Factorization Homology and 4D TQFT

In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $\mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of"stringnets with boundary conditions"as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].


Introduction
The notion of factorization homology for topological manifolds was introduced by Ayala and Francis (see [AF2019, AFR2018]) following earlier work of Beilinson and Drinfeld and many others. The main idea of this construction is quite natural: it allows one to construct invariants of n-dimensional manifolds by "gluing" local data associated to balls embedded in M . A simple example of such a construction is the usual homology H * (M, A), where A is an abelian group. More general form of factorization homology uses as input the following algebraic data: • An object A in a symmetric monoidal ∞-category V (this is the object assigned to the ball) • A structure of an algebra over the operad of (framed) n-disks on A; this is used to define the gluing of local data As an output, the factorization homology of an n-dimensional manifold M with coefficients in A (denoted ∫ M A) gives again an object of V.
Unfortunately, the precise definition of factorization homology has some drawbacks. First, it is given in the language of ∞-categories, so it is rather technical. More importantly, factorization homology is defined by suitable universality properties, so this definition is not very explicit; in fact, even existence of such an object is non-trivial.
The main goal of our note is giving an explicit construction of factorization homology in two special cases: (1) n = 1, A is a spherical fusion category.
In both cases, we take the target category V to be the (2, 1) category Rex of essentially small finitely co-complete k-linear categories and right exact functors, as defined in [BZBJ2018]; the symmetric monoidal structure on Rex is given by Kelly product. In particular, it contains the (2, 1) category of small k-linear abelian categories as a full subcategory. We will discuss category Rex in more detail in Section 2.
The n = 1 case is rather simple: the only non-trivial 1-manifold is S 1 , and it is easy to show that for a spherical fusion category A, ∫ S 1 A = Z(A) is the Drinfeld center of A (as an abelian category; the monoidal structure on the Drinfeld center requires additional construction). Yet we include this case as it is necessary to understand the n = 2 case.
The n = 2 case has been studied in the papers of Ben-Zvi, Brochier, and Jordan [BZBJ2018, BZBJ2018b].
In both cases, we show that one can give an explicit definition of factorization homology ∫ M A using suitable colored graphs in dimension n + 1 modulo an equivalence relation generated by local moves; this follows the general ideas first suggested by Walker in [Wal2006]. For n = 1, such local relations were first explicitly written by physicists Levin and Wen in [LW2005], who dubbed such graphs on surfaces "stringnets". A rewriting of this notion in a more mathematical language can be found in [Kir], where it is shown that the stringnets and their boundary conditions coincide with the 2-1 part of Turaev-Viro (2 + 1)-dimensional TQFT.
1 nis Sullivan, and Pavel Safronov for helpful discussions. We woudl also like to thank the referees for their many suggestions.
Notation and conventions. Throughout the paper, the word "manifold" stands for a smooth manifold which admits a finite open cover such that any finite intersection of the open subsets forming the cover is diffeomorphic to affine space. It is known that this is equivalent to requiring that M is diffeomorphic to the interior of a compact manifold with boundary.
Throughout the paper, we fix an algebraically closed field k of characteristic 0. All abelian categories considered in this paper will be locally finite k-linear categories (see [EGNO2015, Section 1.8]). In particular, we denote by Vec the category of finite-dimensional vector spaces over k.
We will be denoting by ⊠ the Deligne tensor product of abelian categories, see [EGNO2015, Section 1.11]; it is well defined for locally finite k-linear categories and coincides with the Kelly tensor product in Rex (see [BZBJ2018, Section 3.2] for references).
We will be heavily using notions of fusion category, pivotal and spherical fusion category, and premodular category. We refer the reader to [EGNO2015] for definitions and basic properties of such categories. In most of our formulas and computations, we will be suppressing the associativity and unit morphisms, as well as the pivotal morphism V ≃ V * * .
We also be using graphical presentations of morphisms. We use the convention that morphisms go from top to bottom, and the braiding of a braided category is represented by the ' \' strand going over the ' / ' strand; see Appendix for more details.
Note. While this paper was in preparation, we have received a preprint of Juliet Cooke [Coo2019], which contains the result very similar to ours. Yet the proof and exposition are different, so we hope that many readers will find our work useful.

Factorization homology overview
In this section, we give a brief summary of the theory of factorization homology. We only try to cover as much as is necessary for our purposes, referring the reader to review [AF2019] and original papers cited there for details.
To keep things simple, we will only consider the theory for oriented manifolds, ignoring other possible choices of framing structures. All manifolds considered here will be smooth finitary, i.e. those that are interiors of compact manifolds with (possibly empty) boundary.
We define the symmetric monoidal category Disk or n as the category whose objects are finite disjoint unions of copies of R n (or, equivalently, open unit ball B n ) and morphisms are orientation-preserving embeddings. The set of embeddings is considered as a topological space, with compact-open C ∞ topology, so Disk or n becomes a topological category, and thus, an ∞-category. The monoidal structure is given by the disjoint union.
Given a symmetric monoidal ∞-category V, an n-disk algebra in V is a symmetric monoidal functor of ∞-categories Disk or n → V. In particular, this defines an object A ∈ Obj(V) (namely, the image of the standard unit n-disk). Abusing the language, we will also call A a "disk algebra".
Given such a disk algebra A, one defines for any oriented n-manifold M the factorization homology M A ∈ Obj V as a certain colimit, over all embeddings of collections of oriented disks into M . Note that existence of such factorization homology is not guaranteed: in order for it to be defined, we need V to have sufficiently many colimits. We do not reproduce the definition here; instead, we state some of the properties of this construction, and give a list of properties (Theorem 2.4) which characterizes factorization homology ( [AF2015], [AFT2017]). We refer the reader to the original papers for details and proofs.
Theorem 2.1. So defined factorization homology satisfies the following properties: (1) For an open n-ball B n , we have (3) It sends disjoint union to tensor product in V: Let us now restrict our attention to the special case when the target category V is the (2, 1) category Rex of finitely co-complete k-linear categories with right exact functors, as defined in [BZBJ2018]. We won't repeat the definition, listing instead the properties of this category; we refer the reader to [BZBJ2018, Section 3] for proofs.
(1) The category Rex is equivalent (as a (2, 1) category) to the category Pr c of compactly generated presentable k-linear categories with compact and cocontinuous functors [BZBJ2018, Section 3.1] (2) Category Rex is closed under small colimits [BZBJ2018, Proposition 3.5] (3) Category Pr c (and thus Rex) includes as a full subcategory the 2-category of small k-linear abelian categories, with right exact functors (4) Category Rex has a symmetric monoidal structure, given by so-called Kelly product. If A, B are abelian category such that Deligne tensor product A ⊠ B (see [EGNO2015, Section 1.11]) is defined, then Kelly tensor product coincides with Deligne tensor product [BZBJ2018, Section 3.2]. From now on, we use notation ⊠ for the Kelly tensor product (5) Let A be an E 1 algebra in Rex. Then we can define the notion of left (respectively, right) A-modules in V and define relative tensor product M ⊠ A N as an object of V. In the special case when A is a multitensor category (i.e. a locally finite k-linear abelian rigid monoidal category), and M, N are abelian categories, the relative Kelly product M ⊠ A N is also an abelian category and coincides with the balanced tensor product (see Section 3 below) whenever the latter is defined [BZBJ2018, Remark 3.15, Corollary 3.18]. (6) Balanced braided tensor categories are 2-disk algebras in Rex (see [BZBJ2018, Section 3.3]). Using these properties, it was shown in [BZBJ2018] that factorization homology with values in V is welldefined (they were working in the case n = 2; however, it is trivial to see that the same also applies in case n = 1) We will now formulate the final property of the factorization homology, called the excision property. For simplicity, we will only do so in the case when the target category V is the (2, 1) category Rex, even though it also holds for any factorization homology.
Before formulating the excision property, we need the following lemma about state some simple corollaries of the properties above.
Lemma 2.2. Let n = 1 or 2 and let A be an n-disk algebra in V = Rex. Then (1) For any n − 1-dimensional oriented manifold N , let A(N ) = ∫ N ×I A, where I = (0, 1) is the open interval. Then A(N ) has a canonical structure of an E 1 algebra in Rex, with the multiplication coming from embedding (N × I) ⊔ (N × I) → N × I ("stacking").
(2) Let M be an n-dimensional manifold with boundary; we denote by M o the interior of M . Let N be a boundary component of M . Assume that we are given a diffeomorphism of a neighborhood of N in M o with N × (0, 1); we will call such an isomorphism a collaring or a collared structure at N . Then this gives on ∫ M o A a natural structure of a module over A(N ).
One of the goals of this paper is to show that in some special cases, this algebra and module structure coincide with so-called skein algebra (respectively, skein module), see [Wal2006], [JF2015].
We can now state the final property of factorization homology.
Theorem 2.3. In the assumptions of Lemma 2.2, factorization homology satisfies the following excision property: Let M 1 , M 2 be n-manifolds with boundary, and M o i the interior of M i . Let N 1 , N 2 be connected components of the boundary of M 1 , M 2 respectively, together with a diffeomorphism N 1 ≃ N 2 (where bar stands for opposite orientation). Moreover, assume we are given collared structure at N 1 , N 2 as in Lemma 2.2.
Let M be the manifold obtained by gluing together M 1 with M 2 using ϕ; choice of collared structures gives a smooth structure on M .
Then one has an equivalence of categories where ⊠ A is the relative tensor product in Rex.
Finally, the following properties uniquely characterize factorization homology (this is a recasting of [BZBJ2018, Theorem 2.5], in turn based on [AF2015], [AFT2017]): Theorem 2.4. Let A be a 2-disk algebra in Rex. Then the functor ∫ − A satisfies, and is characterized by, the following properties: (1) If U is contractible, then there is an equivalence in Rex, (2) The E 1 "stacking" structure on A(N ) in Lemma 2.2 is unique (any two diffeomorphisms Y ≃ N × I respecting the fibre structure N × I proj → I induces equivalent E 1 -algebra structures on ∫ Y A).

Module categories, balanced tensor product, and center
In this section, we review the results about balanced tensor product of module categories. Our main goal is to give two constructions of the center of an C-bimodule category M -Z C (M) (Definition 3.1) and hTr C (M) (Definition 3.7), and show that when C is pivotal multifusion, they are equivalent (Theorem 3.10).
Recall our convention that all categories considered in this paper are locally finite k-linear. Most of the time, they will be abelian; however, in some cases we will need to use k-linear additive (but not necessarily abelian) categories. For such a category A, we will denote by Kar(A) the Karoubi envelope (also known as idempotent completion) of A. By definition, an object of Kar(A) is a pair (A, p), where A is an object of A and p ∈ Hom A (A, A) is an idempotent: p 2 = p. Morphisms in Kar(A) are defined by Throughout this section C is a pivotal category, though in the definitions C is only required to be monoidal. When C is multifusion, we use the conventions and notation laid out in the Appendix. In particular, Irr(C) is the set of isomorphism classes, Irr 0 (C) are those simples appearing as direct summands of the unit 1, {X i } will be a fixed set of representatives of Irr(C), d R i is the (right) dimension of X i , and we will be using graphical presentation of morphisms.
We assume that the reader is familiar with the notions of module categories; for a left module category M over C, we will denote the action of A ∈ C on M ∈ M by A ⊳ M . Similarly, we use M ⊲ A for right action. In this paper, all module categories are assumed to be semisimple (as abelian categories).

4
This section is organized as follows: Subsection 3.1 provides the definition and some properties of Z C (M), Subsection 3.2 does so for hTr C (M), and Subsection 3.3 shows that when C is pivotal multifusion, these definitions are essentially the same.

Z C (M).
The following definition is essentially given in [GNN2009, Definition 2.1] (there C is assumed to be fusion).
Definition 3.1. Let C be a finite multitensor category, and let M be a C-bimodule category. The center of M, denoted Z C (M), is the category with the following objects and morphisms: Objects: pairs (M, γ), where M ∈ M and γ is an isomorphism of functors γ A ∶ A ⊳ M → M ⊲ A, A ∈ C (half-braiding) satisfying natural compatibility conditions. Morphisms: In particular, in the special case M = C, this construction gives the Drinfeld center Z(C).
Remark 3.2. Equivalently, the center Z C (M) can be described as the category of C-bimodule functors C → M; see [GNN2009] for details.  is given by: (Note the sum on the right in (3.2) is over Irr 0 (C) and not Irr(C).) The other adjunction isomorphism for I∶ M ⇌ Z(M) ∶ F , Hom M (M 1 , M 2 ) ≃ Hom Z(M) (I(M 1 ), (M 2 , γ)) is given by a similar formula, essentially obtained by rotating all the diagrams above.
Note that the isomorphisms here differ slightly from that of [Kir]. We punt the proof to the Appendix (on page 30).
An important special case is when M = M 1 ⊠ M 2 , where M 1 is a right module category over a pivotal multifusion category C, and M 2 is a left module category over C. In this case, by [ENO2010, Proposition 3.8], one has that Z C (M 1 ⊠ M 2 ) is naturally equivalent to the balanced tensor product of categories: where the balanced tensor product is defined by the universal property: for any abelian category A, we have a natural equivalence where F un, F un bal stand for category of k-linear additive functors (respectively, category of k-linear additive Cbalanced functors); see details in [ENO2010, Definition 3.3].
Under the equivalence (3.4), the natural functor Recall that a functor F ∶ A → B, where B is abelian and A additive (not necessarily abelian) is called dominant if any object of B appears as a subquotient of F (X) for some X ∈ Obj A. Similarly, we say that a full subcategory A ⊂ B is dominant if any object of B appears as a subquotient of some X ∈ Obj A. In the case when A is a full additive subcategory in a semisimple abelian category B, this immediately implies that the Karoubi envelope of A is equivalent to B (in particular, this implies that Kar(A) is abelian).  For applications, we will need to consider centers over a full, dominant, monoidal subcategory C ′ ⊆ C. Equivalently, C ′ is a pivotal category whose Karoubi envelope is multifusion.
Lemma 3.6. Let C ′ be a pivotal locally finite k-linear additive category whose Karoubi envelope C = Kar(C ′ ) is multifusion. Let M be a C-bimodule category, and hence naturally a C ′ -bimodule category (as before, we assume that M is a semisimple abelian category). Then there is a natural equivalence In particular, for right, left C-module categories M 1 , M 2 , there is a natural equivalence Proof. The equivalence is given as follows: objects (M, γ) in Z C (M) are naturally objects in Z C ′ (M) by forgetting some of the half-braiding, i.e. (M, γ C ′ ); morphisms f ∶ (M, γ) → (M ′ , γ ′ ) are naturally morphisms f ∶ (M, γ C ′ ) → (M ′ , γ C ′ ). We need to check that this is an equivalence.
The functor is essentially surjective: any half-braiding over C ′ can be completed to a half-braiding over C. To see this, let γ be a half-braiding over C ′ . Let X ∈ Obj C Obj C ′ , and let it be a direct summand of some Y ∈ Obj C ′ , X ι ⇌ p Y . Then we define the extension of γ to X by γ X = (id M2 ⊲ p) ○ γ Y ○ (ι ⊳ id M1 ). It is easy 6 to check, using the semisimplicity of C, that γ X is independent on the choice of Y and p, ι. It is also easy to check that the resulting extension is indeed natural in X.
For morphisms, it is clear that this functor is faithful. To show fullness, consider f ∈ Hom Z C ′ (M) ((M 1 , γ 1 ), (M 2 , γ 2 )). We need to check that it also intertwines half-braiding with X ∈ C, but this follows easily from the definition of the extension of half-braiding given above.
Note since γ has a unique extension to all of C, this proof actually shows that the equivalence is an isomorphism.
Note that in the proof above, we do not use the rigidity of C, but we need it to conclude the second statement concerning balanced tensor products.

hTr C (M).
Next we define the other notion of center.
Definition 3.7. Let C be monoidal, and M a C-bimodule category. Define the horizontal trace hTr C (M) as the category with the following objects and morphisms: Objects: same as in M Morphisms: , the sum is over all (not necessarily simple) objects X ∈ C, and ∼ is the equivalence relation generated by the following: For Composition is given by When the context is clear, we will drop the subscript hTr = hTr C . We will write [ϕ] ∈ Hom hTr(M) (M 1 , M 2 ) for the morphism represented by ϕ ∈ Hom X M (M 1 , M 2 ) for some X.
It can be shown that in a certain sense this definition is dual to the definition of center given above and is closely related to the notion of co-center as described in [DSPS2013, Section 3.2.2]. However, we will not be discussing the exact relation here.
It is easy to see that the category hTr(M) is additive but not necessarily abelian.
There is a natural inclusion functor hTr∶ M → hTr(M) which is identity on objects, and on morphisms it is the natural map Hom Proof. Straightforward exercise left to the reader.
We also consider C ′ ⊆ C as in Lemma 3.6, but here we do not need rigidity nor semisimplicity on C: Lemma 3.9. Let C ′ be monoidal, and let C = Kar(C ′ ) be its Karoubi envelope. Let M be a C-bimodule category, and hence naturally a C ′ -bimodule category. Then there is a natural equivalence In particular, for right, left C-module categories M 1 , M 2 , there is a natural equivalence Proof. The equivalence is given by the identity map on objects, and for two objects M 1 , M 2 ∈ Obj M, the map on morphisms is given by completing the bottom arrow: It remains to prove that the bottom arrow is an isomorphism. Let us first observe the following. Let X, Y ∈ Obj C ′ , and suppose X is . Surjectivity: Essentially, we need to show that any morphism in hTr C (M) can be "absorbed" into is in the image. Injectivity: Essentially, we need to show that relations can also be "absorbed" into hTr C ′ (M). Let [ϕ] ∈ Hom hTr C ′ (M) (M 1 , M 2 ) that is sent to 0. By the observation above, we may represent it by some We want to be able to replace the A j 's with objects in It is a simple matter to verify that  In particular, for right, left C-modules M 1 , M 2 , we have Before proving the theorem, we will need the following lemma.
Lemma 3.11. The natural linear map is an isomorphism.
Proof. To prove the statement, we define a linear map ; once again see Notation 10.3 in the Appendix for definition of α. It is easy to check the following properties: (1) G is well-defined on morphisms (i.e. it preserves the equivalence relation): this follows from Lemma 10.7.
, then it appears as a direct summand of G(M ); the projection to Y is, up to a factor,  This immediately implies the statement of the theorem by the universal properties of Karoubi envelopes.
By Lemma 3.6 and Lemma 3.9, we extend the above theorem to C ′ ⊆ C: Corollary 3.12. Let C ′ be a pivotal category whose Karoubi envelope C = Kar(C ′ ) is multifusion. Let M be a C-bimodule category, and hence naturally a C ′ -bimodule category. Then we have . We compare these constructions for M and its Karoubi envelope: Lemma 3.13. Under the same hypotheses as Corollary 3.12, In particular, if M ′ is a dominant submodule category of M, then Proof. The natural inclusion M → Kar(M) is a full, dominant functor of C ′ -bimodules, and it is easy to see that the corresponding functor hTr(M) → hTr(Kar(M)) is also full and dominant. It follows that the induced functor on their Karoubi envelopes is an equivalence.

Colored Graphs in Turaev-Viro theory
In this section, we recall the definition of colored graphs (called stringnets in [Kir]) in Turaev-Viro theory. This is intended to serve as a reminder only; proofs are omitted. Details and proofs can be found in [Kir].
Throughout this section, all surfaces are assumed to be oriented. We denote by A a spherical fusion category. We will be heavily using graphical presentation of morphisms in A; we give a summary of our notation and conventions in the Appendix.
For a finite graph Γ embedded in surface Σ, we denote by E(Γ) the set of edges. Note that edges are not oriented. Let E or be the set of oriented edges, i.e. pairs e = (e, orientation of e); for such an oriented edge e, we denote byē the edge with opposite orientation.
If Σ has a boundary, the graph is allowed to have uncolored one-valent vertices on ∂Σ but no other common points with ∂Σ; all other vertices will be called interior. We will call the edges of Γ terminating at these one-valent vertices legs.
Definition 4.1. Let Σ an oriented surface (possibly with boundary) and Γ ⊂ Σ -an embedded graph as defined above. A coloring of Γ is the following data: • Choice of an object V (e) ∈ Obj A for every oriented edge e ∈ E or (Γ) so that V (e) = V (e) * .
• Choice of a vector ϕ(v) ∈ ⟨V (e 1 ), . . . , V (e n )⟩ (see Appendix (10.2)) for every interior vertex v, where e 1 , . . . , e n are edges incident to v, taken in counterclockwise order and with outward orientation (see Appendix Figure 4). We will denote the set of all colored graphs on a surface Σ by Graph(Σ).
Note that if Σ has a boundary, then every colored graph Γ defines a collection of points B = {b 1 , . . . , b n } ⊂ ∂Σ (the endpoints of the legs of Γ) and a collection of objects V b ∈ Obj A for every b ∈ B: the colors of the legs of Γ taken with outgoing orientation. We will denote the pair (B, {V b }) by V = Γ ∩ ∂Σ and call it boundary value. We will denote Graph(Σ, V) = set of all colored graphs in Σ with boundary value V.
We can also consider formal linear combinations of colored graphs. Namely, for fixed boundary value V as above, we will denote In particular, if ∂Σ = ∅, then the only possible boundary condition is trivial (B = ∅); in this case, we will just write VGraph(Σ).
It follows from result of Reshetikhin and Turaev that for every colored graph Γ in a disk D ⊂ R 2 , one can define its "evaluation" where e 1 , . . . , e n are the edges of Γ meeting the boundary of D (legs), taken in counterclockwise order and with outgoing orientation; in particular, in the case when Γ is a star graph, with one vertex colored by ϕ ∈ ⟨V (e 1 ), . . . , V (e n )⟩, then ⟨Γ⟩ = ϕ.
We call a formal linear combination of colored graphs Γ = ∑ c i Γ i ∈ VGraph(Σ, V) a null graph if there exists an embedded disk D ↪ Σ such that all graphs Γ i meet boundary of D transversally, all Γ i coincide outside of D (as colored graphs) and We will say Γ is null with respect to D. We can now give the main definition of this section.
where N is the subspace spanned by all null graphs (for all possible embedded disks).
As an example, it was shown in [Kir] that We can now define the category of boundary conditions. Definition 4.3. Let N be an oriented 1-dimensional manifold, possibly non-compact. Suppose first N has no boundary. DefineẐ TV (N ) as the category whose objects are finite subsets B ⊂ N together with a choice of object V b ∈ Obj A for every point b ∈ B; we will use the notation V = (B, {V b }) for such an object, and B is called the set of marked points of V. Define the morphisms inẐ TV (N ) by where V * , V ′ means the boundary condition obtained by putting points b ∈ B on the "top" N × {1}, colored by objects V * b for outgoing legs (and thus colored by V b for incoming legs), and putting points b ′ ∈ B ′ on the "bottom" N × {0}, colored by objects V b ′ for outgoing legs.
This category is additive and k-linear. We denote by It is immediate from the definition that where I is an open/closed interval. It has been shown in [Kir] that Z TV (S 1 ) = Z(A) is the Drinfeld center of A. We will reprove it (in a slightly different way) as a special case of a more general result later.

Skeins in Crane-Yetter Theory
In this section, we give a definition of colored graphs/skeins in Crane-Yetter theory, mirroring closely the previous section, and we will reuse many definitions. This definition essentially coincides with those given in [JF2015], [Coo2019]; we use framed graphs instead of ribbons and coupons.
Throughout this section, all 3-manifolds are assumed to be oriented, and may be non-compact and/or with boundary. A will be a skeletal premodular category; see appendix for a summary of notation and conventions.
We will consider finite framed graphs Γ in a 3-manifold M , that is, Γ is a smoothly embedded graph in M with finitely many edges, and each edge comes with a transversal ray field along it (that is, each point p on an edge is assigned a ray ρ p in T p M emanating from the origin, varying smoothly with p); the transversal ray field ρ p is the framing of the edge. We also impose the condition that edges are not tangent to each other at a vertex (this is necessary for the "infinitesimal spheres" discussion below). From here on, we will simply refer to finite framed graphs as graphs.
Graphs are allowed to intersect the boundary ∂M transversally; each point of intersection of Γ with ∂M should be a vertex of Γ, and they are the boundary vertices of Γ. Other vertices of Γ are the interior vertices. Furthermore, the framing on Γ induces at each boundary vertex b a ray in T b (∂M ), a framing on b. This makes the boundary ∂M an extended surface, a surface together with a configuration of finitely many framed points.
For each interior vertex v, the "infinitesimal sphere" at v also acquires an extended surface structure as follows. The space of rays emanating from the origin in T v M is a sphere S 2 v , which we call the infinitesimal sphere. An edge e leaving v has a tangent vector ν e at v, which gives us a point ν e ∈ S 2 v . The framing on e at v is a ray ρ v in T v M ; the quarter plane spanned by ν e and ρ v in T v M defines a ray in T νe S 2 v , i.e. a framing of ν e . The collection of such framed points ν e is the extended surface structure that S 2 v inherits from the graph (ν e are distinct by the extra condition of non-tangency of edges at vertices).
Given an input premodular category A, and given an extended sphere S where each marked point p i is colored with an object V i ∈ A, the Reshetikhin-Turaev construction functorially yields a vector space . In particular, this vector space is (non-canonically) isomorphic to ⟨V 1 , . . . , V k ⟩.
Definition 5.1. A coloring of a graph Γ ⊂ M is the following data: • Choice of an object V (e) ∈ Obj A for each oriented edge e ∈ E or (Γ), so that V (e) = V (e) * .
• Choice of a vector ϕ(v) ∈ Z RT (S 2 v ; V (e 1 ), . . . , V (e n )), for each interior vertex v, where e i are the edges incident to v, taken with outward orientation (pointing away from v).
If M has boundary, then we can color each boundary vertex of Γ with the color of the incident edge (taken with outgoing orientation). The pair (B, {V b }) of the set of boundary vertices with a coloring is the boundary value of Γ. We will denote Graph(M, V) = set of all colored graphs in M with boundary value V and similarly consider formal linear combinations: It follows from result of Reshetikhin and Turaev that for every colored graph Γ in a ball D ⊂ R 3 , one can define its "evaluation" where e 1 , . . . , e n are the edges of Γ meeting the boundary of D (legs), taken with outgoing orientation; in particular, in the case when Γ is a star graph in the unit ball in R 3 , with one vertex at the center colored by ϕ ∈ Z RT (S 2 v ; V (e 1 ), . . . , V (e n )), then ⟨Γ⟩ = ϕ. 1 We call a formal linear combination of colored graphs Γ = ∑ c i Γ i ∈ VGraph(M, V) a null graph if there exists en embedded closed ball D ↪ M such that all Γ i meet ∂D transversally, all Γ i coincide outside of D as colored graphs, and ⟨Γ⟩ D = c i ⟨Γ i ⟩ D = 0 (Note D is allowed to touch the boundary ∂M .) We will say Γ is null with respect to D.
We can now give the main definition of this section: We can now define the category of boundary conditions. Definition 5.3. Let Σ be an oriented surface, possibly non-compact. Suppose first Σ has no boundary. DefineẐ CY (Σ) as the category whose objects are finite subsets B ⊂ Σ, together with a framing and coloring V b ∈ Obj A for each point b ∈ B; we will use the notation V = (B, {V b }) for such an object (suppressing the framing), and we call B the set of marked points of V. Define the morphisms inẐ CY (Σ) by where V * , V ′ means the boundary condition obtained by putting points b ∈ B on the "top" Σ × {1}, colored by objects V * b for outgoing legs (and thus colored by V b for incoming legs), and putting points b ′ ∈ B ′ on the "bottom" N × {0}, colored by objects V b ′ for outgoing legs.
Z CY (Σ) is additive and k-linear. We denote by It is immediate from the definition that for a 2-disk D 2 , Z CY (D 2 ) ≃ A.

Generalities of Skein Modules and Categories of Boundary Values
In this section, we consider properties of skein modules and categories of boundary values that are common for both the Turaev-Viro theory and Crane-Yetter theory. Subsection 6.1 is focused on the space of relations (i.e. the null graphs N ⊂ VGraph(Y, V)), in particular how they are generated. In subsection 6.2, we exhibit a "stacking" monoidal structure on the category of boundary values of manifolds of the form P × (0, 1), and show it to be pivotal.
Throughout this section, n = 1 or 2. We will use Z,Ẑ to denote either Z TV ,Ẑ TV (when n = 1) or Z CY ,Ẑ CY (when n = 2), so that Z(n-manifold) is a category, and Z((n+1)-manifold; V) is a vector space. A is spherical fusion for n = 1, and is premodular for n = 2. Denote by I = (0, 1), the open interval.
6.1. Skein Modules. Recall that a null graph in Y is null with respect to some (n + 1) ball D, and D is allowed to touch the boundary ∂Y . In future applications, it will be convenient to only consider balls D that do not meet ∂Y , such balls can be displaced by ambient isotopy but balls meeting ∂Y may not. Boundary vertices are univalent, so graphs have simple behavior near the boundary. If we exclude balls D that meet ∂Y , the resulting space of null graphs N ′ will be strictly smaller than N , but not by much; the following lemma says we just need to include equivalence of graphs under ambient isotopy rel boundary: Lemma 6.1. Let Y be an (n + 1)-manifold, possibly with boundary or non-compact, and let V ∈ ObjẐ(∂Y ) be a fixed boundary value. Define N ′ ⊂ N ⊂ VGraph(Y, V) to be the subspace generated by graphs that are null with respect to a ball that does not meet the boundary ∂Y . Define N ′′ ⊂ VGraph(Y, V) to be relations obtained by ambient isotopy, i.e. generated by graphs Γ 1 − Γ 0 , where Γ t = ϕ t (Γ), ϕ t is a compactly-supported ambient isotopy fixing ∂Y . Then N = N ′ + N ′′ .
Proof. It suffices to show that N ⊂ N ′ + N ′′ . Let Γ = ∑ c i Γ i be a null graph with some boundary value V, null with respect to a ball D ⊂ Y , and suppose D meets the boundary ∂Y . We would like to shrink D to not meet ∂Y while maintaining that Γ be null with respect to it. Clearly if D does not meet any point in V then we can do this, and then Γ ∈ N ′ .

13
Suppose D does contain some boundary vertex b ∈ V. For each i, apply a small ambient isotopy ϕ t i supported in a small neighborhood of b so that the resulting graphs ϕ 1 i (Γ i ) agree in a (possibly smaller) Then we can push D slightly inwards away from the boundary at b, and note that this new graph Γ ′ = ∑ c i ϕ 1 i (Γ i ) will be null with respect to the deformed D. This reduces the number of points in V that D contains, so after performing this finitely many times, we are back to the case considered above where D does not contain any boundary vertices. Thus we see that repeated applications of isotopies (i.e. relations in N ′′ ) takes Γ to another graph Γ ′ ∈ N ′ ; in other words, Γ ∈ N ′′ + N ′ .
The following lemma says that isotopies can be broken into a sequence of "smaller" ones: Lemma 6.2. Let ϕ t be an isotopy of diffeomorphisms ϕ t ∶ Y → Y that is supported on some compact set K. Let {U i } be a finite open cover. Then there exists a sequence of isotopies ϕ t j such that each ϕ t j is supported on some U aj ∩ K, and the isotopies concatenate to give a piecewise-smooth isotopy from ϕ 0 to ϕ 1 .
Proof can be found in [EK1971, Corollary 1.3]. In other words, given two diffeomorphisms ϕ 0 , ϕ 1 that are isotopic, there is another sequence of isotopies that takes ϕ 0 to ϕ 1 such that each is supported on a subset of Y . One can make the new isotopies as close to the original isotopy as needed.
Finally, we show that the subspace of null graphs are spanned by those that are null with respect to "small" balls. More precisely, Proposition 6.3. Let Y be an (n + 1)-manifold, possibly with boundary or non-compact. Let {U i } be a finite open cover of Y . Let V ∈ ObjẐ(∂Y ) be a fixed boundary value. Define N i ⊂ N ⊂ VGraph(Y, V) to be the subspace of null graphs in Y with boundary value V that are null with respect to some closed ball D contained in U i . Then the space of null graphs is generated by N i 's, i.e.

N = N i
Proof. Let Γ = ∑ c j Γ j ∈ N be a null graph. By Lemma 6.1, Γ can be written as a sum of null graphs Γ ′ + Γ ′′ , where Γ ′ = ∑ c ′ j Γ ′ j is a sum of graphs, each Γ ′ j is null with respect to some ball not meeting ∂Y , and Consider one such Γ ′′ j , and suppose that ϕ t ∶ Y → Y is an ambient isotopy supported on a compact subset K ⊂ Y , such that (Γ ′′ j ) t = ϕ t (Θ) for some graph Θ. By Lemma 6.2, there is a sequence of isotopies ϕ t k , such that each ϕ t k is supported on some U a k ∩ K, and the isotopies concatenate to give a piecewise-smooth isotopy from ϕ 0 to ϕ 1 . Then Γ ′′ Now consider a term Γ ′ j in Γ ′ , and suppose it is null with respect to some ball D not meeting ∂Y . There exists an ambient isotopy of identity ϕ t ∶ Y → Y that moves D into some open set U a . Then ϕ 1 (Γ ′ j ) ∈ N a . But by the same argument as above, ϕ 1 (Γ ′ j ) − Γ ′ j ∈ ∑ N i . Hence, we conclude that Γ ′ = ∑ c j Γ ′ j ∈ ∑ N i , and we are done.

Categories of Boundary Values.
Lemma 6.4. Let X 1 , X 2 be n-manifolds without boundary, possibly non-compact. Let ϕ ∶ X 1 → X 2 be an orientation-preserving embedding. Then ϕ induces an obvious inclusion functor that sends objects to their image under ϕ, and sends morphisms to their image under ϕ × id I . This descends to the Karoubi envelopes ϕ * ∶ Z(X 1 ) → Z(X 2 ) Furthermore, an isotopy ϕ t ∶ X 1 → X 2 induces a natural isomorphism from ϕ 0 * to ϕ 1 * , and isotopic isotopies induce the same natural isomorphisms.
Lemma 6.5. Under the same hypothesis above, Proof. The proof forẐ is clear: the inclusions of X 1 and X 2 into X 1 ⊔ X 2 together induceẐ(X 1 ) ⊠Ẑ(X 2 ) → Z(X 1 ⊔ X 2 ), and this is easily seen to be an isomorphism of categories. The equivalence for Z then follows by universal property, and the fact that the Deligne-Kelly tensor product of two finite semisimple abelian categories is also a finite semisimple abelian category.
Finally we discuss the "stacking" monoidal structure of some special n-manifolds. Let P be a (n − 1)manifold without boundary, possibly disconnected (with finitely many components) or non-compact. For n = 1, P is just a collection of points. For n = 2, P is a collection of open intervals and circles.
Let I = (0, 1), and let m ∶ I ⊔ I → I be x 2 on the first I and (x + 1) 2 on the second I. This is part of an A ∞ -space structure, as defined in [Sta1963]: m is not associative, but there is a "straight line" isotopy m t 3 ∶ I ⊔ I ⊔ I → I from m 0 3 = m ○ (m ⊔ id I ) to m 1 3 = m ○ (id I ⊔m), relating two extreme ways of including three intervals into one.
Proposition 6.6. There is a monoidal structure onẐ(P × I) given as follows: • The tensor product is ⊗ ∶=m * ∶Ẑ(P × I) ⊠Ẑ(P × I) →Ẑ(P × I) • The unit 1 is the empty configuration. (Left, right unit constraints are given in proof.) • The associativity constraint α is the natural isomorphism that is induced bym t 3 . Similarly, there is a monoidal structure on Z(P × I).
Proof. Left unit constraint l A ∶ A⊗1 → A is given by a "straight line" graph, likewise for right unit constraint. That α satisfies the pentagon relations follows from the fact that any two inclusions I ⊔4 ↪ I are isotopic, and any two isotopies are themselves isotopic. The result for Z(P × I) follows from universal property.
Proposition 6.7. The monoidal structure onẐ(P × I) and Z(P × I) given in Proposition 6.6 is pivotal.
Remark 6.8. The input category A has to be spherical, but the resulting categories Z(P × I) may not be; in future work, we will show that Z(S 1 × I) is pivotal but not spherical.
Proof. It suffices to prove this forẐ(P × I), since its Karoubi envelope will inherit the pivotal structure.
Denote by υ the map that takes P × I × [0, 1], squeezes it in half along the I direction, bends it like an accordion so that the left side collapses, and puts it back in P × I × [0, 1] so that the top and bottom are now attached to the top (see Figure 3).
The left evaluation and coevaluation morphisms for V are obtained by applying υ and η to id V , respectively. Similarly, the right evaluation and coevaluation morphisms for V are obtained by applying υ ′ and η ′ to id V , respectively. It is easy to see that these morphisms have the required properties.
Given a morphism f ∈ HomẐ (P ×I) (V, V ′ ) represented by a graph Γ, it is easy to check that its left and right duals are given by applying the rotation Θ to Γ, and keeping all orientations and labels of the edges of Γ.
The pivotal structure is essentially the identity morphism, but with one vertex on each vertical line labeled by δ, the pivotal structure of A.
Example 6.9. We pointed out at the end of Section 4 that Z TV (I) ≃ A. Giving Z TV (I) the stacking monoidal structure above, we see that this equivalence is a tensor equivalence respecting the pivotal structure.
Example 6.10. Similarly, we had Z CY (I × I) ≃ A. I × I can stack in two ways, along the first copy of I (horizontal stacking) or the second (vertical stacking). They both give monoidal structures equivalent to A's.
Proposition 6.11. The E 1 -algebra structure of Z(P × I) is unique in the sense of Theorem 2.4; that is, any automorphism of P × I induces an E 1 -algebra self-equivalence on Z(P × I).
Proof. It is not hard to see that it suffices to consider P connected. For P = * , I, S 1 , the space of selfdiffeomorphisms of P is connected, so any fiber-preserving automorphism of P × I is isotopic to the identity, so the induced functor of the automorphism is an equivalence of E 1 -algebras.
Next we consider (left) module categories over Z(P × I). First, I is a left module over the A ∞ space I, as follows. Let f ∶ I → (1 2, 1) ⊂ I be some inclusion that is identity near 1. The embedding n = (⋅ 2)⊔f ∶ I⊔I → I gives left multiplication, and it is associative up to some isotopy, that is, the two inclusions n 0 3 ∶= n ○ (id I ⊔n) and n 1 3 ∶= n ○ (m ⊔ id I ) are isotopic via some isotopy n t 3 . It is not hard to see that any two such left module structures are equivalent. Now let X be a collared n-manifold, i.e. we have an embedding P × I ↪ X, where the 0 end in I escapes to infinity in X. Crossing with P , we can upgrade the above left module structure on I to X, obtaining a left multiplicationñ ∶ P × I ⊔ X → X and an isotopyñ t 3 fromñ ○ (id P ×I ⊔ñ) toñ ○ (m ⊔ id X ). (See Lemma 2.2.) Proposition 6.12. Given a collared n-manifold X, there is a leftẐ(P × I)-module category structure on Z(X) given by ⊳ ∶=ñ * ∶Ẑ(P × I) ⊠Ẑ(X) →Ẑ(X) and the associativity constraint is given by the natural isomorphism induced by the isotopyñ t 3 . Such a structure is unique up to equivalence.
Similarly there is a left Z(P × I)-module category structure on Z(X).
Proof. Similar to Proposition 6.6.
There is a similar story for right module structure, where X is a collared n-manifold so that 1 escapes to infinity.

Excision for Z TV , Z CY
In this section, we prove the main result of the paper, that Z TV and Z CY satisfy excision. As in the previous section, essentially the same proof works for both the Turaev-Viro and Crane-Yetter theory, so we adopt the same notation as before, namely Z,Ẑ stands for either of the theories.
Let X be an n-manifold without boundary, with finitely many components, possibly non-compact. To present X as the quotient of some n-manifold X ′ by some gluing, consider a smooth function f ∶ X → S 1 = R 2Z, together with a trivialization of P -bundles P × I ≃ f −1 (I) for some (n − 1)-manifold P . Take X ′ to be the "preimage of (0, 3) under f "; more precisely, pullback f along the universal covering map R → R 2Z to getf ∶X → R, and take X ′ =f −1 ((0, 3)) (see figure below). So X is obtained from X ′ by gluing the parts over (0, 1) and (2, 3).
The natural gluing map X ′ → X is the composition X ′ ⊂X → X. We can also embed X ′ in X as follows: consider a following sequence of maps X ′ → P × I ⊔ X ′ ⊔ P × I → X ′ → X; the first map is just the obvious inclusion, the second is the left and right module maps "squeezing" X ′ into itself, and the third map is the natural quotient map. It is easy to see that the composition is an embedding, in fact a diffeomorphism onto X f −1 (0.5). We denote this composition by i.
Proof. Consider a map Ψ ∶ X ′ × [0, 1] → X × [0, 1] described as a composition of operations given by the figures below (with further explanations later): The first figure depicts X ′ ×[0, 1], with the [0, 1] factor going in the vertical direction. The foliation depicted consists of the obvious horizontal leaves X ′ ×{r}; we depict the foliation only to better explain the operations we perform below. The left and right parts are the P × I portions that would glue to give X. Operation (1) "pinches" the left vertical side down and the right vertical side up. Operation (2) "squeezes" the bottom left and top right portions. Operation (3) glues the two vertical sides.
Below we depict a graph Γ in X ′ × [0, 1] with incoming boundary value A ⊳ M and outgoing boundary value N ⊲ A, representing an element of Hom hTr(Ẑ(X ′ )) (M, N ). It is sent to a graph Ψ(Γ) in X × [0, 1] with incoming boundary M and outgoing boundary N , representing an element of HomẐ (X) (M, N ).
1 Note that operation (1) creates corners in the top left and bottom right, so Ψ is not exactly a smooth map; however, it is an embedding when restricted to X ′ × (0, 1), and can easily be slightly perturbed to be a smooth embedding. As we ultimately care about the images of graphs Ψ(Γ) up to isotopy, we will not bother with the details of this perturbation nor the non-smoothness of Ψ at the corner.
The only points in X × [0, 1] that are hit more than once are in f −1 (0.5); we call this the seam. In the figure above, the seam is depicted as the vertical dotted line in the right most figure. The seams is also the image of the top left and bottom right boundary pieces (the parts labeled A). The image of Ψ X ′ ×(0,1) is exactly X × (0, 1) seam.
We claim that the following map is well-defined: It is not hard to see that the assignment Γ ↦ Ψ(Γ) yields a well-defined map Hom Â Z(X ′ ) (M, N ) → HomẐ (X) (i * (M ), i * (N )); a graph Γ = ∑ c i Γ i that is null with respect to some ball D would have image Ψ(Γ) null with respect to Ψ(D). We need to check that the relations ∼ in Hom hTr(Ẑ( ψ ∈ HomẐ (P ×I) (B, A). We see that We leave checking that composition is respected as a simple exercise.
We want to show that i * is an equivalence, and will be considering Ψ −1 applied to graphs. It is not clear that this is well-defined, e.g. moving parts of a graph in X × [0, 1] across the seam could result in different graphs with different boundary conditions in X ′ × [0, 1]. However, the relation Θ ○ (ψ ⊳ id M ) − (id N ⊲ ψ) ○ Θ essentially takes care of this ambiguity.
Suppose a graph Γ in X ×[0, 1] intersects the seam L 0 transversally, in that the edges meet L 0 transversally and no vertices are on L 0 . Then Γ defines a boundary value at the seam: the marked points are the points of intersection, and coloring is the color associated to the edge taken with right-ward orientation (that is, in direction of ν). In particular, the boundary value of i * (Γ) in the figure above is A. If Γ intersects L α transversally, then we can also define its boundary value at L α similarly; to be precise, it is the boundary value of ζ −α (Γ) at the seam. Lemma 7.3. Let Γ be a graph in X × [0, 1] that represents a morphism in HomẐ (X) (i * (M ), i * (N )) for some M, N ∈ Obj hTr(Ẑ(X ′ )). Choose some α such that Γ is transverse to L α , and suppose it defines the boundary value A α . We see that Then as a morphism in Hom hTr(Ẑ(X ′ )) (M, N ), Ψ −1 α (Γ) is independent of such a choice of α. Proof. Clear from the picture.
We come to the main "topological" result of the paper: Theorem 7.4. The extension i * ∶ hTr(Ẑ(X ′ )) →Ẑ(X) is an equivalence.
Proof. It was already evident from the object map that hTr(Ẑ(X ′ )) →Ẑ(X) is essentially surjective -it only misses objects that have marked points on f −1 (0.5), but such an object is isomorphic to an object with points moved slightly off of f −1 (0.5).
Let 0 < β < 0.5 be such that i * (M ) and i * (N ) do not have any marked points in f −1 ((0.5−β, 0.5+β)) ⊂ X; denote J = (0.5 − β, 0.5 + β). Consider the open cover {U 1 , U 2 } of X × [0, 1], where U 1 = f −1 (J) and U 2 = X × [0, 1] L 0 . By Proposition 6.3, the space of null graphs is generated by graphs that are null with respect to balls D contained in either U 1 or U 2 , thus it suffices to show that Φ sends such graphs to 0. By the previous observation, it suffices to check that there exists an α that doesn't intersect such D.
For D ⊂ U 2 , such L α exists by Sard's theorem -for small enough α, L α does not intersect D, so it suffices to consider transversality with Γ, which is a generic condition. Now suppose D ⊂ U 1 . Since there are no marked points on the boundary in U 1 , by Lemma 6.1, we may assume that D does not meet the boundary. As we pointed out, the vector field ν defining the isotopy ζ α has the property that it will displace D off of L 0 . So if ζ α (D) does not intersect L 0 , we can take L −α+ε , where small ε is chosen to get transversality with Γ, and we are done.
Combining the topological result above with the algebraic results of Section 3, we have the main result of the paper: Theorem 7.5. There is an equivalence In particular, when X = X 1 ∪ X 2 as in Remark 7.1, Proof. We claim that Z(P × I) is multifusion; we justify this claim later. By Proposition 6.7,Ẑ(P × I) is pivotal. In reference to the notation in Section 3, take C ′ =Ẑ(P × I), C = Z(P × I), M ′ =Ẑ(X ′ ), M = Z(X ′ ).
Then we have Z Z(P ×I) (Z(X ′ )) ≃ Kar(hTrẐ (P ×I) (Z(X ′ ))) by Corollary 3.12, ≃ Kar(hTrẐ (P ×I) (Ẑ(X ′ ))) by Lemma 3.13, ≃ Kar(Ẑ(X)) by extending i * from Theorem 7.4 to Kar = Z(X) 19 The second statement follows from the first by applying Lemma 6.5 and (3.4). Now we need to justify Z(P × I) being multifusion. This is true for P = { * } and for P = I. By Example 8.2, which uses the argument above for P = I, we have Z(S 1 × I) ≃ Z(A) as a k-linear abelian category (but not as a monoidal category, see Remark 8.3); in particular, this implies that it is semisimple with finitely many simple objects. Since by Proposition 6.7, the stacking monoidal structure on Z(P × I) is rigid and pivotal, this shows that Z(P × I) is a pivotal multifusion category.
So Z(P × I) is pivotal multifusion for any connected P ; the claim follows for a disjoint union of finitely many such P 's.
Corollary 7.6. In each of the two cases below • n = 1, A a spherical fusion category • n = 2, A a premodular category for an n-manifold X the category Z(X) of boundary values for colored graphs constructed above coincides with the factorization homology ∫ X A.
Proof. We verify that Z(−) satisfies the three characterizing properties laid out in Theorem 2.4.
(1) For both cases of n, A defines an n-disk algebra in V = Rex, thus defining factorization homologies ∫ − A which coincide with Z(−) on the n-disk.
(2) Proposition 6.11 proves the uniqueness of the E 1 -algebra structure on Z(P × I).
Corollary 7.7. In the assumptions of Corollary 7.6, Z(X) is a finite semisimple category.
Proof. Any connected X can be built from I n by a sequence of gluings of collared manifolds. For example, for n = 2, gluing opposite edges of a square gives an annulus, and gluing boundaries of the annulus together gives the torus. By Theorem 7.5, the corresponding category Z(X) thus can be constructed from the Deligne product of several copies of Z(I n ) ≃ A by repeatedly applying the center construction, replacing a category M by Z Z(P ×I) (M). Since it was shown in the proof of Theorem 7.5 that Z(P × I) is pivotal multifusion, it now follows from Proposition 3.5 that applying the center construction always gives a finite semisimple category. Thus, Z(X) is a finite semisimple category.

Examples and Computations
In this section, we present some examples and computations using the results obtained so far.
Here the A loop is given a trivial (e.g. always horizontal) framing. It is clear from this picture that EndÂ(1) is commutative.
By Proposition 6.6, Ann = S 1 × I has a horizontal stacking operation that, under the equivalenceÂ ≃ Z CY (Ann) above, is given by a map Hom A1 This stacking operation gives rise to the monoidal structure that is defined in Proposition 6.6, where we take P = S 1 .
Remark 8.3. Note that the stacking operation in Example 8.2 does not result in the usual tensor product on the Drinfeld center Z(A) (the latter can be defined as the functor assigned to pair of pants in Turaev-Viro theory). It is explored in more detail in [YHT2020], where the tensor product is defined purely in terms of A (i.e. without recourse to topology); it is shown that the stacking tensor product is typically not spherical (but pivotal) and not fusion (but multifusion).
Next we will be concerned with relating Z CY of a surface Σ with that of a punctured one Σ 0 , that is, Σ 0 = Σ {p}. We will think of Σ as obtained from Σ 0 by gluing with an open disk, "sealing" the puncture: Σ = Σ 0 ∪ D 2 , implicitly choosing some collared structure on Σ 0 and D 2 .
RecallÂ ∶= hTr(A) from Example 8.2. There is a right action of HomÂ(1, 1) on the morphisms of Z CY (Σ 0 ), by "pushing in" from the puncture, i.e. HomẐ It is easy to see that for Γ ∈ HomẐ CY (Σ0) (Y, Y ′ ) and f, g ∈ HomÂ(1, 1), HomÂ(1, 1). (Note: D and simples X i are of A, and not of Z(A).) π is an idempotent in HomÂ(1, 1), and hence also acts as an idempotent on HomẐ CY (Σ0) (Y, Y ′ ). Proposition 8.4. Let Σ 0 = Σ {p} as above. Consider the categoryB consisting of the same objects aŝ Z CY (Σ 0 ), but morphisms given by Then the restriction toB of the inclusion functor corresponding to i ∶ Σ 0 ↪ Σ is an equivalence: Proof. First note thatB is indeed closed under composition of morphisms because π is idempotent. It is clear that i * B is essentially surjective. To prove fully faithfulness, consider two objects Y, Y ′ ∈Ẑ CY (Σ 0 ). Abusing notation, we also denote i We construct an inverse map to i * . Let U be a small open neighborhood of p in Σ, and let N = U × [0, 1] ⊂ Σ × [0, 1] be a small open neighborhood of the pole. Choose U small enough so that it does not contain any marked points of Y, Y ′ . Consider a graph Γ ∈ Graph(Σ × [0, 1]; Y * , Y ′ ). Define j(Γ) as follows: if Γ intersects the pole, then use an isotopy supported in N to push Γ off of it, resulting in a new graph Γ ′ . Now Γ ′ can be considered a graph in Graph(Σ 0 × [0, 1]; Y * , Y ′ ). Then we define j(Γ) = Γ ′ ⊲ π.
We need to check that j is well-defined. Firstly, the (linear combination of) graphs Γ ′ ⊲ π is independent of the choice of isotopy -this follows from the sliding lemma (Lemma 10.5). More generally, it means that for any isotopy ϕ of Σ × [0, 1] supported on N , j(Γ) = j(ϕ(Γ)). Finally, it is easy to see that j is inverse to i * . For example, i * ○ j amounts to adding a trivial dashed circle, which is equivalent to 1 by Lemma 10.6.
Corollary 8.5. Z CY (S 2 ) ≃ Z Mü (A), the Müger center of A, and in particular, when A is modular, Proof. Think of the disk D 2 as a punctured sphere, so by Proposition 8.4, we have thatẐ CY (S 2 ) ≃B, wherê B is the category with the same objects asẐ CY (D 2 ) ≃ A, but morphisms are, for A, A ′ ∈ A, In particular, when A = A ′ = X i a simple object, it follows from [Müg2003, Corollary 2.14] that simple objects that are not transparent, i.e. not in the Müger center, are killed: It follows thatB coincides with the Müger center, which is already abelian, and so Z CY (S 2 ) ≃ Kar(B) = Z Mü (A).

Crane-Yetter and the Elliptic Drinfeld Center
In [YHT2019], the second author constructed a category similar to the Drinfeld center, but instead the objects have two half-braidings that satisfy some compatibility. In this section, we show that this category is the category of boundary values on the once-punctured torus.
We note that all morphisms depicted using graphical calculus are over A, but they may represent morphisms in a different category. In particular, dashed lines do not need an orientation and in makes sense to use the circular α instead of the semicircular one.
For the reader's convenience, we recall the definition and some properties of the elliptic Drinfeld center: Definition 9.1. Let A be a premodular category. The category Z el (A) consists of objects of the form (A, λ 1 , λ 2 ), where λ 1 , λ 2 are half-braidings on A that satisfy We call the relation (9.1) "COMM". The morphisms Hom Z el (A) ((A, λ 1 , λ 2 ), (A ′ , µ 1 , µ 2 ) are morphisms of A that intertwine both half-braidings, i.e. Hom where α is defined in Lemma 10.7.
On morphisms, f ∈ Hom A (A, A ′ ), We refer to [YHT2019] for the functorial isomorphisms giving the adjunction. Furthermore, I el is dominant.
Theorem 9.4 ([YHT2019], Theorem 4.3). When A is modular, there is an equivalence where Γ is the half-braiding on I(X) in Theorem 3.3, and Ω = c −1 Proposition 9.5. Let T 2 0 be the once-punctured torus. There is an equivalence The left most figure shows how Z CY (D 2 ) ≃ A is a module category over Z CY (I × I) ≃ A in four ways; we think of the 1,2 edges as acting on the left, 3,4 edges as acting on the right. The actions are just usual left and right multiplication. By Theorem 7.4, the first "glue 1,3" arrow induces an equivalenceẐ CY (Ann) ≃ hTrẐ CY(I ×I) (Ẑ CY (D 2 )) ≃ hTr A (A) (see also Example 8.2). Again by Theorem 7.4, the second "glue 2,4" arrow induces an equivalencê Z CY (T 2 0 ) ≃ hTrẐ CY(I ×I) (Ẑ CY (Ann)) ≃ hTr A (hTr A (A)). Let us give a more explicit description of the last equivalence. For A, A ′ ∈ Obj A, Under the equivalence, a morphism ϕ ∈ Hom A (B 1 ⊗ B 2 ⊗ A, A ′ ⊗ B 2 ⊗ B 1 ), shown on the left in the figure below, is sent to the graph in T 2 0 × [0, 1] shown on the right: Now we define a functor hTr A (hTr A (A)) → Z el (A). On objects, it sends A ↦ I el (A). On morphisms, the morphism in (9.2) is sent to It is clear that this assignment respects composition of morphisms. The following sequence of isomorphisms shows that this functor is fully faithful: Since Z el (A) is abelian, we have that the extension to the Karoubi envelope is an equivalence: ) ≃ Z el (A) and we are done. But before we end the proof, we provide an explicit inverse functor that will be useful later: on objects, (9.4) (A, λ 1 , λ 2 ) ↦ im(P (A,λ 1 ,λ 2 ) ), where P (A,λ 1 ,λ 2 ) ∶= 1 D 2 where the equality of diagrams follows from the COMM requirement (9.1), and the dashed line represents a weighted sum over simples (see Appendix). On morphisms, Thus we have a 2-commutative diagram In particular, when A is modular, we have an equivalence incl. * ∶ Z CY (D 2 ) ≃ → Z CY (T 2 0 ). Our next task is to upgrade this equivalence to an equivalence of left Z CY (Ann)-modules. In the rightmost figure in (9.6), the gray area is a collar neighborhood of the puncture of T 2 0 . By Proposition 6.12, there is a (left) Z CY (Ann)-module structure onẐ CY (T 2 0 ): on objects, while on morphisms, the module structure, employing the equivalences of (9.2) and (8.1), is given as follows: (9.7) The D-labeled strand originally goes around the annulus in Ann × [0, 1]; after inserting into T 2 0 × [0, 1], it wraps around like the gray area in (9.6)). This extends to a left Z CY (Ann)-module structure on Z CY (T 2 0 ). Similarly, there is a leftẐ CY (Ann)-module structure onẐ CY (D 2 ) (which extends to Z CY ): In light of (9.5), the following theorem is an upgrade of Theorem 9.4: Theorem 9.6. Let A be modular. There is an equivalence of left Z CY (Ann)-modules Proof. Under the equivalence Z el (A) ≃ Z CY (T 2 0 ), it is easy to see that the equivalence of Theorem 9.4 can be rewritten as where the isomorphism is essentially given byP ′ A andP ′ A from Lemma 10.9 (with M = C), which is easily seen to be natural in A.
Then we see that where we use the sliding lemma (Lemma 10.5) for both equalities, and isotopies to move the strands around. The final diagram is what one obtains if we apply ψ to ϕ ∈ Hom A (A, A ′ ) first and then send it to Z CY (T 2 0 ). Hence, the equivalence does respect the module structure and we are done.
Finally, we state the main result of this section: Theorem 9.7. Let A be modular. Let Σ be a connected compact oriented surface with b boundary components and genus g, and let S 0,b = S 2 (D 2 ) ⊔b be a genus 0 surface with b boundary components. Then In particular, Z CY (closed surface) ≅ Z CY (S 2 ) ≅ Vec and Z CY (once-punctured surface) ≅ Z CY (D 2 ) ≅ A.
Proof. Suppose g > 0, so that we can present Σ as a connect sum Σ ′ #T 2 , where Σ ′ is a connected compact oriented surface with b boundary components and genus g − 1. We think of the connect sum as Σ = Σ ′ 0 ∪ Ann (T 2 0 ), where Σ ′ 0 = Σ ′ {pt} is a punctured surface. Then by Theorem 7.5 and Theorem 9.6, . Thus, by induction on the genus, we have Z CY (Σ) ≃ Z CY (S 0,b ).
The final statements follow from the b = 0, 1 cases and Corollary 8.5.
Remark 9.8. Here is an alternative proof to Theorem 9.7 and Theorem 9.6, pointed out by Jin-Cheng Guu, which avoids passage to the elliptic Drinfeld center and may be of independent interest. Recall the well-known equivalence when A is modular [Müg2003]. This can be interpreted as an equivalence which, using the equivalences established in Example 8.2, is given by By similar reasoning as in the proof of Theorem 9.6, this can be shown to be a Z CY (Ann)-bimodule equivalence. Thus, performing a surgery (replacing an annulus with two disks or vice versa) does not affect Z CY of a surface. In particular, this yields Z CY (Σ) ≃ A ⊠b where b is the number of boundary components in Σ.

Appendix: Pivotal Multifusion Categories Conventions
This appendix is dedicated to notation and basic results about pivotal multifusion categories. It is adapted from [Kir], modified to accommodate for the non-spherical non-fusion case. We also point the reader to [EGNO2015, Chapter 4] and [ENO2005] for further reference.
Let C be a k-linear pivotal multifusion category, where k is an algebraically closed field of characteristic 0. In all our formulas and computations, we will be suppressing the associativity and unit morphisms; we also suppress the pivotal morphism V ≃ V * * when there is little cause for confusion.
We denote by Irr(C) the set of isomorphism classes of simple objects in C, and denote by Irr 0 (C) ⊆ Irr(C) the subset of simple objects appearing in the direct sum decomposition of the unit object 1; it is known that 1 decomposes into a direct sum of distinct simples, so End(1) ≅ ⊕ l∈Irr0(C) End(1 l ). We fix a representative X i for each isomorphism class i ∈ Irr(C); abusing language, we will frequently use the same letter i for denoting both a simple object and its isomorphism class. Rigidity gives us an involution − * on Irr(C); it is known that l * = l for l ∈ Irr 0 (C). For l ∈ Irr 0 (C), we may use the notation 1 l ∶= X l to emphasize that it is part of the unit.
For k, l ∈ Irr 0 (C), let C kl ∶= 1 k ⊗ C ⊗ 1 l , so that C = ⊕ k,l∈Irr0(C) C kl . Any simple X i is contained in exactly one of these C kl 's, or in other words, there are unique k i , l i ∈ Irr 0 (C) such that 1 ki ⊗ X i ⊗ 1 li ≠ 0. Since C * kl = C lk , we have that k i * = l i . When C is spherical fusion, the categorical dimension is a scalar, defined as a trace, but here the nonsimplicity of 1 and non-sphericality complicates things. To avoid confusion, denote by δ ∶ V → V * * the pivotal morphism. The left dimension of an object V ∈ Obj C is the morphism Similarly, the right dimension of V is the morphism Note that these are vectors and not scalars, since 1 may not be simple. It is easy to see that d R V = d L V * = d L * V . When C is spherical, we will drop the superscripts.
When V = X i is simple, we can interpret its left and right dimensions as scalars as follows. We have X i ∈ C kili , so Hom(1, X i ⊗X * i ) = Hom(1, 1 ki ⊗X i ⊗X * i ) ≃ Hom(1 ki , X i ⊗X * i ), and likewise Hom(X i ⊗X * i , 1) ≃ Hom(X i ⊗X * i , 1 ki ), so d L Xi factors through 1 ki , and hence we may interpret d L Xi as an element of End(1 ki ) ≅ k. Similarly, d R Xi may be interpreted as an element of End(1 li ) ≅ k. We denote these scalar dimensions by d L i , d R i , and fix square roots such that d L i = d R i * . The dimensions of simple objects are nonzero. The dimension of C kl is the sum By [ENO2005, Proposition 2.17], this is the same for all pairs k, l ∈ Irr 0 (C), and by [ENO2005, Theorem 2.3], they are nonzero.
We will show a morphism ϕ ∈ ⟨V 1 , . . . , V n ⟩ by a round circle labeled by ϕ with outgoing edges labeled V 1 , . . . , V n in counter-clockwise order, as shown in Figure 4. By (10.4) and the fact that z n = id, this is unambiguous. We will show a morphism ϕ ∈ ⟨V 1 , . . . , V n ⟩ l by a semicircle labeled by ϕ and l as shown in Figure 4; in contrast with a circular node, a semicircle imposes a strict ordering on the outgoing legs, not just a cyclic ordering.
We will make two additional conventions related to the graphic presentation of morphisms. When C is spherical, the orientation of such a dashed line is irrelevant.
Notation 10.3. Let C be spherical. If a figure contains a pair of circles, one with outgoing edges labeled V 1 , . . . , V n and the other with edges labeled V * n , . . . , V * 1 , and the vertices are labeled by the same letter α (or β, or . . . ) it will stand for summation over the dual bases: where ϕ α ∈ ⟨V 1 , . . . , V n ⟩, ϕ α ∈ ⟨V * n , . . . , V * 1 ⟩ are dual bases with respect to pairing (10.6). When C is not spherical, the pairing is no longer compatible with z from (10.4), so such notation can only make sense with semicircles: where ϕ α ∈ ⟨V 1 , . . . , V n ⟩ l , ϕ α ∈ ⟨V * n , . . . , V * 1 ⟩ l are dual bases with respect to the pairing (10.6). The following lemma illustrates the use of the notation above.
Lemma 10.4. For any V 1 , . . . , V n ∈ C, we have Proof of this lemma is straightforward: first show it for simple X, then for direct sums; interested reader can find a proof for spherical C in [Kir].
Lemma 10.5. The following is a generalization of the "sliding lemma": Proof. Let Irr kl = Irr(C kl ), and let Irr k * ∶= ⋃ l Irr(C kl ), i.e. the set of simples X i such that 1 k ⊗ X i = X i . Then = k∈Irr0(C) i∈Irr k * (C) The second equality is proved in a similar manner.
The following lemma is used to prove that Figure 1 is a half-braiding and the functor G in the proof of Theorem 3.10 respects composition: