A topological origin of quantum symmetric pairs

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z/2Z-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Main examples are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra.


Introduction
In this paper we introduce and study the operad of involutive little disks. We show that its operations naturally encode the symmetries of quantum groups and quantum symmetric pairs. Quantum groups are of course well-studied and known to have myriad connections to other parts of mathematics. Only in recent years, however, have quantum symmetric pairs come to the forefront of research on quantum groups.
A quantum symmetric pair consists of a quantum group together with a subalgebra that quantizes the subgroup of fixed-points of some given involution. In the 90s, examples of quantum symmetric pairs were constructed on a case by case basis by M. Noumi, T. Sugitani and M. Dijkhuizen using solutions to the reflection equation [Nou96,NS95,NDS97]. This equation, sometimes called the 'boundary Yang-Baxter equation', was introduced by I. Cherednik in the context of particle scattering on a half line [Che84] and E. Sklyanin's study of quantum integrable systems with non-periodic boundary conditions [Skl88,KS92]. The motivation of the work of Noumi-Sugitani-Dijkhuizen was to study q-orthogonal polynomials (e.g. Askey-Wilson polynomials [AW95], Macdonald polynomials [Mac01] and Koornwinder polynomials [Koo92]) as zonal spherical functions of quantum symmetric spaces; this field is sometimes called 'quantum harmonic analysis' [Koo90,Koo91,Dij96]. A complete classification of quantum symmetric pairs was later achieved by G. Letzter [Let99,Let02,Let03]. S. Kolb then extended the theory of quantum symmetric pairs to include Kac-Moody algebras, with examples such as twisted q-Yangians and q-Onsager algebras [Kol14].
In recent years the reflection equations appeared in a more methodical way in the theory of quantum symmetric pairs. S. Kolb and J. Stokman showed that an invertible solution to a type A reflection equation is exactly a character of the reflection equation algebra O q (SL n ). From such a character they construct a quantum symmetric pair, streamlining the approach of Noumi-Sugitani-Dijkhuizen [KS09]. Furthermore, M. Balagovic and S. Kolb showed that any pair in Letzter's classification carries a canonical solution to a reflection equation, called a universal K-matrix [BK,Kol17]. Such solutions can be used for applications to low-dimensional topology in the spirit of the celebrated Reshetikhin-Turaev knot invariants [tD98,tDHO98].
A categorical approach to the reflection equations, as well as Letzter's classification of quantum symmetric pairs, allowed for applications to geometric representation theory. D. Jordan and X. Ma used quantum symmetric pairs to construct representations of the double affine braid group and of the double affine Hecke algebra of type C ∨ C n [JM11]. M. Ehrig and C. Stroppel showed that translation functors in parabolic category O in type D categorify the action of a quantum symmetric pair [ES13]. Work by H. Bao and W. Wang on type AIII/AIV quantum symmetric pairs, see Table 1, shows such quantum symmetric pairs admit a canonical basis [BW16]. Furthermore, they have a geometric interpretation through the geometry of partial flag varieties of type B/C [BSWW16], and are Schur-Weyl dual to Hecke algebras of type B [BWW16].
One of the fundamental properties of the Drinfeld-Jimbo quantum group is that it is a quasi-triangular Hopf algebra: it has a universal R-matrix providing solutions to the Yang-Baxter equation. We can rephrase this by saying that the category of finite dimensional modules of a quantum group is a braided monoidal category. In this paper we introduce a categorical framework, called a Z 2 -braided pair, which exactly encodes the structure of the universal K-matrix of a quantum symmetric pair; analogous to how a braided monoidal category encodes the R-matrix.
It is well known that braided monoidal categories are exactly the categorical algebras over the so-called E 2 -operad [Fie,Dun97,Wah01]. 1 Recall that the E 2 -operad, or operad of little two-dimensional disks, is a topological operad whose operations are parametrized by the different embeddings of a disjoint union of disks into a larger disk. In the identification of categorical E 2 -algebras with braided monoidal categories the braiding (i.e. the universal R-matrix) is interpreted as an operation rotating two disks in the plane. In this paper we propose an interpretation for the universal K-matrix of Balagovic and Kolb in terms of Z 2orbifold disks rotating around each other, see Figure 3. This naturally leads us to consider an operadic perspective on quantum symmetric pairs in which the E 2 -operad is replaced by an Z 2 -orbifold analogue. We therefore introduce the Z 2 D 2 -operad of involutive little twodimensional disks, and study its categorical algebras. Our main result is that Z 2 -braided pairs are exactly the categorical Z 2 D 2 -algebras.
Recall that an E 2 -algebra A can be used to construct invariants of surfaces by computing the so-called factorization homology of the surface with coefficients in A [Lur17,AF15]. In [BZBJ1,BZBJ2], the authors compute the surface invariants of factorization homology with coefficients in the braided monoidal category of quantum group representations. These invariants produce quantizations of the character variety of the surfaces (the moduli space of local systems on that surface). In §1.3 we explain forthcoming work in which we compute the factorization homology of Z 2 -orbifold surfaces with coefficients in the Z 2 -braided pair of representations of a quantum symmetric pair. Thereby, we will construct invariants of surfaces Σ with an involution and actions of the associated orbifold braid groups B n [Σ/Z 2 ]. We expect that these invariants give quantizations of the character varieties of orbifold surfaces with isolated Z 2 -singularities.
1.1. Summary of results. Central to our results is the notion of a Z 2 -braided pair. A Z 2 -braided pairs consists of a braided monoidal category, a module category and some extra structure which we now recount. See §3 for the definitions of braided monoidal category and module category. (3) A Z 2 -symmetric pair is a Z 2 -braided pair such that σ 2 = id and κ 2 = id.
We will call the natural isomorphism κ the Z 2 -cylinder braiding as its topological interpretation is the braid around the singular point in the orbifold cylinder [D 2 /Z 2 ] (see Figure 1). Crucially, Definition 1.1 implies that the Z 2 -cylinder braiding κ and the braiding σ satisfy the Φ-twisted reflection equation (1.10.1).
Remark 1.2. The notion of a Z 2 -braided pair is very close to the notion of a Φ-braided module category of [Kol17,Remark 3.15] and braided module category of [Bro13]. However, in the definition of Z 2 -braided pair we had to impose additional requirements to match the topology of the Z 2 D 2 -operad. For a precise comparison see Remarks 3.2 and 3.6.
Our main result interprets the axioms in Definition 1.1 via topological operads Z 2 D n . These operads have two colours D and D * , that represent the free quotient (D n ∐ D n )/Z 2 where Z 2 swaps the disks, and the orbifold quotient D n /Z 2 where Z 2 acts by rotations (see Figure 4). The operations in Z 2 D n are parametrized by equivariant embeddings of disjoint unions of the orbifold disks D and D * into each other. Consider a categorical Z 2 D n -algebra (A, M), where A corresponds to D and M corresponds to D * . Disjoint unions of the disk D embed into D, which determines a multiplication structure on A. Moreover, D embeds into D * , but not conversely. This determines an A-module structure on M. Our main result is the following classification, relating the categorical Z 2 D n -algebras to Definition 1.1. (1) A Z 2 D 1 -algebra in Cat is exactly a Z 2 -monoidal pair.
(2) A Z 2 D 2 -algebra in Cat is exactly a Z 2 -braided pair.
(3) Let n ≥ 3. A Z 2 D n -algebra in Cat is exactly a Z 2 -symmetric pair.
Remark 1.4. We stated the main theorem in terms of Cat for simplicity. We prove the main theorem for -linear categorical algebras since -linear categories provide a more natural setting for representation theory, see e.g. [EGNO15]. However, our proofs are readily adaptable to Cat, where they yield the above result.
Convention. We refer to plain categories with a tensor product as monoidal categories, whereas we refer to -linear categories with a tensor product as tensor categories. 2 Remark 1.5. We use ∞-categories to define Z 2 D n -algebras in general. Since Cat is a only 2-category we will delay discussing the ∞-categorical foundations to §6. The reader only interested in applications to representation theory can safely avoid reading that section.
Remark 1.6. Recall that for n ≥ 3 a categorical E n -algebra is the same thing as a categorical E ∞ -algebra (i.e. a symmetric monoidal category). This is an instance of the Baez-Dolan stabilisation hypothesis [BD95]. As one would expect from the stabilisation hypothesis, categorical Z 2 D n -algebras also stabilise when n ≥ 3. The main results [BK,Corollary 9.6] and [Kol17, Corollary 3.14] state that the quantum symmetric pair coideal subalgebra is quasi-triangular. As a corollary we then have: Corollary 1.8. Any quantum symmetric pair together with the standard ribbon element of the quantum group satisfies the hypothesis of Proposition 1.7.
In order to prove Theorem 1.3 we establish various coherence theorems which may be of independent interest. A priori, to specify a categorical Z 2 D n -algebra one needs to provide an infinite amount of data, corresponding to the infinitely many operations in the operad Z 2 D n , the isotopies between these operations, the isotopies between isotopies, and so on. Theorem 1.3 states all this data can be reduced to the finite list of functors, isomorphisms and equations in Definition 1.1. Concretely, one needs to show that all the functors and natural isomorphisms one can construct using these generators make the correct diagrams commute, as specified by the operad Z 2 D n . This is exactly the content of a coherence theorem. We can state our coherence results informally as follows: Theorem 1.9. Let (A, M) be a Z 2 -monoidal/braided/symmetric pair. By a coherence diagram we mean a diagram in A or M constructed using the functors and natural isomorphisms that are part of Definition 1.1.
(1) Any coherence diagram in a Z 2 -monoidal pair commutes. (Theorem 4.19) (2) A coherence diagram in a Z 2 -braided pair commutes if the underlying braids agree. 3 (Theorem 4.18) (3) Any coherence diagram in a Z 2 -symmetric pair commutes. (Theorem 4.20) 1.2. Quantum symmetric pairs and the reflection equations. We now provide some further background on quantum symmetric pairs and the reflection equations.
An infinitesimal symmetric pair (g, g θ ) consists of a complex semisimple Lie algebra g and a sub Lie algebra g θ ⊂ g fixed by some involutive automorphism θ : g → g. Irreducible symmetric pairs 4 were classified by S. Araki [Ara62]. The classification of type A symmetric pairs is recorded in Table 1. A quantum symmetric pair (U q (g), U (q,c) (g θ )) should then be a quantization of the pair (U(g), U(g θ )). 3 The underlying braid of a natural isomorphism is obtained by interpreting instances of the braiding σ and κ as generators of the cylinder braid group B cyl n (Definition 2.3). 4 A symmetric pair is called irreducible if it cannot be obtained as the symmetric pair associated to a non-trivial product of two symmetric pairs. Table 1. The classification of irreducible symmetric pairs in type A. Here w 0 denotes the involution induced by the longest word in the Weyl group.
Notation. Notation for quantum symmetric pairs varies in the literature. Some authors denote the coideal subalgebra by B c . The c denotes a multi-parameter as there can be multi-parameter families of quantizations of U(g).
It turns out that requiring U (q,c) (g θ ) to be a sub-Hopf algebra of U q (g) is too strong a requirement, rather this quantization will be a (left) coideal subalgebra i.e. a subalgebra satisfying ∆(U (q,c) (g θ )) ⊂ U (q,c) (g θ ) ⊗ U q (g).
1.2.1. The reflection equation and cylinder braids. The first constructions of quantum symmetric pairs in [NS95,NDS97,Dij96] depended crucially on solutions to various versions of the reflection equation: where R denotes a given solution of the Yang-Baxter equation. Like the Yang-Baxter equation, the reflection equation has an interpretation in low-dimensional topology, see Figure 2. Similar to the Reshetikhin-Turaev invariants, one can use solutions of the reflection equation to construct knot invariants [tD98,tDHO98]. Solutions to the gl n -reflection equation were studied by J. Donin, A. Mudrov and P.P. Kulish [DKM03] and completely classified by A. Mudrov [Mud02]. It turns out such solutions can be viewed as characters of the reflection equation algebra O q (G). In [KS09], characters of the reflection equation algebra are used to reconstruct the type AIII/AIV quantum symmetric pairs of Noumi-Sugitani-Dijkhuizen.
Remark 1.10. The reflection equation algebra O q (G) is an (equivariant) quantization of the Semenov-Tian-Shansky Poisson bracket on O(G) [Mud06]. This algebra goes by many names: Majid's braided dual of U q (g) [Maj95], the quantum-loop algebra [AS96] and is isomorphic to the locally finite part of U q (g) via the Rosso form [KS09, Proposition 2.8].
1.2.2. The reflection equations revisited. Recall that various versions of the reflection equation appeared in the works [Nou96,NS95,NDS97,Dij96]. It was only later realised by M. Balagovic and S. Kolb, through their algebraic construction of universal K-matrices, that there is a general framework of twisted reflection equations which unifies all different reflection equations [BK,Remark 9.7]. For a quasi-triangular Hopf algebra H with coideal subalgebra B one fixes an additional Hopf algebra involution φ of H, such that (φ ⊗ φ)(R) = R, called the twist. A (φ-twisted) universal K-matrix is a universal solution to the φ-twisted reflection equation: Remark 1.11. The twist φ for a given quantum symmetric pair is determined by the Dynkin data that characterises it in Araki's classification [Ara62]. See Table 1 for twists in type A.
Remark 1.12. Quantum symmetric pairs for which the twist is trivial, φ = id, have universal K-matrices that provide solutions to the untwisted reflection Equation (1.9.1). This explains why in [KS09] characters of O q (G) could only be used to construct quantum symmetric pairs of type AIII/AIV : such characters solve the untwisted reflection equation.
The twist is naturally built into the categorical framework of Z 2 -braided pairs via the involution Φ. Recall from Figure 1 that we can interpret the Z 2 -monodromy in the operad Z 2 D 2 as a coloured braid. The twisted reflection equation is then naturally interpreted in the operad Z 2 D 2 as an isotopy of coloured braids, see Figure 3. Therefore, we can now recognise that the twists φ, which had an algebraic origin but lacked intrinsic topological meaning, are an integral part of the topological interpretation of quantum symmetric pairs.
1.3. Outlook: factorization homology and quantum symmetric pairs. The constructions in this paper are the first step in a program where quantum symmetric pairs are used for applications to low-dimensional topology. The next steps will involve (developing and) applying a general framework called factorization homology.
Factorization homology was first introduced by J. Lurie [Lur09b,Lur17] as a topological variant of the chiral homology of Beilinson-Drinfeld [BD04], and was further developed by J. Francis, D. Ayala and H. Tanaka [AF15,AFT17]. Nowadays there is a large body of literature on factorization homology and related ideas, see e.g. [And10,Gwi12,GTZ14,MW12,CG17,Hor17]. See [Gin15] for a survey. The formalism of factorization homology, in its simplest form, takes as input an E n -algebra A in a symmetric monoidal higher category C and associates to every framed n-manifold M an invariant denoted M A. We will informally explain the assignment. Following [AF15], we view the manifold as glued from all the framed disks embedding into M, and view the E n -algebra A as a functor A : Disk fr n → C from some higher category of framed disks into C. The invariant is then defined as follows: We record some key properties of factorization homology: (1) Factorization homology is uniquely characterised by an excision property. After decomposing a manifold along a collared boundary, one can compute the global invariant of the manifold in terms of the invariants of the pieces [AF15, §3.5]. 5 (2) The invariant is functorial with respect to embeddings. In particular Diff fr (M) acts on M A. Moreover, for a categorical E n -algebra A and X 1 , . . . , X n ∈ A the braid group B n (M) acts on an associated object M (X 1 ⊗ . . . ⊗ X n ) ∈ M A.
(3) There are many versions of factorization homology e.g. for oriented manifolds and manifolds with singularities [AF15, AFT17]. (4) For a given E n -algebra A, the factorization homology of n-manifolds with coefficients in A defines a fully extended n-dimensional TQFT [Sch14]. Most relevant to our work are the applications of factorization homology to representation theory via the factorization homology of braided tensor categories as developed in [BZBJ1,BZBJ2]. In [BZBJ1,BZBJ2], the authors compute invariants of genus g framed and oriented surfaces using factorization homology with coefficients in the braided tensor category of quantum group representations. For example, the invariant assigned to the annulus is the category of representations of the reflection equation algebra O q (G) and the invariant assigned to the torus is the category of strongly equivariant quantum D-modules on G. Via Key Property (2) the authors obtain braid group actions of oriented surfaces. Such surface braid group actions were previously constructed by 'generators and relations' methods in [Jor09] and used to construct representations of the type A double affine Hecke algebra (abbreviated DAHA). The type A DAHA, due to I. Cherednik [Che05], is a quotient of the group algebra of the torus braid group B n (T ) by additional Hecke relations. The braid group actions constructed in [BZBJ1,BZBJ2] recover the surface braid group actions of [Jor09] and provide an intrinsic topological explanation for their existence.
To connect these developments to our work on quantum symmetric pairs and the Z 2 D 2operad we make the following observation. The colimit formula (1.12.1) defining factorization homology is motivated by the fact that a n-manifold M is covered by disks D n . Correspondingly, a Z 2 -orbifold surface [Σ/Z 2 ] with isolated singularities is covered by the orbifold disks D and D * , which appear in the definition of the operad Z 2 D 2 . Hence it is natural to associate invariants to such orbifold surfaces constructed from a Z 2 D 2 -algebra via a formula like Equation (1.12.1). In the follow-up paper [Wee2] we make this idea precise by introducing Γ-equivariant factorization homology, where Γ can be an arbitrary finite group.
Recall that in [JM11] representations of the type C ∨ C n DAHA were constructed using quantum symmetric pairs of type AIII/AIV. These constructions were made using a generators and relations approach alike [Jor09], and similarly lacked a topological interpretation.
The DAHA of type C ∨ C n arises as a quotient of the orbifold surface braid group B n [T 2 /Z 2 ], where the Z 2 action on the torus T 2 is induced by rotating the fundamental domain. In future work we hope to recover the DAHA representations in [JM11] from the braid group actions arising in the equivariant factorization homology of the orbifold torus [T 2 /Z 2 ] with coefficients in a quantum symmetric pair. Moreover, the general construction of equivariant factorization homology will allow us to immediately generalise the constructions in [JM11] to other quantum symmetric pairs, not necessarily of type AIII/AIV. This provides important motivation for our work on quantum symmetric pairs and factorization homology.
1.4. Organisation. The contents of this article are laid out as follows.
In Section 2 we introduce and study the operads Z 2 D n . We first recall the definition of orbifold configuration spaces and then define the operads Z 2 D n . We compute the homotopy type of the spaces of operations of the operads Z 2 D n in terms of orbifold configuration spaces.
Section 4 is a technical section where we prove Theorem 1.9. The reader uninterested in methods for proving a coherence result can safely skip the section. To prove the theorem we follow the 'strictification implies coherence' approach of Joyal and Street [JS93].
Section 5 is devoted to proving Theorem 1.3, though two further results from Section 6 are needed to complete the proof. We begin by defining the 2-category Rex and the Deligne-Kelly tensor product. We then use the coherence results of Section 4 to assign categorical Z 2 D n -algebras for n = 1, n = 2, n ≥ 3 to respectively Z 2 -monoidal, Z 2 -braided and Z 2 -symmetric pairs. We also construct assignments in the opposite direction.
Finally, in Section 6 we recall the definitions in [Lur17] of algebras over ∞-operads. We show that in the special case of Z 2 D n -algebras in Rex we recover the definition of categorical Z 2 D n -algebras we gave in §5. We conclude by finishing the proof of Theorem 1.3 by showing that the assignments constructed in §5 are inverse equivalences.
A note to the reader. It is well known that categorical E 2 -algebras are braided monoidal categories. For a proof in the strict setting see [Wah01], and for a proof using ∞-categories and Dunn Additivity see [Lur17]. As there is no Dunn Additivity for Z 2 D 2 -algebras we had to directly prove that categorical Z 2 D 2 -algebras are Z 2 -braided pairs. A subset of our arguments can be used to give a direct proof that categorical E 2 -algebras are braided monoidal categories. As we are unaware of such a proof in the literature, this may be of independent interest.
Acknowledgements. First and foremost, I wish to thank David Jordan for his invaluable guidance. I would also like to thank Stefan Kolb for patiently answering many questions about quantum symmetric pairs. This work was supported by the Engineering and Physical Sciences Research Council [grant number 1633460].
2. The involutive little disks operad 2.1. Orbifold configuration spaces. Orbifold configuration spaces will be fundamental to our understanding of the Z 2 D n -operads. We now recall the definitions of configuration spaces for manifolds and global quotient orbifolds and their associated braid groups.
Definition 2.1. Let Σ be some surface.
(1) The configuration space of k ordered points in Σ is denoted F n (Σ) and defined as The symmetric group S k acts freely on F k (Σ) by swapping points.
(2) The configuration space of k unordered points in Σ is denoted C k (Σ) and defined to be the quotient space The braid group on n strands of Σ is denoted B k (Σ) and defined to be the fundamental group π 1 C k (Σ).
Recall that for a given group π an Eilenberg-MacLane space of type K(π, 1) is a topological space S that has trivial homotopy groups except π 1 (S) ∼ = π.
Example 2.2. We recall the following classical examples of Eilenberg-MacLane spaces: ( The group B cyl k of cylinder braids admits a presentation with generators σ 1 , . . . , σ k , κ and relations We now give a definition of orbifold configuration spaces. The definition is a slight adaptation of the 'orbit configuration spaces' of M. Xicoténcatl [Xic97]. 6 Definition 2.4. Let Γ be a finite group acting smoothly and faithfully on a surface Σ. Denote Σ free ⊂ Σ the subset of smooth points i.e. the subset of points where Γ acts freely. (1) The configuration space of k ordered smooth points in the orbifold [Σ/G] is denoted F k [Σ/G] and defined as There is a natural free Γ ×k ⋊ S k action on and defined to be the quotient space The braid group on k strands in the orbifold [Σ/Γ] is denoted B k [Σ/Γ] and defined to be the fundamental groupπ 1 C n [Σ/Γ].
The following two Z 2 -global quotients will be our central examples throughout: Example 2.5. Fix a dimension n ≥ 1. Let D n denote the open unit disk in R n . Let Z 2 act on R n via the sign representation. We have an induced action on D n .
(1) Let D * denote the topological space D n equipped with the Z 2 action given by the sign action.
(2) Let D denote the topological space D n ∐ D n equipped with the Z 2 action given by combining the sign action with swapping the two disks. The spaces for n = 2 are illustrated in Figure 4.
Lemma 2.6. Let D and D * be the orbifolds defined above for dimension n = 2. 6 The difference between our definition and [Xic97] is that we restrict to smooth points.
As the proofs are identical we will do both at once. Let Σ be either D or D * . The pro- Notation. We will use the following notation throughout without further comment: (1) To differentiate between the two copies of D n in D we will write (2) We will view maps whose source is a disjoint union of spaces as collections of component maps, e.g. viewing f : . An embedding f : D n → D n is called rectilinear if it is of the form f (x, y) = (λx + t x , λy + t y ) for some λ, t x , t y ∈ R. We will call an embedding f : D → D, and f : D → D * rectilinear if the restrictions f r and f b are rectilinear. We will say an embedding f : We will now define a topological coloured operad Z 2 D n with two colours D and D * and which has spaces of operations Operadic composition is defined in the obvious way: compose the embeddings. Concretely, one inserts a configurations of disks into disks; an example in dimension two is illustrated in Figure 5.
Remark 2.7. Equivariance requires an embedding to map a Z 2 -fixed point to a Z 2 -fixed point. Therefore, It remains to address the topology on the sets of operations 2.6.1 and 2.6.2. For a rectilinear embedding f : D n → D n we write r(f ) for the radius of the disk f (D n ). We define maps . . , f k (0 b ) that record the positions of the centers and the radii. These maps are clearly injections. We induce the subspace topology on the sets of operations, where R k has the Euclidean topology. This endows Z 2 D n with the structure of a topological operad.
Definition 2.8. The topological coloured operad Z 2 D n , described above, with two colours D and D * and operation spaces 2.6.1 and 2.6.2 is called the involutive little n-disks operad. * * 2.3. The homotopy type of the spaces of operations. The connection between configuration spaces and the Z 2 D n -operad is established in the following easy but nevertheless important result.
Proposition 2.9. The maps that forget the radii define homotopy equivalences.
Proof. Easy adaptation of the usual proof for the E n -operad and F k (R 2 ). 7 Remark 2.10. By Proposition 2.9 we can encode isotopies in Z 2 D n (D ∐k , D * ) up to homotopy as paths in F k [D / Z 2 ]. Paths in F k [D * /Z 2 ] are naturally drawn as coloured braids, see e.g. Figures 1 and 3.
In dimension one the configuration spaces F k [D/Z 2 ] and F k [D * /Z 2 ] are particularly easy to describe. To a point p ∈ F 1 [D/Z 2 ] we assign a value (2.10.1) and to a configuration we assign a permutation σ x ∈ S k recording the order of the disks in D 1 b , see Figure 6. More precisely, σ x is the unique permutation in S k for which the inequalities Lemma 2.11. The map is a homotopy equivalence.
Similarly, to a point x ∈ F 1 [D * /Z 2 ] we can assign a value Lemma 2.12. The map defines a homotopy equivalence.

Z 2 -braided pairs
Before we state our definitions we fix notation and remind the reader of some standard definitions in categorical representation theory, for more details and background see [EGNO15]. A monoidal category is a category A together with a tensor product functor ⊗ : A × A → A, a unit object 1 ∈ A and natural isomorphisms for X, Y, Z ∈ A satisfying the Mac Lane triangle and pentagon axioms. A braided monoidal category is a monoidal category A together with a natural isomorphism σ, called the braiding, for X, Y ∈ A satisfying the Joyal-Street hexagon axioms. A monoidal functor between monoidal categories A and B consists of a functor Ψ : A → B together with natural isomorphisms for M ∈ M and X, Y ∈ A satisfying a unit and associativity axiom. We call a module category M pointed if it has a distinguished object 1 M ∈ M.
Notation. For a monoidal category A we have the opposite monoidal category A ⊗-op , which is the category A with the reversed order tensor product ⊗ op i.e. X ⊗ op Y := Y ⊗ X.
(1) An anti-involution Φ of a monoidal category A consists of a monoidal functor (Φ, Φ 2 , Φ 0 ) : A → A ⊗-op and a monoidal isomorphism t : (2) A Z 2 -monoidal pair consists of a monoidal category A together with an anti-involution Φ, and a pointed A-module category M. Remark 3.2. The definition of a Z 2 -braided pair is a Z 2 -equivariant version of the braided module categories of A. Brochier [Bro13]. A braided module category over a braided monoidal category A consists of an A-module category M together with a natural isomorphism γ : M ⊗ X → M ⊗ X for X ∈ A, M ∈ M satisfying axioms BP1 and BP2, but where the functor Φ = id A is trivial and replacing Φ 2 by the braiding σ in axiom BP2.
Essentially all examples of Z 2 -braided pairs come from the representation theory of quasitriangular Hopf algebras, and their coideal subalgebras. Figure 7. Axiom BP1 and a graphical interpretation in terms of coloured braids. Figure 8. Axiom BP2 and a graphical interpretation in terms of coloured braids.
Notation. For an algebra A we denote with A-mod the category of A-modules, and A-mod f.d. the category of finite dimensional A-modules.
Recall that for a quasi-triangular Hopf algebra H the category H-mod is naturally a braided monoidal category. A coideal subalgebra B ⊂ H is a subalgebra so that ∆(B) ⊂ B ⊗ H. The category B-mod is then naturally a module category over H-mod. To obtain a Z 2 -braided pair (H-mod, B-mod) we need some further structure on H and B.
Definition 3.3. [Kol17, Definition 2.7] Let H be a quasi-triangular Hopf algebra with universal R-matrix R ∈ H, a Hopf algebra involution φ : The raison d'etre of universal K-matrices is that they provide solutions to the twisted reflection equation Recall that a balancing on a braided monoidal category A is a natural isomorphism θ : for all M, N ∈ A. A ribbon Hopf algebra is a quasi-triangular Hopf algebra H together with a ribbon element i.e. a central invertible element ν ∈ H such that the natural isomorphism θ : id H-mod ⇒ id H-mod defined by acting with ν gives a balancing on H-mod.
Proposition 3.5. Let H be a ribbon Hopf algebra, together with a be a Hopf algebra involution φ that preserves R and v, and a quasi-triangular coideal subalgebra B. Then (H-mod, B-mod) is canonically equipped with the structure of a Z 2 -braided pair.
Proof. The category H-mod is braided monoidal since H is quasi-triangular. The involution φ : H → H defines a functor Φ : M → M φ , where M φ is the H-module where the H-action is twisted by φ. Since φ is an Hopf algebra involution that preserves R, the functor (Φ, id, id 1 ) is a braided monoidal functor. We equip Φ with the structure of an anti-involution via The balancing θ then gives a monoidal isomorphism t : Φ 2 ∼ = id A by Remark 3.4. Since φ(ν) = ν we also have Φ(θ) = θ Φ . As B is a right coideal subalgebra the category B-mod inherits the structure of a right module category over H-mod with pointing 1 ∈ B-mod. Action by the universal K-matrix defines the Z 2 -cylinder braiding.
As explained in the introduction, examples are provided by quantum symmetric pairs.
Remark 3.6. Given a braided monoidal category A, the notion of a Z 2 -braided pair (A, M) is very close to the notion of a Φ-braided module category of [Kol17]. Recall that a Φ-braided module category over A consists of an A-module category M, a braided monoidal equivalence (Φ, id, id 1 ) : A → A and a family of natural isomorphisms κ M,X : M ⊗ X → M ⊗ Φ(X) for X ∈ A and M ∈ M satisfying the axioms BP1 and BP2 in Figures 7 and 8. 9 The difference is that in a Z 2 -braided pair Φ is required to be anti-monoidal and involutive. Given a Φ-braided module category M over A such that the underlying functor Φ is strictly involutive, then the pair (A, M) defines a Z 2 -braided pair exactly if A admits a balancing. 10 Definition 3.7. A Z 2 -symmetric pair is a Z 2 -braided pair (A, M) such that σ 2 = id, κ 2 = id.
Examples of Z 2 -symmetric pairs are provided by infinitesimal symmetric pairs.
Example 3.8. Let g be a complex semisimple Lie algebra and let θ : g → g be an involution with fixed-points g θ . We can decompose θ = Jφ(−)J −1 for an outer automorphisms φ and 9 Where Φ 2 is replaced by σ in BP2. 10 In that case we can define the anti-involution via (Φ, σ, id 1 ) with t : Φ 2 ∼ = id being the balancing.

Coherence results
A coherence theorem asserts all diagrams of a certain class commute. Most famously, Mac Lane's coherence theorem for monoidal categories states any diagram 12 in a monoidal category A commutes. An important consequence is that the a-priori different (parenthesized) tensor products X 1 ⊗ . . . ⊗ X n of objects X 1 , . . . , X n ∈ A are canonically isomorphic. In a braided monoidal category A coherence is more subtle. It is not true that all diagrams commute, for example σ 2 = id in general. Thinking of σ as the simple braid on two strands provides intuition: the double braid and the trivial braid are different braids. This can be made precise: one associates a diagram of braids to a diagram in A by interpreting σ as the generators of a braid group B n . The Joyal-Street Coherence Theorem then states that a diagram in a braided monoidal category commutes if the associated diagram of braids commutes [JS93]. We will now prove a coherence theorem (1.9) for Z 2 -braided pairs. 4.1. Coherence in the strict setting. We will now precisely state the coherence theorem, and prove it in a strict setting. Recall that a monoidal category is called strict if α = id, ρ = id and λ = id. (1) We call the Z 2 -monoidal pair strict if A is a strict monoidal category with a strict anti-involution Φ, and M is a strict module category. 13 (2) We call a Z 2 -braided pair (or Z 2 -symmetric pair) strict if the underlying Z 2 -monoidal pair is strict.
We have the following simple, but important result: (1) The braiding σ satisfies the Yang-Baxter equation for all X, Y, Z ∈ A.
(2) The Z 2 -monodromy κ satisfies the twisted reflection equation for any M ∈ M and X, Y ∈ A.
Proof. The proof that the braiding satisfies the Yang-Baxter equation is standard, see e.g.
[EGNO15, 8.1.10]. To prove part (2) we observe that in the following diagram 11 Here G is the Lie group integrating g. Note that decompositions of involutions of semisimple Lie algebras are not unique in general.
the subdiagrams commute by naturality of κ, the BP2 axiom and the fact that Φ is a braided monoidal functor. Commutativity of the outer sides follows and expresses that κ satisfies the twisted reflection equation.
Before stating the coherence theorem we introduce some notation and definitions. Any structural isomorphism f in A is a natural isomorphism between functors with domain A ×n and codomain A for some n ≥ 0. To a structural isomorphism f with a given presentation as a vertical/horizontal composition we can associate its underlying braid on n strands, denoted β f ∈ B n , by interpreting instances of the braiding σ as simple braids, and doubling strings for instances of id ⊗ . See Figure 9 for two illustrated examples.
Remark 4.4. Note that the assignment of the underlying braid depends on the presentation of the structural isomorphism. Similarly a structural isomorphism f in M is a natural isomorphism between functors with domain M × A ×n and codomain M for some n ≥ 0. To such f together with a given presentation we can associate its underlying cylinder braid on n strands, denoted β f ∈ B cyl n , by interpreting instances of the braiding σ and Z 2 -cylinder braiding κ as the generators σ and κ in B cyl n , and doubling strings for instances of id ⊗ . Remark 4.5. By axioms BP1 and BP2 of Figures 7 and 8 one can change the presentation of a structural isomorphism in M so that only one string is involved in an instance of the Z 2 -cylinder braiding at a time but without changing the underlying braid.
Proposition 4.6. Let (A, M) be a strict Z 2 -braided pair and consider two parallel structural isomorphisms f, g : T 1 ⇒ T 2 in A (or M). Then f = g if β f = β g .
Proof. As both cases are analogous we will only give the proof for f and g in M, The nontrivial components of the presentations of f and g consist of instances of σ, κ and id ⊗ , id Φ . By Remark 4.5 we can change the presentation of f and g so that the underlying braids only braid two strings at a time without changing the underlying braid. 14 As β f = β g one can relate the presentations of the braids β f and β g via repeated applications of the cylinder braid relations 2.3.1 and 2.3.2. Using that Φ is braided and strict we can then also rewrite the presentations of f and g into each other by applying the cylinder braid relations. By Lemma 4.2 it follows that f = g.
We will now prove coherence for general Z 2 -braided pairs by showing that one can replace a Z 2 -braided pair (up to appropriate equivalence) by a strict Z 2 -braided pair.

4.2.
Strictification of the braided monoidal category. The strictification of A will move in two steps: first strictifying the tensor product on A and then strictifying the involution Φ.
Notation. Let A be a monoidal category, recall that A can be seen as a right A-module category in a canonical way. We denote this module category by A A . For two right Amodule categories M 1 and M 2 we denote by Fun A (M 1 , M 2 ) the category of right A-module functors, and morphisms of right A-module functors. For definitions see [EGNO15,ch. 7].
Definition 4.7. Let A be a monoidal category. We denote by A st the strict monoidal category Fun A (A A , A A ). The tensor product on objects is given by composition of functors (denoted by •), with unit object the identity functor. The tensor product on morphisms is given by horizontal composition of natural transformations (denoted by * ). Convention. We will use the following fact implicitly throughout: If F is a monoidal functor so that the functor underlying F is an equivalence, then any quasi-inverse F −1 is naturally monoidal, and the natural isomorphisms 1 ⇒ F −1 F and F F −1 ⇒ 1 are monoidal isomorphisms [EGNO15, 2.4.10]. Moreover, the natural isomorphisms can be chosen to satisfy the triangle identities of an adjunction [Mac97, §4.4].
We now transport the anti-involution along the equivalence to the strict category.
Lemma 4.9. 1. Let α : F 1 → F 2 be a monoidal isomorphism, and let H, G be monoidal functors. The natural isomorphism id H * α * id G : HF 1 G ⇒ HF 2 G is monoidal. 2. Let F : A → B be an equivalence of monoidal categories, Φ : A → A ⊗-op a monoidal functor and t : Φ 2 ⇒ id A an isomorphism of monoidal functors. Then there exists a antiinvolution Ψ : B → B ⊗-op , π : Ψ 2 ⇒ id B such that Ψ • F ∼ = F • Φ are monoidally isomorphic and the diagram

commutes.
14 Besides Remark 4.5 we are using the analogous observation about the hexagon axioms for the braiding.
Proof. The first statement is an easy verification. For the second claim we define Ψ := F ΦF −1 . As a composite of monoidal functors it is monoidal. Similarly, as a composite of adjoint equivalences it is an adjoint equivalence. One easily verifies the natural isomorphism π can defined as the composite Next we will strictify the anti-involution using a method adapted from [Gal].
Definition 4.10. Let A be a strict monoidal category, with an anti-involution (Ψ, ψ, ψ 0 ), π : Ψ 2 ⇒ id A . The category A Z 2 has as objects quadruples where X i ∈ A and η i are isomorphisms in A such that the left diagram in Figure 10 commutes consists of a pair f i : X i → Y i of morphisms in A such that the right diagram of Figure 10 commutes for i = 0, 1. Figure 10. Object and morphism diagrams in A Z 2 .
We will sometimes refer to X i (resp. η i ) the object (resp. morphism) components of X.
Lemma 4.11. The category A Z 2 is strictly monoidal with tensor product is a strict anti-monoidal functor such that Ψ 2 st = id is a monoidal isomorphism. Proof. A straightforward calculation.
We can now prove a strictification theorem for monoidal categories with an anti-involution.
Proposition 4.12. 1. The functor defines an equivalence of categories. 2. The maps equip Ω with the structure of a monoidal functor.
That Ω(f ) defines a morphism is clear. To show Ω is an equivalence we will check it's fully faithful and essentially surjective. Faithfulness is clear. For fullness, let f : Ω(X) → Ω(Y ) be any morphism. We have by definition of f being a morphism. Then f = Ω(f 0 ). Finally, Ω is essentially surjective since for any X we have the isomorphism id η 0 : X → Ω(X 0 ).
The assertions of (2) are all easy checks that follow from π, ψ, ψ 0 being natural isomorphisms and monoidality of π and Ψ.
We can also transport the braiding σ on A to the strictification. (1) Let A and A be monoidal categories and let (F, µ) : A → A be a monoidal equivalence. If A is braided, there is a unique braiding on A making F braided monoidal. (2) A monoidal equivalence F is braided monoidal iff its quasi-inverse F −1 is braided monoidal.   Notation. We denote the quasi-inverse of (L, L 2 , L 0 ) by (R, R 2 , R 0 ) and let η : 1 ⇒ RL and ε : LR → 1 be the monoidal isomorphisms of the equivalence L. Similarly, we have η : 1 ⇒ RL, ε : LR ⇒ 1.
Proof. Since L is full and faithful we can define isomorphisms (R 2 ) M,X : . Then by naturality We wish to show the upper inner hexagon commutes. The other subdiagrams commute by definition of R 2 , naturality of L 2 and Lemma 4.15. The outside of the diagram commutes since ε : LR ⇒ id is monoidal. Therefore, the inner hexagon commutes. Faithfulness of L implies R 2 has the required property.
Note that the category M st is also a strict module category over A Z 2 st with action F · X := F • X 0 , for X = (X 0 , X 1 , η 0 , η 1 ) ∈ A Z 2 st , and F ∈ M st . Lemma 4.15 holds mutatis mutandis for the composite ΩL and L. To strictify the Z 2 -braided pair it remains to transport the Z 2 -monodromy along the equivalences to A st and A Z 2 st . Proof. Let U ∈ M st and S, T ∈ A st . Since R is full and faithful the diagram defines the components K U,S uniquely. Naturality of η, R 2 and κ implies that R(K) • R(α * β) = R(α * Ψ(β))•R(K). Faithfulness of R then implies naturality of K. To show K defines a Z 2 -monodromy we need to check it satisfies axioms BP1 and BP2. We consider the diagram in Figure 11. We wish to show the inner octagon commutes. The outer diagram commutes since κ is a Z 2 -monodromy. The other subdiagrams commute by naturality, Lemma 4.16, definition of K, and R being braided monoidal. We conclude the inner octagon commutes. By faithfulness of R we find that K satisfies axiom BP1. Similarly, one can write down a diagram to show K also satisfies BP2. We leave this to the reader, or see [Wee1]. It remains to show K satisfies the diagram in the Proposition. Let M ∈ M, X ∈ A and consider the diagram The subdiagrams commute by naturality, the triangle identity, the definition of R 2 , and the definition of K. Therefore the outer diagram commutes, which is what we wished to show.
Finally, we transport the Z 2 -cylinder braiding from A st to A Z 2 st . Namely, let S = (S 0 , S 1 , η S 0 , η S 1 ) ∈ A Z 2 st and F ∈ M st we define K F, S as the composite Figure 11. A diagram chase to prove K satisfies axiom BP1.
One easily verifies that K is a Z 2 -cylinder braiding. Thus (A Z 2 st , M st ) is a strict Z 2 -braided pair.
4.4. Coherence as a corollory to strictification. Our strictification results now yield the coherence theorem as an easy corollary. Proof. We will provide the prove when f and g are structural isomorphisms with a given presentation in M. The case of structural isomorphisms in A is analogous and left to the reader. We consider the diagram with sides L(f ) and L(g) in M st . The presentation of f and g consists of a vertical composition of horizontal compositions of the a, κ, r, etcetera. Below each arrow of the presentation we build a rectangle using the diagrams in Proposition 4.8, Lemma 4.9.2, Proposition 4.12.3, Lemma 4.15, Lemma 4.16 and Proposition 4.17 so that we obtain a prism of natural isomorphisms. The lower face of the prism consists of two structural isomorphisms f st and g st in A Z 2 st , whereas the upper face of the prism is our original diagram containing f and g. It is clear from these rectangles that if β f = β g , then also β fst = β gst . By assumption β f = β g so that Proposition 4.6 implies that f st = g st . Then L(f ) = L(g) follows since the prism has vertical faces consisting of commutative diagrams of isomorphisms, and we just proved that the lower face commutes. By faithfulness of L we conclude f = g.
We also have the following two coherence results: Theorem 4.19. Let (A, M) be a strict Z 2 -monoidal pair, any two parallel structural isomorphisms are equal Proof. This is clear in a strict Z 2 -monoidal pair since all horizontal composition of identities are identities. Now we deduce coherence from the strictification results, following the technique of the proof of Theorem 4.18 Theorem 4.20. Let (A, M) be a strict Z 2 -symmetric pair, any two parallel structural isomorphisms are equal Proof. Note that the strictification (A Z 2 st , M st ) is a strict Z 2 -symmetric pair. We leave the proof, an easy adaptation of the proofs of Proposition 4.6 and Theorem 4.18, to the reader.

Classifying categorical algebras
Notation. Let be a field. We denote with Vect the category of -vector spaces.
We now analyse categorical algebras in Rex, a setting of -linear categories well suited to higher algebra. By a -linear category we mean a category enriched over Vect. We introduce Rex following the exposition given in [BZBJ1].
Notation. We denote with Vect f.d. the -linear category of finite dimensional -vector spaces.
Notation. For a (higher) category C and X, Y ∈ C we sometimes write C(X, Y ) for Hom C (X, Y ).
Recall that a functor is called right exact if it preserves finite colimits. A category is essentially small if it is equivalent to a small category. 15 Definition 5.1. Rex is the 2-category of -linear essentially small categories that admit finite colimits with morphisms right exact functors and 2-morphisms natural isomorphisms.
Example 5.2. For any -algebra A the category A-mod f.d. is in Rex.
For C, D, E k-linear we let Bilin(C × D, E) denote the category of -bilinear functors from C × D to E that preserve finite colimits in each variable separately.  Convention. From now on any functor between categories in Rex will be assumed to be -linear and right exact without further comment.
Two operations in Z 2 D n are called isotopic if there is a path in the space of operations connecting them; we will refer to this path as an isotopy. If β : g ⇒ g ′ is an isotopy of operations and (γ i : f i ⇒ h i ) k i=1 a collection of isotopies then we call the associated isotopy β * (γ 1 ∐ · · · ∐ γ k ) : g • (f 1 ∐ · · · ∐ f k ) ⇒ g ′ • (h 1 ∐ · · · ∐ h k ) the horizontal composition. A 2-isotopy between two isotopies is a homotopy between the two paths in the space of operations.
Notation. For a permutation σ ∈ S k we abuse notation by also writing σ for the induced permutation maps σ : D ∐k → D ∐k and σ : A ⊠k → A ⊠k where A ∈ Rex. More generally, for a collection of operations the map (resp. functor) σ permutes the codomains of the f i (resp. F i ).
Definition-Proposition 5.5. An Z 2 D n -algebra F in Rex is a choice of categories A, M ∈ Rex together with assignments (1) For every operation f assigned functors F (f ) as follows where A ⊠0 := Vect f.d. .
(2) For every isotopy between compositions of operations (g, (f i ) k i=1 , h) and permutation σ ∈ S k an assigned natural isomorphism as follows Subject to the conditions that: (i) identity operations get assigned identity functors, (ii) for every operation f the constant isotopies such that there exists a 2-isotopy between the composite isotopies filling the square then the following equation holds between the assigned natural isomorphisms Remark 5.6. We will view Z 2 D n as an ∞-operad to define Z 2 D n -algebras in general. Definition-Proposition 5.5 unpacks that definition for Rex. The proof is delayed until §7.
Various special cases of condition (iii) determine the 'pseudofunctoriality' of Z 2 D n -algebras: Lemma 5.7. Let F be a Z 2 D n -algebra in Rex.
(1) Let α : f ≃ g be an isotopy between operations, then F ( are assigned the same natural isomorphism denoted F (α).
(5) The constant isotopy (6) The two composites of natural isomorphisms between F m Convention. A right exact functor F : Vect f.d. → C in Rex is uniquely determined by the image of ∈ Vect f.d. . We will identify F with F ( ) ∈ C without further comment.
In the coming three subsections we will construct assignments of -linear categorical Z 2 D nalgebras for n = 1, n = 2, n ≥ 3 out of -linear Z 2 -monoidal, Z 2 -braided, Z 2 -symmetric pairs, and vice versa. That these assignments are inverse equivalences is shown in Section 6.
Remark 5.8. We delay proving the assignments are equivalences because we don't want to write down all the data here that determines an isomorphism of Z 2 D n -algebras in Rex.
5.1. Z 2 -tensor pairs versus Z 2 D 1 -algebras. By a tensor category we will a monoidal category A such that A ∈ Rex and ⊗ ∈ Bilin(A × A, A).
Remark 5.9. By definition of the Deligne-Kelly product ⊠ the tensor product ⊗ corresponds to a functor ⊗ : A ⊠ A → A. A tensor category is thus exactly an E 1 -algebra in Rex.
A right module category over a tensor category A is an A-module category M such that M ∈ Rex and act ∈ Bilin(M × A, M).
Definition 5.10. A Z 2 -tensor pair consists of a tensor category A with an anti-involution Φ and a right A-module category M with a pointing 1 M ∈ M.
Remark 5.11. One obtains a Z 2 -monoidal pair from a Z 2 -tensor pair by forgettingenrichment. As forgetting -enrichment is faithful 16 Theorems 4.18, 4.19, 4.20 automatically yield coherence theorems for their -linear tensor category analogues.
Proposition 5.12. To a Z 2 D 1 -algebra F in Rex there is an associated Z 2 -tensor pair (A F , M F ).
Proof. We set A F := A and M F := M. Consider the fixed operations Table 2. The operations are assigned functors that for which we introduce the following suggestive notation, respectively: ⊗, id A , Φ, 1, act, id M and 1 M . Note that by Definition-Proposition 5.5 (i) the functors id A and id M are indeed the identity functors on A and M.
Next we need to show there exist natural isomorphisms α, λ, ρ, Φ 2 , Φ 0 , t, r, a that are part of the definition of a Z 2 -tensor pair. One can define all the isomorphisms analogously, therefore we only define α leaving the others to the reader. Consider the composite operations f ⊗ •(f ⊗ ∐id D ) and f ⊗ •(id D ∐f ⊗ ). Observe that the two composites define isotopic operations since they lie in the same connected components lying over (0, 0, 0, id) ∈ {0, 1} ×3 × S 3 in the notation of Lemma 2.11. Choose an isotopy f α between them, then by Lemma 5.7.1 we have an associated natural isomorphism F f α ) : Using the natural isomorphisms of Lemma 5.7.5 we can define a natural isomorphism α by requiring that the diagram

commutes.
To show that our functors and natural isomorphisms equip (A, M) with the structure of a Z 2 -tensor pair, we need to check that the natural isomorphisms satisfy the defining equations of a Z 2 -tensor pair. Since all such equations can be treated similarly, we will verify one case in detail leaving the others to the reader. We will verify that α, λ and ρ satisfy the triangle axiom of a monoidal category. By definition the triangle axiom requires that for every X, Y ∈ A the diagram commutes. This is equivalent to asking that the equation holds. Let us consider the isotopy id f ⊗ * (id id D ∐ f ρ ) and choose a composite isotopy id f ⊗ * (f λ ∐ id id D ) • f α * id id D ∐f 1 ∐id D . Note that both these isotopies are paths between the same operations, and recall that by Lemmas 2.9 and 2.11 the space of operations Z 2 D n (D 3 , D) 16 Two morphisms in a -linear category are equal iff they are equal in the underlying plain category.
contracts onto a discrete space. We conclude the two isotopies are 2-isotopic, so that their assigned natural isomorphisms are equal by Lemma 5.7.3. Now consider the diagram and note that the arrows ∼ = are unambiguous by Lemma 5.7 part 6. Then all inner squares commute by Lemma 5.7 and the definitions of λ, ρ and α. The inner triangle commutes by Lemma 5.7 parts 3 and 4. Hence the outer sides of the diagram commute which expresses Equation (5.12.1) holds. Operation Visualisation Proof. We will abbreviate F (A,M) by F . First we define the assignment of functors to embeddings. To do that we introduce some notation. Recall we can assign a tuple (ε 1 , . . . , ε k ) ∈ {0, 1} ×k and a permutation σ ∈ S k to operations via the Lemmas 2.9, 2.11 and 2.12. We denote Φ 0 := id A and Φ 1 := Φ, and ⊗ 0 := Vect f.d.

1
− → A, ⊗ 1 := id A and ⊗ n := ⊗•(⊗ n−1 ⊠id A ) : We can then define the associated functors to operations in Z 2 D n (D ∐k , D) by and define the associated functors to operations in Z 2 D n (D ∐k , D * ) by and define the associated functors to operations in Z 2 D n (D * ∐ D ∐k , D * ) by Note this assignment satisfies Definition-Proposition 5.5 (i).
Next we will assign the natural isomorphisms. To an isotopy between operations g • (σ f • (∐f j )) and h we will assign a structural isomorphisms between the assigned functors F (g) • (σ • (⊠F (f j )) and F (h). Combining Theorem 4.19 and Remark 5.11 we know there is at most one structural isomorphism between two given functors. Hence to unambiguously specify the assignment it suffices to show that there exists a structural isomorphism. We will only discuss one particular case of an assignment, leaving the others to the reader since they are all analogous. Consider operations g • (σ F • (∐ j f j )) and h with codomain D that are isotopic. Note that then all the f j must also be operations with codomain D. We consider their images under the maps of Lemma 2.9 and 2.11: σ j (n) mod 2 for any j, 1 ≤ n ≤ k j .
Hence F (h) and F (g) • (σ f • (⊠ j F (f j ))) multiply a tuple of objects (X 1 , . . . , X k ) in A in the same order, and with the same distribution of Φ's (modulo 2). Then we can define a structural isomorphism between F (h) and F (g)•(⊠ j F (f j )) via repeated applications of α, φ, and t (and possibly φ 0 , λ and ρ in case one or more of the k j = 0). Note that our assignment of natural isomorphisms satisfies Definition-Proposition 5.5 (ii). It remains to check condition (iii). Consider a diagram of isotopies and operations as in condition (iii) together with a 2-isotopy. The two sides of Equation (5.5.1) are structural isomorphisms by construction of our assignments. Thus the two sides are parallel structural isomorphisms. Then they must agree by Theorem 4.19 combined with Remark 5.11. 5.2. Z 2 -braided pairs versus Z 2 D 2 -algebras.
Definition 5.14. A Z 2 -braided tensor pair is a Z 2 -tensor pair (A, M) together with a braiding σ on A, such that Φ is braided monoidal, and a natural isomorphism such that the diagrams of Figures 7 and 8 commute for all M ∈ M, X, Y ∈ A.
To study categorical Z 2 D 2 -algebras we will make use of an ordering on R 2 that subdivides the spaces of operations.
Definition 5.15. Let D and D * be of dimension n > 1.
Remark 5.16. Please note that the assignments of Definition 5.15 do not define continuous maps. (1) To a path γ in F k [D/Z 2 ] there is a canonically associated braid [γ] ∈ B n such that Proof. We will prove part two, and leave part one to the reader as the proof is analogous. Consider the non-continuous projection and observe that the set-theoretic fibers of p define contractible subspaces . Given a path γ in F k [D * /Z 2 ] with endpoints γ 0 ∈ F i 0 and γ 1 ∈ F i 1 we choose a path e 0 lying in F i 0 from x i 0 to γ 0 and a path e 1 lying F i 1 from γ 1 to x i 1 . Recall that we have the quotient map we then define [γ] := [q(e 1 * γ * e 0 )] ∈ π 1 C k [D * /Z 2 ], q(x 0 ) 2.6 = B cyl k . Since the subspaces F i are contractible one easily verifies the assignment is independent of the choice of e 0 and e 1 and respects composition and inversion.
Remark 5.18. Consequently, Lemma 5.17 together with the homotopy equivalences of Proposition 2.9 allows us to associate braids to isotopies of operations in Z 2 D 2 . Moreover, note that 2-isotopic isotopies are assigned the same braids.
Proposition 5.19. To a Z 2 D 2 -algebra F in Rex there is an associated Z 2 -braided tensor pair (A F , M F ).
Proof. We set A F := A and M F := M. Consider the fixed operations f ⊗ , f Φ , f act , in Figure  12, the identity operations id D and id D * and the unique operations f 1 ∈ Z 2 D 2 (∅, D) and f 1 M ∈ Z 2 D 2 (∅, D * ). These operations are by F assigned functors that we denote respectively ⊗, Φ, act, id A ,id M , 1, and 1 M . Note that by Definition-Proposition 5.5 (i) the functors id A and id M are indeed the identity functors on A and M.
Next we need to show there exist natural isomorphisms α, λ, ρ, σ, Φ 2 , Φ 0 , t, r, a, κ that are part of the definition of a Z 2 -braided tensor pair. Our strategy is as follows. To define a structural isomorphism η, we choose an isotopy γ such that the associated braid [γ] is equal to the underlying braid β η , and then define η as the induced isomorphism of F (γ). Let us treat the case of σ, leaving the others to the reader. Consider the 'swap' permutation τ = (12) ∈ S 2 . We choose the isotopy i σ from f ⊗ to the composite operation f ⊗ • (τ • (id D ∐ id D )), given by rotating the disks 180 degrees around each other. By definition of Z 2 D 2 -algebra the assignment We then define σ := F (i σ ).
It remains to check that the natural isomorphisms satisfy the defining equations of a Z 2braided tensor pair. As in the proof of Theorem 5.12 one easily deduces that the equations hold from the existence of various 2-isotopies. For example, consider axiom BP 1. The isotopy of coloured braids drawn there can easily be 'fattened' to give a 2-isotopy in Z 2 D 2 (D ∐2 ∐ D * , D * ). The existence of this 2-isotopy implies a, Φ 2 , κ and σ satisfy axiom BP1. We leave the details to the reader. Proof. We will abbreviate F (A,M) by F . Via Definition 5.15 we can assign a permutation σ ∈ S k and a tuple (ε 1 , . . . , ε k ) ∈ {0, 1} ×k to an operation f . We then assign functors F (f ) to operations via the formulae in Equations 5.13.1 -5.13.3. Next we assign natural isomorphisms to isotopies. To an isotopy γ between operations g • (σ f • (∐f j )) and h we will assign a structural isomorphisms between the assigned functors F (g) • (σ • (⊠F (f j )) and F (h), as we did in the proof of Theorem 5.13. Via Lemma 5.17 and Remark 5.18 we have an associated braid [γ] to γ. Recall that by Theorem 4.18 there is at most one structural isomorphism between F (g) • (σ • (⊠F (f j )) and F (h) that has underlying braid [γ]. Hence to specify the assignment it suffices to show that there exists such a structural isomorphism. Existence is an easy case-by-case analysis similar to the one in Theorem 5.13 that we leave to the reader.
Note that our assignment of natural isomorphisms satisfies Definition-Proposition 5.5 (ii). It remains to check condition (iii). Consider a diagram of isotopies and operations as in condition (iii) together with a 2-isotopy. The two sides of Equation (5.5.1) are structural isomorphisms by construction of our assignments and by definition of structural isomorphism. Moreover, the assigned braids of both sides are equal by Lemma 5.17 and Remark 5.18. Hence, Equation (5.5.1) holds by Theorem 4.18 combined with Remark 5.11. 5.3. Z 2 -symmetric pairs versus Z 2 D n -algebras.
Definition 5.21. A Z 2 -braided tensor pair (A, M) is called a Z 2 -symmetric tensor pair if for all M ∈ M, X, Y ∈ A Proposition 5.22. Let n ≥ 3. To a Z 2 D n -algebra F in Rex there is an associated Z 2symmetric tensor pair (A F , M F ).
Proof. As the proof is very similar to Proposition 5.19 we will be brief. The n-dimensional analogues of the operations, isotopies and 2-isotopies in the proof of Proposition 5.19 define the structure of a Z 2 -braided pair on A F := A and M F := M. Denote the isotopies i σ and i κ that are mapped by F to the natural isomorphisms σ and κ. For n ≥ 3 there now exist 2-isotopies enforcing the equations σ 2 = id and κ 2 = id.
Theorem 5.23. Let n ≥ 3. To a Z 2 -symmetric tensor pair (A, M) there is an associated Z 2 D n -algebra F (A,M) in Rex.
Proof. As the proof is very similar to Theorems 5.13 and 5.20 we will be brief. Via Definition 5.15 we can assign a permutation σ ∈ S k and a tuple (ε 1 , . . . , ε k ) ∈ {0, 1} ×k to an operation f . We then assign functors F (f ) to operations via the formulae in Equations (5.13.1) -(5.13.3). For an isotopy γ between operations g • (σ f • (∐f j )) and h it is not hard to show that there exists a structural isomorphism between the assigned functors F (g)•(σ •(⊠F (f j )) and F (h). Theorem 4.20 combined with Remark 5.11 then imply this assignment is unique, and satisfies Definition-Proposition 5.5.

General algebras
To define algebras in the (2, 1)-category Rex over the topological operad Z 2 D n we employ the language of (∞, 1)-categories. Our model of (∞, 1)-categories is given by quasicategories. 17  One obtains the associated ∞-operad by taking the (homotopy) coherent nerve N of O ⊗ . 20 To apply the homotopy coherent nerve to a topological category one needs to change enrichment to simplicial sets via the singular chains functor.
Convention. We will change enrichment from topological to simplicial without comment.
It is worthwhile to remind ourselves of some of the low-dimensional simplices of NC. Example 6.3. Let C be a simplicially enriched category, and denote • (resp. * ) the horizontal composition in C of vertices (resp. edges). The coherent nerve NC is a simplicial set with vertices being the objects of C, 1-simplices being the morphisms of C, 2-simplices being diagrams of the form an extra 2-morphism/edge α 0123 : f 03 ≃ f 23 • f 12 • f 01 ∈ C(X 0 , X 3 ) 1 and two 3-morphisms/2-simplices in C(X 0 , X 3 ) 2 with boundaries is a homotopy equivalence.
(3) For any X 1 , . . . , X k ∈ O ⊗ there is a X ∈ O ⊗ [k] and q-coCartesian morphisms ρ i ! : X → X i lying over ρ i . Definition 6.8. Let p : O ⊗ → NFin * be an ∞-operad. A Z 2 D n -algebra in O ⊗ is a map of ∞-operads from Z 2 D n to O ⊗ i.e. a functor F : N ⊗ (Z 2 D n ) → O ⊗ such that p • F = q and F sends q-coCartesian morphisms lying over inert morphisms to p-coCartesian morphisms.
Since Rex is a (2,1)-category the operadic nerve allows a simple description as it is 3coskeletal. Let us briefly recall this notion. Denote ∆ n ⊂ ∆ the full subcategory generated by δ * id f commutes. 26 We will now prove Definition-Proposition 5.5 by showing Z 2 D n -algebras in N ⊗ (Rex) are exactly determined by the list of data specified there.
functor F is a map of ∞-operads. Indeed, this is clear from Example 6.6 since these q-coCartesian morphisms are assigned either id A or id M which are of course p-coCartesian.
We will conclude by showing that the assignments in section 5 are inverse equivalences. Theorem 6.14.
(1) The assignments Proof. The three proofs are identical, so we will prove all three statements at once. First observe that the composite assignment by construction gives back (A, M) on the nose. Conversely, let us show that a Z 2 D nalgebra F is isomorphic to the Z 2 D n -algebra F (A F ,M F ) as objects in Alg Z 2 Dn (Rex). We will abbreviate F (A F ,M F ) by G. Since N ⊗ (Rex) is 3-coskeletal, we find that a morphism η : F → G is completely determined by specifying: • for each 1-simplex f : X → Y in N ⊗ (Z 2 D n ) a 1-simplex η(f ) : F (X) → G(Y ) in N ⊗ (Rex), • for each 2-simplex α : h ≃ f • g in N ⊗ (Z 2 D n ) 2-simplices • so that for every 3-simplex in N ⊗ (Z 2 D n ), in the notation of Definition-Proposition 6.13, Equations (6.14.1) to (6.14.3) hold.
(F (δ) * id) • η F (γ) = (id * η F (β)) • η F (α), (6.14.1) (η F (δ) * id) • η G (γ) = (id * η G (β)) • η F (α), (6.14.2) (G(δ) * id) • η G (γ) = (id * η G (β)) • η G (α). (6.14.3) By Property (2) in Proposition 6.7 it suffices to define η on operations and η F , η G on isotopies between operations. We set η(f ) := F (f ) for all operations, and set η F (α) := F (α) for isotopies. To define η G (id f ) we must construct a natural isomorphism F (f ) ⇒ G(f ). The operation f lies in the contractible fiber of one of the projections and note that there exists an operation f 0 in the fiber that is a horizontal/vertical composite of the fixed operations f ⊗ , f act , f Φ , f 1 , f 1 M of Proposition 5.12 (or 5.19 or 5.22). Let us choose a path in the fiber from f to f 0 . This defines an isotopy γ f : f ≃ f 0 which we view as a 1-simplex in N ⊗ (Z 2 D n ). We define η G (id f ) to be the composite natural isomorphism More generally, we define η G for an isotopy α : h ≃ g • f as the horizontal composition It remains to show η defines an isomorphism. By writing out the definitions of G and η in terms of F one shows Equations (6.14.1) to (6.14.3) hold. For example, Equation (6.14.1) follows immediately from functoriality of F . For the details see [Wee1]. To show that η is an isomorphism it suffices to show its components η(id X ) for all X ∈ N ⊗ (Z 2 D n ) are equivalences [Joy, Theorem C]. This clearly holds as η(id X ) = id F (X) .