hopf pairing, see [Shi19]. With the convention F ⊗ = I , the statement remains truefor g = 0 for an appropriate marking (we need to cut the sphere into two disks along its equator).Of course, there is now also an isomorphism H ∗ F C ( Σ g ) ∼ = Ext ( I, F ⊗ g ) ∗ , but this is not canonical. Under(3.5), the projective action of the mapping class group of Σ g on H ∗ B Σ g ,Γ ◦ g C corresponds to the one constructedin [LMSS20] on Ext( I, F ⊗ g ). We will, however, not spell out the details of this comparison here. Example 3.13 (Drinfeld doubles in finite characteristic and the Dijkgraaf-Witten modular functor) . For afinite group G , the Drinfeld double D ( G ) is a ribbon factorizable Hopf algebra with underlying vector space k ( G ) ⊗ k [ G ], where k ( G ) is the space of k -valued functions and k [ G ] is the group algebra of G . The multiplicationin D ( G ) is given by ( δ a ⊗ b )( δ c ⊗ d ) = δ a δ bcb − ⊗ bd for all a, b, c, d ∈ G , where δ a is the k -valued function on G supported in a ∈ G with value 1 ∈ k ; we refer to [Kas95, Chapter IX]for details. The category Mod k D ( G ) of finite-dimensional D ( G )-modules is modular, and it is non-semisimpleif and only if | G | divides the characteristic of k .Whenever Mod k D ( G ) is semisimple, it gives rise to a topological field theory [FQ93, Mor15] that is partlybased on [DW90] and therefore often called Dijkgraaf-Witten theory . Therefore, we call the modular functor F Mod k D ( G ) with values in chain complexes the Dijkgraaf-Witten modular functor .In this example, we prove that the evaluation of F Mod k D ( G ) on a closed surface Σ is explicitly given by F Mod k D ( G ) ( Σ ) ≃ N ∗ ( PBun G ( Σ ); k ) , (3.6)where N ∗ ( PBun G ( Σ ); k ) are the (normalized) k -chains on the groupoid PBun G ( Σ ) of G -bundles over Σ (for theclosed torus, this is [SW19, Proposition 3.3]). Under the equivalence, the mapping class group action on theleft hand side corresponds to the obvious action on the right hand side. Similar statements hold for surfaceswith boundary.An in-depth discussion of F Mod k D ( G ) is beyond the scope of this article, and we will only sketch the proofof (3.6): First recall the equivalence Mod k D ( G ) ≃ Mod k G//G as linear categories, where
Mod k G//G is thecategory of finite-dimensional modules over the loop groupoid
G//G of G , i.e. the action groupoid of G actingon itself by conjugation (there is naturally a ribbon structure on Mod k G//G such that this equivalence is anequivalence of ribbon categories). Also recall
G//G ≃ PBun G ( S ). If Σ is an extended surface with projectiveboundary label X in Mod k D ( G ), then we may see Σ : S −→ S as a bordism from the incoming boundary S to the outgoing boundary S . Thanks to Mod k D ( G ) ≃ Mod k PBun G ( S ), the label X gives rise to projectiveobjects X j ∈ Mod k PBun G ( S j ) for j = 0 , PBun G ( S ) PBun G ( Σ ) PBun G ( S ) r r that induces the pullback functors Ch PBun G ( S ) k Ch PBun G ( Σ ) k Ch PBun G ( S ) k , r ∗ r ∗ where Ch PBun G ( M ) k denotes the category of functors from the groupoid PBun G ( M ) of G -bundles over a manifold M to Ch k . We can now define the derived pull push functor Z L G ( Σ ) := L r r ∗ : Ch PBun G ( S ) k −→ Ch PBun G ( S ) k , (3.7)where L r is the homotopy left Kan extension along r . We also define the auxiliary complexes G ( Σ, X ) := h X ∨ , Z L G ( Σ ) X ∨ i , where h− , −i denotes the morphism spaces in Mod k PBun G ( S ) understood degreewise. Whenever Σ is closed,we find G ( Σ ) = N ∗ ( PBun G ( Σ ); k ) by definition. Hence, in order to prove (3.6), it remains to prove that G isactually equivalent to the modular functor F Mod k D ( G ) . To this end, we need the following statements:17 For composable bordisms Σ : S −→ S and Σ ′ : S −→ S , the pull push maps (3.7) satisfy Z L G ( Σ ′ ◦ Σ ) ≃ Z L G ( Σ ′ ) Z L G ( Σ ) as follows from a straightforward analogue of the Beck-Chevalley property for pull pushmaps [Mor15, SW20] to the derived setting. This implies that G satisfies excision for the gluing of disjoint surfaces (we do not make a statement about self-sewing). • On a connected surface of genus zero, G agrees with the modular functor, i.e. it is given by the morphismspaces of Mod k D ( G ) ≃ Mod k G//G and the monoidal product as in (2.1). This fact is proven by anexplicit computation of the corresponding pull push map (3.7): We write the homotopy left Kan extensionas a homotopy colimit over the homotopy fiber of r . Since the homotopy fiber of any restriction functor PBun G ( Σ ) −→ PBun G ( S ) from a connected surface Σ to a non-empty collection S of boundary componentshas discrete homotopy fibers (this follows from the holonomy classification of bundles and the long exactsequence for homotopy groups), the homotopy fibers of r are also discrete, and all homotopy colimitsneeded for the homotopy left Kan extension are just coproducts of vector spaces. This allows us to verifythe claim directly.From these two statements, we can already conclude G ( Σ ) ≃ F Mod k D ( G ) ( Σ ) for every closed surface and therebydeduce (3.6). The proof that, under the equivalence, the Map ( Σ )-action on F Mod k D ( G ) ( Σ ) corresponds to thetopological one on N ∗ ( PBun G ( Σ ); k ) is very similar to the vector space valued case. We will not discuss thedetails here. In Section 2 we have defined chain complexes of vector spaces for a surface and the auxiliary datum of amarking. In order to prove the main result, we will have to understand how these quantities depend on themarking. Before investigating this point in the next section, we need to understand how different markings ona surface are related. To this end, we will use and extend work of Bakalov and Kirillov [BK00] who, based on[HT80, Har83], define a groupoid of markings on a surface and prove that this groupoid is connected and simplyconnected. Some key ideas in [BK00] and also the name
Lego-Teichm¨uller game go back to Grothendieck’sresearch proposal
Esquisse d’un Programme [Gro84]. For our purposes, we will need to replace the contractiblegroupoid of markings on a given extended surface by a contractible ∞ -groupoid. This replacement was motivatedin the introduction. Its significance will become clear in the next section. For an extended surface Σ , we denote by M ( Σ ) the groupoid of markings on Σ [BK00]. Its objects are markingson Σ (see Section 2.1). The morphisms of M ( Σ ) are given in terms of four generators (called moves ): • Z (cyclic rotation of the marking), • F (cut deletion), • B (braiding), • S (passing to transversal cut on a torus with one hole).These four moves are subject to a list of relations and generate M ( Σ ) under sewing. We refer to [BK00, Section 4]for the detailed definition of the moves and their relations. Remark 4.1.
Instead of arbitrary markings, one can restrict to fine markings (for a fine marking, the underlyingcut system contains spheres with at least one and at most three holes). This case is also treated in [BK00] andturns out to be not substantially different. A full list of the generators and relations for the groupoid of finemarkings is given in [FS17, Section 2.2].Building on the results of [HT80, Har83], Bakalov and Kirillov prove the following fundamental result aboutthe groupoids of markings:
Theorem 4.2 ([BK00, Theorem 4.24]) . For any extended surface Σ , the groupoid M ( Σ ) of markings on Σ isconnected and simply-connected. We call a category A contractible if the topological space | N A| obtained by geometric realization of its nerve N A is contractible, i.e. equivalent to a point. Put differently, in this case, the ∞ -groupoid obtained by local-izing (in the sense of ∞ -categories) A at all morphisms is a contractible Kan complex. Hence, a groupoid iscontractible if and and only if it is connected and simply connected. Therefore, Theorem 4.2 can be rephrasedby saying that the groupoids of markings are contractible.18or an extended surface Σ , Bakalov and Kirillov also define a groupoid C ( Σ ) of cut systems on Σ [BK00,Section 7.1–7.3]. Its objects are cut systems on Σ (as defined in Section 2.1), and its morphisms are generatedby the moves ¯F and ¯S (corresponding to the moves F and S listed above) which are explained in Figure 4. Thesetwo moves are subject to five relations. . . . . . .. . . . . .. . . . . . ¯F¯S Figure The ¯ F-move can be applied to a cylindrical region with a cut and deletes this cut — provided, of course, thatthis still leaves us with a cut system. The dots are supposed to symbolize that the surface may continue to theleft and the right of the displayed region. The ¯ S-move can be applied to a region of the shape of torus with onehole and one cut. It replaces the one cut with a transversal one. The marked points on the boundary and theorientation of cuts and boundaries are suppressed in the picture. For the ¯ F-move, the cut that is being deletedcan have any orientation and any position of the marked point. For the ¯ S-move, the cut that is being replacedby a transversal one as well as its replacement can have arbitrary orientation and position of the marked point.The other cut (or boundary component if the surface ends there) has some arbitrary orientation and positionof the marked point that is not changed by the ¯ S-move.
In addition to Theorem 4.2, we will also need the following contractibility result (that the proof of Theorem 4.2cited above actually relies on):
Theorem 4.3 ([BK00, Theorem 7.9]) . For any extended surface Σ , the groupoid C ( Σ ) of cut systems on Σ isconnected and simply-connected. Let φ : Σ −→ Σ ′ be a mapping class, then φ sends cut systems on Σ to cut systems on Σ ′ and alsomoves between cut systems on Σ to moves between cut systems on Σ ′ (because moves are only defined based onincidences). This way, φ yields a functor C ( φ ) : C ( Σ ) −→ C ( Σ ′ ) between groupoids which by Theorem 4.3 is evendetermined by its object function. Similarly, any sewing s : Σ −→ Σ ′ yields a functor C ( s ) : C ( Σ ) −→ C ( Σ ′ )which just regards any pair oppositely oriented gluing boundaries of Σ as a cut in Σ ′ . The functors assigned tomapping classes and sewings respect the relations holding in Surf . These considerations carry over from cuts tomarkings (for the action of mapping classes on the charts underlying the marking, one precomposes the chartwith the inverse of the mapping class).
Proposition 4.4.
Cut systems and markings on extended surfaces naturally form symmetric monoidal functors C : Surf −→ Grpd , M : Surf −→ Grpd , where the monoidal product on Surf is disjoint union and the monoidal product on
Grpd is the Cartesian product.
As explained in [BK00, Section 7.4], there is a projection functor π Σ : M ( Σ ) −→ C ( Σ ) (4.1)sending a marking to its underlying cut system. The moves B and Z are sent to identities while F and S aresent to ¯F and ¯S, respectively.By means of this functor, we can see that the markings over a fixed extended surface form a category fiberedin groupoids over the cut systems on that surface. To this end, let us first recall the relevant notions from[Hol08, Definition 3.1] (this definition is based on [DM69]): A functor E : A −→ B is called a category fiberedin groupoids (or said to exhibit A as a category fibered in groupoids over B )191) if all lifting problems of the form 0 N A ∆ N B NE can be solved(2) and if for any diagram a f ←−− b g −−→ c in A and any morphism h : E ( a ) −→ E ( c ) in B making the diagram E ( c ) E ( a ) E ( b ) hE ( f ) E ( g ) commute, there is a unique h ′ : a −→ c with E ( h ′ ) = h .(Note that point (2) uses conventions dual to those in [Hol08, Definition 3.1] because the categories fibered ingroupoids needed in this article correspond to category-valued cosheaves instead of sheaves as in [Hol08].) Lemma 4.5.
For any extended surface Σ , the canonical functor π Σ : M ( Σ ) −→ C ( Σ ) exhibits M ( Σ ) as acategory fibered in groupoids over C ( Σ ) .Proof. It follows from the definition of M ( Σ ) that the lifting problem0 N M ( Σ )∆ N C ( Σ ) . Γ Nπ Σ µ e µ can be solved if µ is one of the moves ¯F or ¯S (because by definition these lift to F and S, respectively, forany given start value). From this, we deduce that the lifting problem can be solved when µ is an arbitrarymorphism in C ( Σ ), which amounts to property (1) above. Contractibility of the groupoids M ( Σ ) and C ( Σ )gives us property (2).For a cut system C on Σ , denote by m Σ ( C ) := π − Σ ( C ) (4.2)the fiber of the projection functor π Σ from (4.1) over C . From Theorem 4.2 and 4.3 we can deduce theequivalence m Σ ( C ) ≃ ⋆ (4.3)of categories.In the sequel, it will be important to reconstruct M ( Σ ) from the fibers (4.2). To this end, we will use the Grothendieck construction , a classical construction in category theory, see e.g. [MM92, Section I.5], that we willalso use for various other constructions later: For a functor F : A −→
Cat from a category A to the categoryof categories, its Grothendieck construction R F is defined as the category of pairs ( a, x ), where a ∈ A and x ∈ F ( a ). A morphism ( a, x ) −→ ( a ′ , x ′ ) is a pair ( f, α ) of a morphism f : a −→ a ′ in A and a morphism α : F ( f ) x −→ x ′ in F ( a ′ ). The category R F comes with a natural functor R F −→ A sending an object( a, x ) ∈ R F to a and a morphism ( f, α ) : ( a, x ) −→ ( a ′ , x ′ ) in R F to f . For later purposes, we record the usefulfact that the Grothendieck construction R F of a functor F : A −→
Cat can be described as the lax colimit ofthe functor F , i.e. Z F = laxcolim a ∈A F ( a ) ; (4.4)this statement can be found e.g. in [JY20, Theorem 10.2.3] or in [HGN17] within a more general framework.One should think of R F as the result of gluing together in a categorical way the categories F ( a ) for a ∈ A according to a prescription determined by A . 20ow from Lemma 4.5 and [Hol08, Theorem 1.2] (or rather the dual version) it follows that the fibers (4.2)form a (pseudo-)functor m Σ : C ( Σ ) −→ Grpd (4.5)whose Grothendieck construction R m Σ comes with a canonical equivalence Z m Σ ≃ −−→ M ( Σ ) (4.6)induced by the inclusions m Σ ( C ) −→ M ( Σ ). This leads to a key observation that we will need later: We canwrite the groupoid of markings on a fixed surface as the result of categorically gluing together the markingsover varying cut systems, where the groupoid of cuts systems gives us the gluing prescription. We will now introduce a new category of cut systems and markings on a fixed extended surface whose objectsare cut systems and markings, respectively, equipped with additional data, namely a coloring. The usefulnessof this definition will become apparent in the next section, where it will allow us to relate marked blocks fordifferent markings.For a connected extended surface Σ with n boundary components, a colored cut system U on Σ is a pair C of a cut system on Σ and a subset S of the set of cuts of C such that | S | + n ≥ . (4.7)A cut that lies in the distinguished subset S of all the cuts will be referred to as a colored cut . A cut which isnot colored will be referred to as uncolored cut . A colored cut system on a non-connected extended surface isdefined as a colored cut system on every connected component.The colored cut systems on an extended surface Σ form a category b C ( Σ ) in the following way: Objects arecolored cut systems on Σ . The morphisms are generated by two types of moves:(U) For any colored cut systems U = ( C, S ) and any proper subset S ′ ⊂ S such that ( C, S ′ ) is still a coloredcut system (meaning that (4.7) must be satisfied), there is a non-invertible morphism U = ( C, S ) S ′ ⊂ S −−−−−→ ( C, S ′ ) , called uncoloring . In other words, there is a morphism that implements forgetting a coloring of subfamilyof cuts if enough colored cuts are left to ensure that requirement (4.7) is met.(AM) Between colored cut systems, we have admissible moves , i.e. a move between the underlying cut systems,as defined for C ( Σ ), such that this move does not affect the colored cuts. A more formal definition of anadmissible move may be given as follows: For a colored cut system U on Σ with underlying cut system C ,denote by Σ U the surface obtained from Σ by cutting at all colored cuts. Then C induces a cut system C U on Σ U . By Proposition 4.4 the re-sewing s U : Σ U −→ Σ gives rise to a functor C ( s U ) : C ( Σ U ) −→ C ( Σ ) (4.8)sending C U to C . With this notation, we define an admissible move U −→ V between colored cut systems U and V on Σ as follows: Such an admissible move only exists when Σ U = Σ V , and in that case, it isdefined as a move µ : C −→ C ′ between the underlying cut systems that is the image of a move C U −→ C V of cut systems on Σ U = Σ V under (4.8).We impose the following relations:(RU) Suppose U = ( C, S ) is a colored cut system and S ′′ ⊂ S ′ ⊂ S proper inclusions of subsets such that ( Γ, S ′ )and ( Γ, S ′′ ) are still colored cut systems, then the composition U = ( Γ, S ) S ′ ⊂ S −−−−−→ ( Γ, S ′ ) S ′′ ⊂ S ′ −−−−−→ ( Γ, S ′′ )of uncolorings corresponding to S ′′ ⊂ S ′ and S ′ ⊂ S , respectively, is equal to the uncoloring U = ( Γ, S ) S ′′ ⊂ S −−−−−→ ( Γ, S ′′ )corresponding to S ′′ ⊂ S . 21RM) For the composition of admissible moves, the relations for the underlying moves that hold in the groupoid C ( Σ ) of cut systems are inherited.(C) Uncolorings and admissible moves commute in the obvious way. Definition 4.6.
For an extended surface Σ , we call the category b C ( Σ ) the category of colored cut systems Σ .The category b C ( Σ ) comes with a canonical functor Q Σ : b C ( Σ ) −→ C ( Σ ) (4.9)which forgets the coloring, sends uncolorings to identities and admissible moves to the underlying moves. Notethat (RM) ensures that this defines really a functor.We can generalize Proposition 4.4, in which we stated that cut systems can be functorially assigned tosurfaces, to colored cuts: Indeed, a morphism f : Σ −→ Σ ′ induces a functor b C ( f ) : b C ( Σ ) −→ b C ( Σ ′ ) whichsatisfies Q Σ ′ b C ( f ) = C ( f ) Q Σ and sends colored cuts to colored cuts and uncolored cuts to uncolored cuts. Forthe definition of b C ( s ) : b C ( Σ ) −→ b C ( Σ ′ ) for a sewing s : Σ −→ Σ ′ , we additionally have to prescribe thatboundary components which are glued to together through the sewing s give rise to a colored cut (assigning an uncolored cut instead would generally violate (4.7)). Proposition 4.7.
Colored cut systems on extended surfaces naturally form a symmetric monoidal functor b C : Surf −→ Cat . The important fact about the categories of colored cut systems is that they are still contractible despite thenewly introduced uncoloring morphisms:
Theorem 4.8.
For any extended surface Σ , the category b C ( Σ ) of colored cut systems on Σ is contractible.Proof. (i) We consider the subcategory U ( Σ ) ⊂ b C ( Σ ) with the same objects, but whose morphisms aregenerated only by uncolorings and define L ( Σ ) as the ∞ -categorical localization of the ∞ -category N b C ( Σ )at U ( Σ ). Now the canonical map N b C ( Σ ) −→ L ( Σ ) induces an equivalence | N b C ( Σ ) | ≃ −−→ | L ( Σ ) | . Moreover, L ( Σ ) is a Kan complex, i.e. an ∞ -groupoid since U ( Σ ) contains all the non-invertible morphismsin b C ( Σ ). For concreteness, we choose in this proof the Dwyer-Kan model [DK80] for localization.(ii) Since the functor Q Σ from (4.9) sends all morphisms in U ( Σ ) to identities, it induces a simplicial map ω Σ : L ( Σ ) −→ N C ( Σ ). In this step, we prove that ω Σ is a Kan fibration, i.e. ω Σ allows for solutions tothe lifting problems Λ nk L ( Σ )∆ n N C ( Σ ) , ι ξ ω Σ σ e σ (4.10)where ι : Λ nk −→ ∆ n for n ≥ ≤ k ≤ n are the horn the inclusions. • For n = 0, this just means that ω Σ is surjective on 0-simplices, which we can easily observe to betrue. • For n = 1, we need to solve the lifting problem0 L ( Σ )∆ N C ( Σ ) ι U ω Σ µ e µ (4.11)(the lifting problem for the inclusion 1 −→ ∆ can be solved by passing to inverse morphisms). Asone easily sees, it suffices to prove this lifting property if µ is a move, i.e. ¯F or ¯S, or an inverse of thesemoves. We will now prove that such a lift indeed exists (we illustrate the strategy by means of anexample in Figure 5): Let µ : C −→ C ′ be a move in C ( Σ ). On each connected component of Σ , thecut system C has a boundary component or a cut that will not be affected by µ in the sense of (AM)22n page 21 (this can be observed for both ¯F and ¯S). This implies that for any U with Q Σ ( U ) = C ,we find a colored cut system V connected to U by a zigzag of uncolorings (generally, the uncoloringsthemselves will not be enough; zigzags of them are needed) such that µ does not affect the coloredcuts of V . As a consequence, µ induces an admissible move b µ : V −→ W . In particular, Q Σ ( b µ ) = µ .Then the zigzag U ←→ V b µ −−→ W in b C ( Σ ) is mapped to µ under Q Σ because Q Σ sends uncolorings to identities, and it gives us a1-morphism e µ : U −→ W in L ( Σ ) with ω Σ ( e µ ) = µ . This shows that the e µ solves the lifting problem.The same argument applies to the inverses of moves. u ′ u ′′ ¯S Figure Example for the construction of the lifts needed in (4.11) in the case of the ¯ S-move. The ¯ S-move can be appliedto a subsurface of the shape of a torus with one boundary component and one cut. We will assume that thesurface actually continues at this boundary component, so it will play the role of a cut (if the surface endsthere, the situation would be easier as we will explain after covering the present situation). Suppose now wewant to lift ¯ S to L ( Σ ) with the start value given by the first picture (colored cuts are drawn in red; orientationof cuts and marked points are suppressed as in Figure 4). If we start with this colored cut system, then ¯ Swill not induce an admissible move because the cut that is replaced with the transversal cut by the ¯ S-move iscolored. But by a zigzag of uncolorings (denoted by u ′ and u ′′ ) we arrive at the third colored cut system fromthe left. Now ¯ S will induce an admissible move. This way, we obtain the desired lift of ¯ S. As just mentioned,if the hole of the torus belongs to a boundary component, the situation simplifies: We forget all colors (whichis allowed thanks to the presence of a boundary component) and lift ¯ S directly. • In order to solve the lifting problem (4.10) for n ≥
2, we first observe that the horn ξ : Λ nk −→ L ( Σ )admits a filler e σ : ∆ n −→ L ( Σ ), i.e. e σι = ξ , because L ( Σ ) is a Kan complex by (i). Moreover, ω Σ e σι = ω Σ ξ = σι . In other words, both ω Σ e σ and σ fill the horn ω Σ ξ : Λ nk −→ N C ( Σ ) in the nerve of the 1-groupoid C ( Σ ). Hence, they are equal, and we can conclude that the lifting problem (4.10) can be solved for n ≥ ω Σ is a Kan fibration.(iii) The fiber of ω Σ : L ( Σ ) −→ N C ( Σ ) over C ∈ C ( Σ ) is given by the ∞ -localization of the nerve N Q − Σ ( C )of the fiber Q − Σ ( C ) of Q Σ : b C ( Σ ) −→ C ( Σ ) over C at all uncolorings in that fiber (this follows from thefact that the localization that led from b C ( Σ ) to L ( Σ ) just happens in the fibers of Q Σ ). Therefore, weconclude | ω − Σ ( C ) | ≃ | N Q − Σ ( Σ ) | . But Q − Σ ( C ) has an initial object, namely the one obtained by coloring all cuts of C . As a result, Q − Σ ( C ) is a contractible category. This implies | ω − Σ ( C ) | ≃ ⋆ . Combining thiswith (ii) and the long exact sequence for homotopy groups, we conclude that | ω Σ | : | L ( Σ ) | ≃ −−→ | N C ( Σ ) | is an equivalence.From (i) and (iii) we obtain | N b C ( Σ ) | ≃ | N C ( Σ ) | . Now the Theorem follows from Theorem 4.3 that asserted that C ( Σ ) is contractible.In this subsection we have, so far, replaced the groupoid of cut systems of an extended surface with a categoryof colored cut systems and proven that the latter category is still contractible. It remains to replace the groupoidof markings with a colored analogue. To this end, we will use the presentation (4.6) of M ( Σ ) as Grothendieckconstruction: Definition 4.9.
For an extended surface Σ , we define the category b M ( Σ ) of colored markings on Σ as theGrothendieck construction b M ( Σ ) := Z (cid:16)b C ( Σ ) Q Σ −−−→ C ( Σ ) m Σ −−−→ Grpd (cid:17) , where Q Σ and m Σ appeared in (4.9) and (4.5), respectively.23roposition 4.7 carries over to colored markings: Proposition 4.10.
Colored markings on extended surfaces naturally form a symmetric monoidal functor b M : Surf −→ Cat . The contractibility of b M ( Σ ) will follow directly from Theorem 4.8 and Thomason’s Theorem [Tho79, Theo-rem 1.2] that states that for a functor F : A −→
Cat , the natural maphocolim a ∈A N F ( a ) ≃ −−→ N Z F (4.12)is an equivalence. Theorem 4.11.
For an extended surface Σ , the category b M ( Σ ) of colored markings on Σ is contractible.Proof. By Thomason’s Theorem (4.12) we obtain N b M ( Σ ) = N Z (cid:16)b C ( Σ ) Q Σ −−−→ C ( Σ ) m Σ −−−→ Grpd (cid:17) ≃ hocolim U ∈ b C ( Σ ) N m Σ Q Σ ( U ) . (4.13)It follows from (4.3) that N m Σ Q Σ ( U ) is equivalent to a point. As a consequence, the right hand side of (4.13)is equivalent to the homotopy colimit of the constant diagram with value ⋆ over b C ( Σ ), but the latter is givenby N b C ( Σ ) which have already proven to be contractible in Theorem 4.8. The ∞ -groupoid of colored markings. The contractibility result from Theorem 4.11 is a substantial part ofthe effort needed for the construction of the modular functor in the next section. Phrased differently, it tellsus that the ∞ -groupoid K ( Σ ) obtained by localizing the category b M ( Σ ) of colored markings on an extendedsurface at all uncolorings is contractible. Having defined a category of colored markings on an extended surface, we may now finally construct the modularfunctor with values in chain complexes. To this end, recall that we have defined in Section 2.2 marked blocksthat do not only depend on the surface and the boundary label, but also on the auxiliary datum of a marking . Inthe vector space valued situation, the standard procedure is to extend the definition to morphisms of markings,i.e. to construct a functor out of M ( Σ ) for each surface Σ . This amounts to relating the structure that ispresent on a modular category with the moves between different markings. Unfortunately, for our differentialgraded marked blocks, this does not seem to be possible directly. The key problem is that Lyubashenko’sS-transformation that is used for the definition of vector space valued marked blocks on the S-move does notseem to generalize directly to differential graded marked blocks. Similar, albeit less severe problems exist forthe F-move.Our categories b M ( Σ ) of colored markings precisely solve this problem: We show that the marked blocksextend to functors b M ( Σ ) −→ Ch k on categories of colored markings (Theorem 5.4) such that all uncolorings aresent to equivalences, i.e. the functors descend to the ∞ -localization K ( Σ ) of b M ( Σ ) at all uncolorings. The ideafor the definition of the functor b M ( Σ ) −→ Ch k is to work with a version of marked blocks which are glued viahomotopy coends at colored cuts and via ordinary coends at uncolored ones. We prove in Proposition 5.1 thatthis is equivalent to the marked blocks we had originally defined in Section 2.2. Having established this ‘mixed’definition of a marked block for any colored marking, we will, roughly, apply the moves F and S only to thoseparts of the marked block which coincide with the classical marked block with values in vector spaces using the‘classical definitions’ there. All the rest of the information is contained in the uncoloring maps. The details ofthis construction will be discussed in the proof of Theorem 5.4.The functors b M ( Σ ) −→ Ch k descend to the category obtained by gluing all categories of colored markings fordifferent surfaces together via the Grothendieck construction (Proposition 5.5). The final remaining step in theconstruction of the modular functor will then be a homotopy left Kan extension (Section 5.2).For presentation purposes, we will first treat anomaly-free modular categories and then comment on theanomalous case in Remark 5.12. This makes sense because the projectivity of the mapping class group actionsin the anomalous case will be of the same type for the linear and differential graded setting.24 .1 Extension of the definition marked blocks to colored markings For a given pivotal finite tensor category C , let X be a family of projective boundary labels for an extendedsurface Σ ; of course, we also allow the case that Σ is closed. If Γ is a marking on Σ , then we have defined inSection 2.2 a marked block B Σ,Γ C ( X ) depending on Γ . This chain complex was defined via an iterated homotopycoend over Proj C with one homotopy coend for each cut in Γ .Let now Λ be a colored marking on Σ . By Definition 4.9 a colored marking Λ is a pair ( U, Γ ) of a coloredcut system U and a marking Γ which both have the same underlying cut system. In other words, Λ arisesfrom Γ by declaring some cuts to be colored cuts in a way prescribed by U . Recall that on each connectedcomponent of Σ , the number of colored cuts plus the number of boundary components has to be at least one;this is required by (4.7).We now define F m C ( Σ, X, Λ ) to be the chain complex that we obtain from B Σ,Γ C ( X ) by replacing all homotopycoends corresponding to uncolored cuts by ordinary coends while the homotopy coends corresponding to thecolored cuts remain unaffected.In formulae, this is expressed as follows: Let Σ Λ be the extended surface obtained by cutting Σ along allcolored cuts of Λ . Then the marking Γ underlying Λ gives rise to a marking Γ Λ on Σ Λ (we are recalling herenotation already established on page 21). If Λ has q colored cuts, we arrive at F m C ( Σ, X, Λ ) ∼ = Z P ,...,P q ∈ Proj C L b Σ Λ ,Γ Λ C ( X, P , P , . . . , P q , P q ) . (5.1)This equality just expresses in formulae the definition of F m C ( Σ, X, Λ ) that was just given in words. The onlynon-trivial fact used here is that replacing homotopy coends by ordinary coends leads to vector space valuedmarked blocks b C (a fact that was explained in Section 2.5). In (5.1) we see the ‘mixed’ definition of markedblocks mentioned in the introduction of this section made precise: Homotopy coends are used for gluing atcolored cuts, ordinary coends at uncolored cuts. This allows us to express parts of this chain complex by meansof the vector spaces b C thanks to the results of Section 2.5. Proposition 5.1.
Let C be a pivotal finite tensor category and X a projective boundary label for an extendedsurface Σ with colored marking Λ with underlying marking Γ . Then the canonical map from homotopy coendsto ordinary coends, applied to all uncolored cuts of Γ , induces a trivial fibration ε Λ : B Σ,Γ C ( X ) ≃ −−→ F m C ( Σ, X, Λ ) . (5.2) Proof.
Using (5.1) and the symbols introduced there, the map in question is the map Z P ,...,P q ∈ Proj C L B Σ Λ ,Γ Λ C ( X, P , P , . . . , P q , P q ) −→ Z P ,...,P q ∈ Proj C L b Σ Λ ,Γ Λ C ( X, P , P , . . . , P q , P q )induced by the augmentation fibration B Σ Λ ,Γ Λ C ( X, P , P , . . . , P q , P q ) −→ b Σ Λ ,Γ Λ C ( X, P , P , . . . , P q , P q ) fromProposition 2.7 which is a trivial fibration because the definition of colored markings ensures that Σ Λ has atleast one boundary component in every connected component. This proves the assertion.The proof showed that the map (5.2) is actually induced by the augmentation fibration from Section 2.5, butonly applied to a selected subfamily of the cuts prescribed by the coloring. Therefore, we will refer to the trivialfibration (5.2) as a partial augmentation fibration . Corollary 5.2 (Uncoloring maps) . Let C be a pivotal finite tensor category and X a projective boundarylabel for an extended surface Σ with colored marking Λ . Any uncoloring Λ −→ Ω induces a trivial fibration F m C ( Σ, X, Λ ) ≃ −−→ F m C ( Σ, X, Ω ) . induced by the canonical map from homotopy coends to ordinary coends applied to all cuts that becomeuncolored through the uncoloring. We refer to this map as uncoloring map.Proof. The uncoloring maps fits into the commutative triangle B Σ,Γ C ( X ) F m C ( Σ, X, Λ ) F m C ( Σ, X, Ω ) ε Λ ε Ω featuring the partial augmentation fibrations from Proposition 5.1. Therefore, the uncoloring map needs to bean epimorphism. By the 2-out-3 property it is also an equivalence.25tep by step, we will now define the marked blocks (5.1) on the morphisms of the category of colored markings.We begin with those morphisms of colored markings that do not change the underlying cut system. This is theeasy part because it can be played back entirely to statements about the marked blocks b C with values in vectorspaces: Lemma 5.3.
Let C be a finite ribbon category and Σ an extended surface with projective boundary label X in C . For a colored cut system U on Σ , the assignment m Σ Q Σ ( U ) ∋ Γ F m C ( Σ, X, ( U, Γ )) extends to a functor L U : m Σ Q Σ ( U ) −→ Ch k .Proof. The functor Q Σ : b C ( Σ ) −→ C ( Σ ) from (4.9) just forgets the coloring and sends U to a cut system C := Q Σ ( U ). The groupoid m Σ Q Σ ( U ) = m Σ ( C ) is by its definition in (4.2) the groupoid of markings on thecut system C . Morphisms are just the morphisms of markings on C that leave the cut system C unaffected.Explicitly, the objects of this groupoid are markings on the genus zero surfaces that we obtain from cutting Σ at all cuts of C . The morphisms are, separately for each of these genus zero surfaces, generated by theZ-move and the B-move [BK00, Section 4.1] subject to their relations given in [BK00, Section 4.7]. From (5.1)it follows that we can define the desired functor L U : m Σ Q Σ ( U ) = m Σ ( C ) −→ Ch k on these moves just asfor vector space valued marked blocks (because under the homotopy coend only vector space valued markedblocks b C appear, see (5.1)). More precisely, the Z-move is sent to the Z-isomorphism induced by the pivotalstructure [FS17, Definition 3.5 (i)], and the B-move is sent to the B-isomorphism induced by the braiding [FS17,Definition 3.5 (ii)]. In [FS17], these definitions were made for fine markings, but they carry over to markingswhich are not necessarily fine. The statement that the Z-isomorphism and the B-isomorphism satisfy the neededrelations in contained in [FS17, Lemma 3.8].In order to define marked blocks on the entire category of colored markings on a given extended surface, wewill use modularity: Theorem 5.4.
Let C be an anomaly-free modular category and Σ an extended surface with projective boundarylabel X in C . Then the functors m Σ Q Σ ( U ) −→ Ch k for U ∈ b C ( Σ ) from Lemma 5.3 induce a functor F m C ( Σ, X, − ) : b M ( Σ ) −→ Ch k (5.3) that sends all morphisms to equivalences, i.e. it descends to the ∞ -localization K ( Σ ) of b M ( Σ ) at all uncolorings.Proof. The category b M ( Σ ) was defined as a Grothendieck construction (Definition 4.9) that can be described asa lax colimit (4.4). By the universal property of the lax colimit, a functor b M ( Σ ) −→ Ch k amounts to functors m Σ Q Σ ( U ) −→ Ch k for U ∈ b C ( Σ ) plus a consistent set of natural transformations (that we will elaborate on in amoment). As the needed functors m Σ Q Σ ( U ) −→ Ch k , we take the functors L U from Lemma 5.3. Additionally,for any morphism f : U −→ V in b C ( Σ ), we need a natural transformation α f filling the triangle m Σ Q Σ ( U ) Ch k . m Σ Q Σ ( V ) L U m Σ Q Σ ( f ) α f L V These transformations need to be compatible with the composition of morphisms in b C ( Σ ). Instead of defining α f for arbitrary morphisms, we can of course also define it on generating morphisms, namely uncolorings (U)and admissible moves (AM), see page 21, and verify that the relations (RU), (RM) and (C) are satisfied.On generators, we make the following definitions:(U) The uncolorings are sent to the uncoloring maps from Corollary 5.2.(AM) The definition on the admissible moves induced by the ¯F-move and the ¯S-move in C ( Σ ) is accomplishedas follows:(¯F) An ¯F-move of cut systems give rise to an admissible move ¯F : U −→ V of colored cut systems if andonly if the deleted cut is not colored. In order to obtain for Γ ∈ m Σ Q Σ ( U ) the needed isomorphism α ¯F : F m C ( Σ, X, ( U, Γ )) −→ F m C ( Σ, X, ( V, m Σ (¯F) Γ )) , (5.4)we can now use the F-isomorphism from [FS17, Definition 3.5 (iii)] which uses the (ordinary) YonedaLemma for the morphism spaces in C . 26¯S) The ¯S-move of a cut system can be applied to a subsurface of Σ of the shape of a torus with oneboundary component and one cut. The move replaces this cut by a transversal one (as depicted inFigure 4). It give rise to an admissible move ¯S : U −→ V of colored cut systems if and only if the cutthat is being replaced is not colored. In order to obtain for Γ ∈ m Σ Q Σ ( U ) the needed isomorphism α ¯S : F m C ( Σ, X, ( U, Γ )) −→ F m C ( Σ, X, ( V, m Σ (¯S) Γ )) (5.5)we can now use the S-isomorphism from [FS17, Definition 3.5 (v)] which makes use of the S-transformation for the canonical coend [Lyu95b]. Note that modularity enters in this step becauseit ensures that one can define the S-transformation.It is seen as follows that the relations are satisfied: It can be easily observed that (RU) and (C) are satisfied.The relations (RM) being satisfied is a statement about vector space valued marked blocks for the anomaly-freecase (similarly to Lemma 5.3), which we will demonstrate for one the five relations in [BK00, Section 7.3] for thedefinition of C ( Σ ), namely the compatibility of ¯F and ¯S. The induced relations for ¯F and ¯S, seen as admissible moves in b C ( Σ ), arise by coloring the cut systems involved this relation while respecting of course the definitionof b C ( Σ ). An example of such a coloring is shown in Figure 6, and we list all other possible colorings in thecaption, but they can all be treated as the one which is shown in the picture. ¯S¯F¯F ¯S Figure Pictorial presentation of a colored version of the compatibility of ¯ F and ¯ S. We would obtain another relationby coloring also the leftmost cut or only the leftmost cut. In any case, the cuts in the middle cannot be colored.
Verifying that α ¯F and α ¯S as defined above in (5.4) and (5.5) satisfy the relation from Figure 6 now amountsto a statement about the F-isomorphism and the S-isomorphism for a vector space valued marked block for atorus with two holes. The latter can be extracted from [FS17, Section 3.2], where it is shown that vector spacevalued marked blocks yields a vector space valued functor defined on the groupoid of (fine) markings.This concludes the proof that we obtain a functor (5.3). The statement that the functor sends all morphismsto equivalences is only non-trivial for the uncolorings. In this case, it follows from Corollary 5.2.In the next step, we prove that the constructions from Theorem 5.4 are natural (in the appropriate sense) inthe labeled extended surface. In order to make this precise, we define for any modular category C the category C - R b M as the Grothendieck construction C - Z b M := Z (cid:18) C - Surf −→ Surf b M −−→ Cat (cid:19) (5.6)(this definition could be made for any label set, of course). The category C - R b M should be interpreted as theresult of categorically gluing together the categories of colored markings over varying C -labeled surfaces. Wedenote by Π : C - Z b M −→ C - Surf (5.7)the projection. Both categories inherit a symmetric monoidal structure from disjoint union such that Π is asymmetric monoidal functor. Proposition 5.5.
For any anomaly-free modular category C , the functors from Theorem 5.4 induce a symmetricmonoidal functor F m C : C - Z b M −→ Ch k . roof. If we write the Grothendieck construction (5.6) as a lax colimit (recall (4.4)), we see that a functor F m C : C - R b M −→ Ch k amounts to functors b M ( Σ ) −→ Ch k (5.8)for each extended surface Σ with projective boundary label X and a consistent system of natural transformations(that we will elaborate on in a moment; afterwards, we also comment on compatibility with the monoidalstructure). For the functors (5.8), we take the functors F m C ( Σ, X, − ) : b M ( Σ ) −→ Ch k provided by Theorem 5.4.Additionally, we need to specify for any morphism f : ( Σ, X ) −→ ( Σ ′ , X ′ ) in C - Surf a natural transformation b M ( Σ ) Ch k b M ( Σ ′ ) F m C ( Σ,X, − ) b M ( f ) ξ f F m C ( Σ ′ ,X ′ , − ) (5.9)such that the transformations ξ f respect the composition in C - Surf . We specify these transformations separatelyfor sewings and mapping classes: • Let s : ( Σ, ( X, P, P )) −→ ( Σ ′ , X ) be a sewing morphism in C - Surf that glues an ingoing to an outgoingboundary component which are both labeled with P (without loss of generality, it suffices to considersewings of this form). From (5.1), we can now read off that there is a canonical map F m C ( Σ, ( X, P, P ) , Λ ) −→ F m C ( Σ ′ , X, b M ( s )( Λ ))for any colored marking Λ in Σ (coming just from the definition of the homotopy coend; compare also tothe sewing maps (2.4)). This map can be easily seen to be natural in Λ , i.e. we get a natural transformation b M ( Σ ) Ch k b M ( Σ ′ ) F m C ( Σ, ( X,P,P ) , − ) b M ( s ) ξ s F m C ( Σ,X, − ) These transformations preserve the composition of sewings strictly. • It is an important observation that the definition of marked blocks for a marked surface (or their general-izations to colored markings) just depends on the incidences of (colored) cuts and markings (the relativelocation of these objects to each other, see also the explanations on page 8) — and these incidences donot change when we act with a mapping class. As a consequence of this observation, for any mappingclass φ : Σ −→ Σ ′ seen as morphism ( Σ, X ) −→ ( Σ ′ , X ′ ) in C - Surf the triangle b M ( Σ ) Ch k b M ( Σ ′ ) F m C ( Σ,X, − ) b M ( φ ) F m C ( Σ,X ′ , − ) commutes, so we can actually use the identity transformation to fill this triangle.Since the transformations corresponding to sewings respect composition and since the transformations corre-sponding to mapping classes are identities, we conclude that the functors F m C ( Σ, X, − ) : b M ( Σ ) −→ Ch k inducea functor C - R b M −→ Ch k .Moreover, the functor C - Surf −→ Surf b M −−→ Cat is symmetric monoidal (Proposition 4.10), and the functors(5.8) as well as the transformations (5.9) are compatible with this monoidal structure. Therefore, the functor C - R b M −→ Ch k just constructed is also symmetric monoidal.28 .2 Homotopy left Kan extension The functor F m C : C - R b M −→ Ch k from Proposition 5.5 is defined on a category of labeled extended surfacesequipped with a colored marking. In order to obtain a functor defined directly on the category of C -labeledsurfaces, we use a homotopy left Kan extension along the functor Π : C - R b M −→ C - Surf from (5.7).Recall that for any category S , the category Ch S k of functors S −→ Ch k can be equipped with the projectivemodel structure. For any functor Φ : S −→ T , we obtain a Quillen pair Φ ! : Ch S k / / Ch T k : Φ ∗ o o by left Kan extension. We denote the homotopy left Kan extension, i.e. the left derivative of Φ ! , by L Φ ! . Definition 5.6.
For any anomaly-free modular category C , we define the functor F C := L Π ! F m C : C - Surf −→ Ch k (5.10)as the homotopy left Kan extension of the functor F m C : C - R b M −→ Ch k from Proposition 5.5 along the functor Π : C - R b M −→ C - Surf from (5.7).
Remark 5.7.
In [FS17, Section 3.3] a Kan extension along an unmarking functor U : mSurf −→ Surf from acategory mSurf of marked surfaces to the category of surfaces is used for the construction of a so-called pinnedblock functor . The use of the Kan extension in Definition 5.6 seems similar, but we should emphasize thatit is really different. The category mSurf in [FS17] is not equivalent to C - R b M (because mSurf actually has nonon-trivial automorphisms), and the unmarking functor U in [FS17] is not a projection functor like Π , butrather a translation of moves to mapping classes.The notation F C ( Σ, X ) suggests a relation to the complexes F m C ( Σ, X, Λ ) from Theorem 5.4 that additionallydepended on a colored marking on Σ . This notation is justified by the next result: Proposition 5.8.
For any anomaly-free modular category C and any extended surface Σ with projectiveboundary label X , there is a canonical equivalence hocolim Λ ∈ b M ( Σ ) F m C ( Σ, X, Λ ) ≃ −−→ F C ( Σ, X ) . (5.11) After the choice of a colored marking Λ on Σ , there is a canonical equivalence F m C ( Σ, X, Λ ) ≃ −−→ F C ( Σ, X ) . (5.12)The proof of Proposition 5.8 will need a standard Lemma. First we establish some notation and terminology:For a functor L : A −→ B and b ∈ B , we denote by L/b the slice category of pairs ( a, f ) of a ∈ A and a morphism f : L ( a ) −→ b . A morphism ( a, f ) −→ ( a ′ , f ′ ) in L/b is a morphism g : a −→ a ′ such that f ′ L ( g ) = f . Dually,we can define the slice category b/L . A functor L : A −→ B is called homotopy final if for each b ∈ B the slicecategory b/L is contractible in the sense that | N ( b/L ) | is equivalent to a point. Lemma 5.9.
For any functor F : B −→
Cat , the forgetful functor π : R F −→ B has the property that thenatural functor K b : F ( b ) −→ π/b for any b ∈ B is homotopy final. A proof can be deduced from the more general statement [L-HTT, Proposition 4.3.3.10] in the context of ∞ -categories. Proof of Proposition 5.8.
By (5.10) and the homotopy colimit formula for the homotopy left Kan extension wearrive at F C ( Σ, X ) = hocolim (cid:18) Π/ ( Σ, X ) −→ C - Z b M F m C −−−→ Ch k (cid:19) , where Π/ ( Σ, X ) −→ C - R b M is the forgetful functor. The natural functor K Σ,X : b M ( Σ ) −→ Π/ ( Σ, X ) (thatappears for an arbitrary Grothendieck construction in Lemma 5.9) induces a maphocolim Λ ∈ b M ( Σ ) F m C ( Σ, X, Λ ) = hocolim (cid:18) b M ( Σ ) K Σ,X −−−−−→ Π/ ( Σ, X ) −→ C - Z b M F m C −−−→ Ch k (cid:19) −→ hocolim (cid:18) Π/ ( Σ, X ) −→ C - Z b M F m C −−−→ Ch k (cid:19) = F C ( Σ, X )29the equality in the first line holds by definition of F m C in Proposition 5.5). For the proof of (5.11), it remainsto prove that this map is an equivalence, but this follows from Lemma 5.9 which states K Σ,X is homotopyfinal, which implies that the map induced between the homotopy colimits is an equivalence (see e.g. [Rie14,Theorem II.8.5.6] for this standard result).For the proof of (5.12), it suffices to prove that the canonical map F m C ( Σ, X, Λ ) −→ hocolim Λ ∈ b M ( Σ ) F m C ( Σ, X, Λ ) (5.13)is an equivalence (because then we can compose with (5.11)). This can be concluded from the contractibility ofthe ∞ -groupoid K ( Σ ) obtained by ∞ -localization of b M ( Σ ) at all uncolorings: The right hand side of (5.13) isthe homotopy colimit of the functor b M ( Σ ) −→ Ch k from Theorem 5.4 which, additionally, has the property thatit sends all uncolorings to equivalences and hence descends to K ( Σ ) without changing the homotopy colimit(because ∞ -localizations are homotopy final [Cis19, Proposition 7.1.10]). It suffices now to prove that the map ⋆ −→ K ( Σ ) selecting Λ is homotopy final, but this follows from [L-HTT, Corollary 4.1.2.6] because K ( Σ ) is acontractible Kan complex by Theorem 4.8. Corollary 5.10.
Let C an anomaly-free modular category. Then for any extended surface Σ with projectiveboundary label X and any marking Γ on Σ , there is a canonical equivalence B Σ,Γ C ( X ) ≃ −−→ F C ( Σ, X ) . Proof.
We observe that by coloring all cuts of Γ we obtain a colored marking Γ c such that F m C ( Σ, X, Γ c ) = B Σ,Γ C ( X ) holds by definition. Now we use (5.12) from Proposition 5.8. Theorem 5.11.
Let C be an anomaly-free modular category. Then the functor F C : C - Surf −→ Ch k (5.14) from Definition 5.6 is a modular functor with values in chain complexes for the category C in the sense ofDefinition 3.4.Proof. Through the construction leading to Definition 5.6, we have established that F C is a functor C - Surf −→ Ch k . Moreover, for labeled extended surfaces ( Σ, X ) and ( Σ ′ , X ′ ), we have Π/ ( Σ ⊔ Σ ′ , X ⊔ X ′ ) ∼ = Π/ ( Σ, X ) × Π/ ( Σ ′ , X ′ ). Since F m C : C - R b M −→ Ch k is symmetric monoidal, we can now conclude that F C is symmetricmonoidal (depending on how we model the homotopy colimits, the structure maps will just be weak equiva-lences).From Corollary 5.10 we may conclude directly that the cylinder category of F C is equivalent to Proj C becausemarked blocks on decorated cylinders are given by the morphism spaces of C by (2.1).It remains to prove that (5.14) satisfies excision: Let s : ( Σ, ( X, P, P )) −→ ( Σ ′ , X ) be a sewing morphismin C - Surf that glues an ingoing to an outgoing boundary component which are both labeled with P . Any fixedmarking Γ on Σ induces a marking Γ ′ on Σ ′ . Now the naturality of the maps from Corollary 5.10 gives us thecommuting square (where they induce the vertical equivalences) R P ∈ Proj C L B Σ,Γ C ( X, P, P ) B Σ ′ ,Γ ′ C ( X ) R P ∈ Proj C L F C ( Σ, ( X, P, P )) F C ( Σ, X ) . ≃ equivalence from Proposition 2.3for excision with marking ≃ induced by evaluation of F C on s The square commutes because the sewing transformations ξ s from the proof of Proposition 5.5 generalize thesewing maps from Proposition 2.3. It follows that the lower horizontal map is an equivalence which provesexcision. Remark 5.12 (The anomalous case) . Theorem 5.11 provides — at least in the anomaly-free case — the modularfunctor needed for the Main Theorem 3.6, and Corollary 5.10 gives us the concrete prescription how to computeit in terms of a marking. The restriction to anomaly-free modular categories throughout Section 5 was madefor presentation purposes because the modifications needed to deal with anomalous case are analogous to theones needed for vector space valued modular functors: Let us recall e.g. from [FS17] that for marked blockswith values in vector spaces, the construction in the anomalous case proceeds precisely as in the anomaly-freecase, but with the groupoid M ( Σ ) of markings on an extended surface Σ replaced with a groupoid M C ( Σ ) whichcontains additional central generators [FS17, Section 3.2] and is no longer contractible. In fact, for a connected30urface Σ , we have M C ( Σ ) ≃ ⋆// Z by a non-canonical equivalence. The groupoid M C ( Σ ) comes with a functor M C ( Σ ) −→ M ( Σ ) sending the central generators to identities. We may see M C ( Σ ) as a central extension of M ( Σ ). Now the vector space valued marked blocks will be defined on M C ( Σ ) instead of M ( Σ ). The centralgenerators will be sent to a scalar multiple of this identity. This scalar is given by ζ g , where ζ ∈ k × is theframing anomaly (Remark 3.8) and g is the genus of Σ . This allows us to interpret a functor out of M C ( Σ ) asa (certain type of) projective functor out of M ( Σ ).In the differential graded setting, it is straightforward to take these central extensions into account as well:The central extension M C ( Σ ) −→ M ( Σ ) induces a central extension b M C ( Σ ) −→ b M ( Σ ) of the category b M ( Σ ) ofcolored markings. As in Theorem 5.4, we will obtain for any projective boundary label X of Σ a functor F m C ( Σ, X, − ) : b M C ( Σ ) −→ Ch k that sends all morphisms to equivalences. By the same arguments as for Proposition 5.5, it induces a symmetricmonoidal functor C - R b M C −→ Ch k on the Grothendieck construction. Via homotopy left Kan extension along C - R b M C −→ C - R M C induced by the functors b M C ( Σ ) −→ M C ( Σ ), we obtain a symmetric monoidal functor C - R M C −→ Ch k . But since M ( Σ ) ≃ ⋆ for every extended surface, we have C - R M ≃ C - Surf . Similarly, C - Z M C ≃ C - Surf c (5.15)(this can be interpreted in the sense that C - R M C provides a model for the central extension C - Surf c ; in fact,we could just define C - Surf c as C - R M C ). This allows us to see the functor C - R M C −→ Ch k as a functor C - Surf c −→ Ch k — and this will give us the modular functor in the anomalous case. It is however not clear thatits values are actually equivalent to the marked blocks. This, however, can be seen with arguments analogousto those in the proof of Proposition 5.12. This completes the proof of the Main Theorem 3.6 in the generalcase. Remark 5.13 (Relation to Lyubashenko’s mapping class group representations and to the Reshetikhin-Turaevconstruction) . Let C be a modular category. By construction, the zeroth homology H F C of the modular functor F C : C - Surf c −→ Ch k is a modular functor with values in vector spaces. This modular functor is (up to sometechnical subtleties that we will explain now) built from Lyubashenko’s mapping class group representations[Lyu95a, Lyu95b, Lyu96]: For an extended surface Σ with projective boundary label X , we have a canonicalisomorphism H B Σ,Γ C ( X ) ∼ = (cid:16) b Σ,Γ C ( X ∨ ) (cid:17) ∗ , (5.16)where B Σ,Γ C denotes marked blocks (Section 2.2) and b Σ,Γ C vector space valued marked blocks (Section 2.5).For the specific marking in Example 2.6, this follows from (2.8) and the self-duality F ∨ ∼ = F (that we alsoused in Remark 3.12), and the general case can be played back to this special case with arguments similar tothose in the proof of Proposition 2.7. If Σ has at least one boundary component per connected component, theisomorphism (5.16) is compatible with the isomorphism H B Σ,Γ C ( X ) ∼ = b Σ,Γ C ( X ) induced by the augmentationfibration (2.10) in the sense that the triangle of isomorphisms H B Σ,Γ C ( X ) (cid:16) b Σ,Γ C ( X ∨ ) (cid:17) ∗ b Σ,Γ C ( X ) (5.16)induced byaugmentation fibration (2.6) and F ∨ ∼ = F commutes. The left hand side of (5.16) is functorial in b M C ( Σ ), but will descend to M C ( Σ ) such that (5.16)is a natural isomorphism of functors defined on M C ( Σ ). Therefore, (5.16) will induce an isomorphism ofmapping class group representations. 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