Circuit algebras are wheeled props
CCIRCUIT ALGEBRAS ARE WHEELED PROPS
ZSUZSANNA DANCSO, IVA HALACHEVA, AND MARCY ROBERTSON
Abstract.
Circuit algebras, introduced by Bar-Natan and the first author, are a generalization of Jones’splanar algebras, in which one drops the planarity condition on “connection diagrams”. They provide a usefullanguage for the study of virtual and welded tangles in low-dimensional topology. In this note, we presentthe circuit algebra analogue of the well-known classification of planar algebras as pivotal categories with aself-dual generator. Our main theorem is that there is an equivalence of categories between circuit algebrasand the category of linear wheeled props – a type of strict symmetric tensor category with duals that arisesin homotopy theory, deformation theory and the Batalin-Vilkovisky quantization formalism. Introduction
In [Jon99], Jones introduced the notion of a planar algebra as an axiomatization of the standard invariantof a finite index subfactor. A planar algebra is an algebraic structure whose operations are parametrized by planar tangles . A planar tangle is a -manifold with boundary, embedded in a “disc with r holes”, where theboundary points of the -manifold lie on the boundary circles of the disc with holes: an example is shownin Figure 1 on the left. Planar tangles form a coloured operad where composition is defined by gluing theouter circle of one tangle into an inner circle of another, as long as the tangle endpoints match (eg. [Jon99],[BHP12, Definition 2.2]). Given a field k of characteristic zero, a planar algebra is a sequence of k -vectorspaces, which admit an action by the operad of planar tangles ([Jon99],[BHP12, Definition 2.4]). Figure 1.
A planar tangle on the left, a wiring diagram on the right.Planar algebras arise in many contexts where a tensor category with a “good” notion of duals is involved,and have played an important role in the theories of subfactors, conformal and quantum field theories andknot and tangle invariants. There is a well-known classification of planar algebras as pivotal categories with asymmetrically self-dual generator ([MPS10], [HP17], [BHP12]). Pivotal categories are rigid tensor categoriesin which every object is isomorphic to its double dual (Section 6).
Circuit algebras were defined by Bar-Natan and the first author in [BD17] as a generalization of planar al-gebras which provides a convenient language for virtual and welded tangles in low-dimensional topology. Theterm, inspired by electrical circuits, was coined by Bar-Natan. Circuit algebras are defined similarly to planaralgebras, but their operations are parametrized by not necessarily planar wiring diagrams (Definition 2.1).Wiring diagrams can be defined similarly to planar tangles, but with embedded submanifolds replaced byabstract -manifolds whose boundary points are identified with points on the boundary circles (Figure 1).This results in a purely combinatorial – no longer topological – structure; we discuss this distinction in detailin Section 2. Date : September 22, 2020. a r X i v : . [ m a t h . QA ] S e p Z. DANCSO, I. HALACHEVA, AND M. ROBERTSON
Wiring diagrams can be composed in the same manner as planar tangles, making the collection of wiringdiagrams into a coloured operad. A circuit algebra is an algebra over this operad: a collection of vectorspaces along with linear maps between them, parametrized by wiring diagrams. In Section 2 we present thedefinition of circuit algebras in detail. In Section 3 we present an example from knot theory, and explain therelationship to planar algebras in more detail.Circuit algebras are more recent and, so far, not as widely studied as planar algebras. In the last fewyears a number of authors have used circuit algebras to study invariants for virtual and welded tangles([BD17], [DF18], [Hal16],[Tub14]). In this paper we expand the definition of [BD17] to include more detailand improve accessibility for a wider audience. In Section 3, we present the example of virtual tangles, whichcan be defined as both a planar algebra and a circuit algebra. Indeed, every circuit algebra has an underlyingplanar algebra, as we prove in Proposition 3.3 :
Proposition.
There exists a pair of adjoint functors
CA PA . There is no expectation, however, that this adjunction should be an equivalence. As just one example,classical tangles from knot theory admit a planar algebra structure, but not a circuit algebra structure.The main result of this paper is a classification result for circuit algebras in terms of linear wheeled props ,analogous to that of planar algebras via pivotal categories. A linear prop is a strict symmetric tensor categorywhose objects are generated by a single object.
Wheeled props [MMS09], are rigid props or, equivalently,strict symmetric tensor categories with duals which are generated by a single object. Wheeled props arisenaturally in deformation theory and the Batalin–Vilkovisky quantization formalism of theoretical physics,invariant theory, and other abstract settings where a generalized trace operation plays a role ([Mer10a],[Mer11], [CFP20], [DM19]). Our main theorem (Theorem 5.5) is the following:
Theorem.
There is an equivalence of categories between circuit algebras and linear wheeled props CA ∼ = wPROP . In Section 4 we give a formal introduction to linear wheeled props as algebras over a monad on diagramsof vector spaces indexed by directed graphs. The key step in this identification is understanding that wiringdiagrams can be identified with directed graphs (Lemma 5.3). In order to make this paper readable to thelargest possible audience, however, we have also included an appendix containing an equivalent, axiomatic,definition of wheeled props (Definition 6.1). To keep our comparison with the algebraic classification of planaralgebras in mind, we point out in Proposition 6.8 that linear wheeled props embed into the category of linearpivotal categories. In summary, we have the following diagram in which the two horizontal adjunctions areequivalences of categories:
CA wPropPA PivCat ∼ = ∼ = . We further discuss the conjectural commutativity of this diagram at the end of Section 6.Throughout this paper we focus our attention on linear wheeled props, that is, wheeled props enriched in k -vector spaces. This choice was made to simplify exposition, but all objects can be defined in any closed,symmetric monoidal category and all arguments still hold. We further note that one goal of this paper is toprovide a bridge between the tensor categories, knot theory and category theory communities. As such, thelevel of detail is intended to make each section accessible to mathematicians working in the other areas. Acknowledgements.
Part of this work was completed while the first and third authors were in residenceat MSRI for the program “Higher Categories and Categorification” in 2020. In addition, we would like tothank Dror Bar-Natan, Scott Morrison and Sophie Raynor for suggestions of key references and Arun Ramfor many helpful comments.
IRCUIT ALGEBRAS ARE WHEELED PROPS 3
Contents
1. Introduction 12. Circuit algebras 33. Planar algebras and virtual tangles 73.1. Circuit algebras and planar algebras 73.2. Virtual tangles 84. Wheeled Props 104.1. Oriented graphs 104.2. Wheeled props 135. Equivalence 165.1. Graphs and Wiring Diagrams 175.2. Equivalence of Categories 186. Wheeled props as tensor categories 196.1. Axiomatic definition 206.2. Examples 236.3. Wheeled props and pivotal categories 27References 282.
Circuit algebras
A circuit algebra, much like a planar algebra, is a family of vector spaces with operations indexed by wiring diagrams : abstract -manifolds with boundary whose boundary points are identified with endpointsalong the circles of a “disc with holes”. We will develop the comparison with planar algebras further inSection 3, but for now we alert the reader to the fact that we are using a definition of planar algebra withoutshading, such as that in [HPT, Definition 2.3] or [BHP12, Section 2]. The key difference between wiringdiagrams and planar tangles is that while planar tangles are inherently topological objects, wiring diagramsare purely combinatorial, and their topological description below is merely for convenience (see Remark 2.3).Throughout this paper, let I denote a countable alphabet, the set of labels . The following definition is anexpanded version of the definition given in [BD17, Section 2]. Definition 2.1. An oriented wiring diagram is a triple D = ( A , M, f ) consisting of:(1) A set A = { A out , A in , A out , A in , . . . , A out r , A in r } of sets of labels, for some non-negative integer r . Thatis, A out i , A in i ⊆ I for each ≤ i ≤ r . The elements of the sets A out i are referred to as outgoing labels and the elements of A in i are incoming labels . The sets A out and A in play a distinguished role: theirelements are called the output labels of the diagram, while the sets A out , A in , ..., A out r , A in r contain input labels of the diagram. We write A out / in i to mean “ A out i and A in i , respectively”.(2) An oriented compact -manifold M , with boundary ∂M , regarded up to orientation-preserving home-omorphism. The connected components of M are homeomorphic to either an oriented circle (withno boundary) or an oriented interval with one beginning and one ending point. We write ∂M out for the set of beginning boundary points of M , and ∂M in for the set of ending boundary points, so ∂M = ∂M out (cid:116) ∂M in .(3) Bijections ∂M out f −→ ∪ ri =0 A out i and ∂M in f −→ ∪ ri =0 A in i . Wiring diagrams have a convenient pictorial representation shown in Figure 2, which illuminates theirrelationship to planar algebras. A disc with r holes , D \ ( ˚ D (cid:116) ˚ D (cid:116) . . . (cid:116) ˚ D r ) , If the sets { A out / in i } are not pairwise disjoint, replace the unions ( ∪ ri =0 A out i ) and ( ∪ ri =0 A in i ) by the set of triples { ( a, i, out / in ) | a ∈ A out / in i , ≤ i ≤ r } . Z. DANCSO, I. HALACHEVA, AND M. ROBERTSON is obtained by removing r disjoint numbered open discs with disjoint boundaries from the interior of a biggerdisc. The boundaries of the removed discs are called the input circles , while the boundary of the big disc isnumbered zero and called the output circle .Assume I is ordered, which is often the case with the labels we use in examples, such as natural numbersor Roman letters. Arrange the elements of A out , then A in in the order induced by the ordering on I atuniform intervals along the output circle, and the elements of A out i , then A in i for i = 1 , ..., r , in order atuniform intervals along the i th input circle. Represent the manifold M and the identification f as immersedcurves in D \ ( ˚ D (cid:116) ˚ D (cid:116) . . . (cid:116) ˚ D r ) . Note that the specific immersion is not part of the data of the wiringdiagram. Figure 2.
An example of an oriented wiring diagram. The labels sets are A out = { a } , A in = { a , a } , A out = { a , a } , A in = { a } , A out = { a , a , a } , A in = { a } , A out = ∅ , A in = { a , a } . The manifold, drawn in brown lines, is a disjoint union of six orientedintervals and two oriented circles. In the picture we don’t draw arrows for circles, since theyare abstract, not embedded: there is only one homeomorphism type of an oriented circle. Definition 2.2.
Given two wiring diagrams D = (cid:16) A = { A out / in , A out / in , . . . , A out / in r } , M, f (cid:17) D (cid:48) = (cid:16) B = { B out / in , B out / in , . . . , B out / in s } , N, g (cid:17) the composition D ◦ i D (cid:48) , for ≤ i ≤ r , is defined whenever A out / in i = B in / out as sets. The resulting composite wiring diagram D ◦ i D (cid:48) = ( A ◦ i B , M (cid:116) ϕ N, f (cid:116) ϕ g ) consists of:(1) the label sets A ◦ i B = { A out / in , A out / in , . . . , A out / in i − , B out / in , . . . , B out / in s , A out / in i +1 , . . . , A out / in r } (2) a compact oriented -manifold M (cid:116) ϕ N , obtained by gluing M and N along the map ϕ , whichidentifies the boundary points in f − ( A out / in i ) and g − ( B in / out ) : ∂M ⊇ f − ( A out / in i ) A out / in i = B in / out g − ( B in / out ) ⊆ ∂N ; ϕf g − (3) a bijection f (cid:116) ϕ g defined to be f on ∂M \ f − ( A out / in i ) and g on ∂N \ g − ( B out / in ) . It is important that A in i is identified with B out and vice versa. IRCUIT ALGEBRAS ARE WHEELED PROPS 5
Composition can be pictorially represented by shrinking the wiring diagram D (cid:48) and gluing it into the i thinput circle of D so that the labels match. Then delete the outer circle of D (cid:48) , as shown in Figure 3. Theordering of the input discs of the composite diagram follows the ordering prescribed in (1) of Definition 2.2.Note that composition may create closed components (circles) in M (cid:116) ϕ N . Figure 3.
An illustration of oriented wiring diagram composition; labels are suppressedfor simplicity, but must match. The deleted outer disc of D (cid:48) is shown as a broken line redcircle. Proposition 2.3.
A wiring diagram ( A , M, f ) is equivalent to a triple ( A , p, l ) , where A is as above, p isa perfect matching (set bijection) between (cid:70) ri =0 A out i and (cid:70) ri =0 A in i , and l ∈ Z ≥ is a non-negative integer,“the number of circles in M ”.Proof. The oriented -manifold M in Definition 2.1 is a disjoint union of a finite number of oriented intervalsand circles. The endpoints of the intervals are identified with the labels, and as such, the only role ofthe intervals is to define a perfect matching – that is, a set bijection – between the sets of incoming andoutgoing labels. Since there is only one homeomorphism type of an oriented circle, one can equivalentlysimply remember the number of circle components of M . (cid:3) This alternative definition illuminates that wiring diagrams are combinatorial – as opposed to topological –objects. This is the main difference between wiring diagrams and planar tangles. We make this combinatorialdescription even more explicit by identifying wiring diagrams with a combinatorial formalism of directedgraphs in Lemma 5.3. Our main reason for presenting the definition using -manifolds is the compositionof wiring diagrams: describing the perfect matching and number of circles resulting from a composition inpurely combinatorial terms is possible, but a headache (we encourage the reader to try). The definition ofcomposition through gluing 1-manifolds is much more elegant and concise.As with planar tangles, one can show that the collection of oriented wiring diagrams together with the ◦ i compositions assembles into a (coloured) operad . A proof of this is a simple exercise along the same lines asthe description of the operad of planar tangles in [Jon99] or the construction of an operad of wires in [Pol10,Section 3]. A concise definition for an oriented circuit algebra is an algebra over this operad; we explainthe notion of operads and algebras over them in more detail in Section 3. Here we unwind this concept andarrive at the following definition; an expanded version of that in [BD17, Definition 2.10]: Definition 2.4. An oriented circuit algebra V consists of a collection of vector spaces indexed by pairsof label sets, { V [ S out ; S in ] } S out ,S in ⊆I , together with a family of linear maps between these, parametrised byoriented wiring diagrams. Namely, for each wiring diagram D = ( A , M, f ) , there is a corresponding linearmap F D : V [ A out ; A in ] ⊗ . . . ⊗ V [ A out r ; A in r ] → V [ A in ; A out ] . This data must satisfy the following axioms:(1) The composition of wiring diagrams corresponds to composition of linear maps in the following sense.Let D = ( { A out / in , . . . , A out / in r } , M, f ) , D (cid:48) = ( { B out / in , . . . , B out / in s } , N, g ) If the sets A out / in i are not disjoint, replace them in the disjoint unions with sets of triples, as before. Z. DANCSO, I. HALACHEVA, AND M. ROBERTSON be two wiring diagrams composable as D ◦ i D (cid:48) . Then the map of vector spaces corresponding to thecomposition D ◦ i D (cid:48) is F D ◦ i D (cid:48) = F D ◦ (Id ⊗ · · · ⊗ Id ⊗ F D (cid:48) ⊗ Id ⊗ · · · ⊗ Id) , where F D (cid:48) is inserted in the i th tensor component.(2) There is an action of the symmetric groups S r on wiring diagrams with r input sets, which permutes(re-numbers) the input sets. The assigment of linear maps to wiring diagrams is equivariant underthis action in the following sense. Let D = ( { A out / in , A out / in , . . . , A out / in r } , M, f ) be a wiring diagram, σ ∈ S r , and let σD = ( { A out / in , A out / in σ (1) , . . . , A out / in σ ( k ) } , M, f ) be the wiring diagram D with the inputsets re-ordered; note that the output set A out / in is fixed. Then the induced linear map F σD is F D ◦ σ − , where σ − acts on V [ A out / in σ (1) ] ⊗ . . . ⊗ V [ A out / in σ ( r ) ] by permuting the tensor factors. Definition 2.5. A morphism of circuit algebras Φ : V → W is a family of linear maps { Φ S out ; S in : V [ S out ; S in ] → W [ S out ; S in ] } S out ,S in ⊆I which commutes with the action of wiring diagrams. That is, forany wiring diagram D = ( A , M, f ) we have a commutative diagram: V [ A out ; A in ] ⊗ . . . ⊗ V [ A out r ; A in r ] ( F V ) D (cid:47) (cid:47) Φ A out A in ⊗ ... ⊗ Φ A out r ; A in r (cid:15) (cid:15) V [ A in ; A out ] Φ A in A out (cid:15) (cid:15) W [ A out ; A in ] ⊗ . . . ⊗ W [ A out r ; A in r ] ( F W ) D (cid:47) (cid:47) W [ A in ; A out ] More concisely put, a morphism of circuit algebras is a map of algebras over the operad of wiring diagrams.The category of all circuit algebras is denoted CA . Example 2.6.
For every pair of sets of labels
S, T ⊆ I , there are left and right identity wiring diagrams .In the perfect matching notation introduced in Remark 2.3:The left identity is Id LS,T = ( { S, T, T, S } , Id , , where: • A out = S , A in = T , A out = T , A in = S ; • and Id refers the perfect matching induced by the set identities Id S and Id T .The right identity is Id RS,T = Id
LT,S . Then, for any wiring diagram D with A out i = S , A in i = T , we have D ◦ i Id RS,T = D . On the other hand if D is such that A out = S , A in = T then Id LS,T ◦ D = D . Consequently,the corresponding linear maps are the identity maps F Id RS,T = Id V [ S ; T ] and F Id LS,T = Id V [ T ; S ] . See Figure 4. Example 2.7.
In a similar vein to the left and right identities, given any pair of permutations σ ∈ S S and τ ∈ S T , there are label permuting wiring diagrams D σ,τ = ( S, T, T, S, ( σ, τ ) , where: • A out = S , A in = T , A out = T , A in = S and • the perfect matching ( σ, τ ) is given by the permutations σ : S = A in → A out = S and τ : T = A out → A in = T .If B is a wiring diagram with A in = T , A out = S then D σ,τ ◦ B is the wiring diagram B with the labels in A in and A out permuted by σ and τ respectively. Similarly, for a wiring diagram C with A in i = S , A out i = T ,the composition C ◦ i D σ,τ is the wiring diagram C with the permutations σ − and τ − applied to the labelsin A out i and A in i , respectively. The linear maps F D σ, id : V [ S ; T ] → V [ S ; T ] give a left S S action on V [ S ; T ] ,and F D id ,τ − gives a commuting right action by S T . See Figure 4. Relabelling wiring diagrams can be constructed the same way given a pair of set bijections for subsets of I . Remark 2.8.
In this paper we have focused on oriented circuit algebras , as they are most useful in thetopological examples and applications that the authors have in mind (see Section 3). However, both planaralgebras and circuit algebras admit many variations including oriented and un-oriented versions, colouredversions and various enrichments.For example, while Definition 2.4 of a circuit algebra describes V as a sequence of vector spaces andlinear maps, the definition makes sense in any closed, symmetric monoidal category. For the unoriented andcoloured versions, one only need modify the definition of wiring diagrams as appropriate. Specifically, todefine unoriented wiring diagrams simply drop the notion of inputs and outputs so that in a wiring diagram D = ( A , M, f ) the labels sets are A = { A , A , . . . , A r } with no in / out distinction. IRCUIT ALGEBRAS ARE WHEELED PROPS 7
Figure 4.
Examples of identity and relabelling wiring diagrams, respectively.3.
Planar algebras and virtual tangles
In this section we make two short detours: one to clarify the relationship between circuit algebras andplanar algebras (Subsection 3.1), and another to present the example of virtual tangles (Subsection 3.2),where circuit algebras provide a useful and simple algebraic framework. We contrast the circuit algebraapproach to virtual tangles with a planar algebra approach, which also illustrates the point of Section 3.1.Section 3.1 is likely most interesting to readers already familiar with planar algebras; nothing in the latterpart of the paper depends on it. Section 3.2 is perhaps most interesting to topologists, though the authorsbelieve it is quite self-contained.3.1.
Circuit algebras and planar algebras.
The main statement of this section is that every circuitalgebra is, in particular, a planar algebra, since planar tangles in discs with holes may be viewed as wiringdiagrams. In more technical terms, there exists a pair of adjoint functors between the category of circuitalgebras CA and the category of planar algebras PACA PA . To make the adjunction precise, we give a brief definition of an oriented planar algebra . There are manyvariations of planar algebras in the literature, for simplicity, we work with oriented planar algebras, whichare algebras over the coloured operad of oriented planar tangles (without shading), in the vein of Definition2.3 of [HPT].Recall that, given a set of colours C , a C -coloured operad P = { P ( c , . . . , c r ; c ) } consists of a collectionof vector spaces P ( c , . . . , c r ; c ) , one for each sequence ( c , . . . , c r ; c ) of colours in C , which is equippedwith an S r -action permuting c , . . . , c r , together with an equivariant, associative and unital family of partialcompositions ◦ i : P ( c , . . . , c r ; c ) × P ( d , . . . , d s ; d ) P ( c , . . . , c i − , d , . . . , d s , c i +1 , . . . , c r ; c ) , whenever d = c i . For full details see [BM07, Definition 1.1].The operad of oriented planar tangles PT = { PT ( s , . . . , s r ; s ) } is a coloured operad where PT ( s , . . . , s r ; s ) is the vector space spanned by isotopy classes of oriented planar tangles of type ( s , . . . , s r ; s ) . Here s i refersto a finite sequence of signs ± , and a planar tangle of type ( s , . . . , s r ; s ) lives in a disc with r holes D \ ( ˚ D (cid:116) ˚ D (cid:116) . . . (cid:116) ˚ D r ) , where along each boundary circle, there is an equidistant sequence of markedpoints, labeled with the sign sequence s i . The planar tangle is an oriented -manifold M embedded in sucha disc with holes, such that the embedding maps ∂M bijectively to the set of marked points with incomingboundary points mapping to positive marked points, and outgoing boundary points to negative markedpoints; cf. [HPT, Definition 2.1].Planar tangles are composed by the shrinking-and-gluing procedure described in the previous sections,where the gluing of the 1-manifolds must be orientation-respecting, meaning the sign sequences must match.This is a partial operadic composition which makes the set of oriented planar tangles a coloured operad.An algebra over an C -coloured operad P is a C -indexed family of vector spaces A = { A ( c ) } c ∈ C togetherwith an action by P ([BM07, Definition 1.2]). An oriented planar algebra is an algebra over the operadof oriented planar tangles: a collection of vector spaces which carry actions of planar tangles, which are Z. DANCSO, I. HALACHEVA, AND M. ROBERTSON compatible with compositions and the symmetric group action, just like the definition of a circuit algebrabased on the notion of wiring diagrams. A map of oriented planar algebras is a morphism of algebras overthe operad of planar tangles – this can be unpacked just like we did for circuit algebras. Denote the categoryof planar algebras by PA .As above, we have that PT ( s , . . . , s r ; s ) denotes the space of oriented planar tangles of type ( s , . . . , s r ; s ) .Let WD ( s , . . . , s r ; s ) denote the vector space of oriented wiring diagrams where, if s i = ( s i, , ..., s i,d i ) , then A in i = { j : s i,j = 1 } and A out i = { j : s i,j = − } . Lemma 3.1.
For every signed sequence of integers ( s , . . . , s r ; s ) , the space of oriented planar tangles PT ( s , . . . , s r ; s ) is a subspace of WD ( s , . . . , s r ; s ) .Proof. Given a planar tangle of type ( s , . . . , s r ; s ) , forget the embedding information on the interior of M to obtain a wiring diagram, where only ∂M is identified with the label sets. (cid:3) Now assume that P and Q are C -coloured operads with inclusions Q ( c , . . . , c r ; c ) ⊆ P ( c , . . . , c r ; c ) foreach ( c , . . . , c r ; c ) . Recall that such data defines a C -coloured sub-operad of P , if the restriction of thesymmetric group actions and ◦ i partial compositions of P agrees with the operad structure of Q . Proposition 3.2.
The operad of oriented planar tangles is a sub-operad of the operad of wiring diagrams.Proof.
The symmetric group S r acts on WD by permuting the indices of the input sets (internal discs), andtherefore restricts to PT ( s , . . . , s r ; s ) ⊆ WD ( s , . . . , s r ; s ) . The ◦ i partial composition of planar tangles, asdescribed in Definition 2.1 [HPT], is precisely the ◦ i partial composition of wiring diagrams when restrictedto the subspace of planar tangles. (cid:3) Let Q be a sub-operad of P , and let Alg ( Q ) and Alg ( P ) denote the categories of algebras over Q and P ,respectively. Then the inclusion φ : Q → P induces an adjunctionAlg ( Q ) Alg ( P ) . φ ! φ ∗ For the further details, see just below Definition 1.2 in [BM07]. The functor φ ∗ is quite intuitive to construct,as follows. An algebra over P is a structure that carries an action by the elements of P ; an algebra over Q is a structure that carries an action by elements of Q ; both of which are compatible with compositions, andsymmetric group actions. Since every element of Q is in particular an element of P , an algebra over P isautomatically an algebra over Q . The left adjoint exists for formal reasons. Proposition 3.3.
There is an adjunction
PA CA φ ! φ ∗ between the category of oriented planar algebras and the category of oriented circuit algebras.Proof. A direct consequence of Proposition 3.2, given the discussion above. (cid:3)
Remark 3.4.
A similar argument will hold for the reader’s favourite variation of planar and circuit algebras(unoriented, coloured, enriched) by appropriate modifications to the underlying operads.3.2.
Virtual tangles.
Virtual tangles – a generalization of classical tangles – were the motivating examplefor the definition of circuit algebras, and so far they have been their main area of application [Bro19, Pol10].An oriented (classical) tangle is a smooth embedding of an oriented -manifold M into a -dimensionalball M (cid:44) → B , such that the boundary is mapped to the boundary of the ball: ∂M (cid:44) → S = δB . Suchembeddings are considered up to boundary-preserving ambient isotopy. In knot theory, tangles are oftenstudied via their Reidemeister theory: project the ball onto a disc to obtain a tangle diagram where theimage of M has only transverse double-points in the interior, called crossings , and no double points on theboundary. Ambient isotopy is captured by three diagrammatic relations, called the Reidemeister , and moves.This leads naturally to a presentation of tangles as a planar algebra [Bar05, Section 5], generated byan overcrossing ! and an undercrossing " , modulo the Reidemeister , and moves shown in Figure 6([BD17]). To summarize, tangles form a planar algebra T = P A (cid:104) ! , " | R , R , R (cid:105) . Generation in a planar algebra means, as one would expect, all possible applications of planar connection diagrams.
IRCUIT ALGEBRAS ARE WHEELED PROPS 9
Figure 5.
An example of a virtual tangle diagram.
Figure 6.
The classical Reidemeister moves, presented as planar algebra relations betweenthe crossings (generators). The relations are imposed in all possible (consistent) strandorientations.Virtual tangles are a generalization of tangles to embeddings into thickened surfaces, rather than into B – see [Kup03] for a proof of this statement in the case of links. Virtual tangles are topologically interesting asa broader family of tangled objects, but they also possess interesting algebraic and combinatorial properties.As one example, their finite type invariants are conjecturally deeply connected with quantizations of Liebialgebras, see [BD16, Introduction] for a brief overview. Like classical tangles, virtual tangles can be described as virtual tangle diagrams modulo Reidemeistermoves, and can be presented as a planar algebra in which one adds a virtual crossing as an additionalgenerator P as well as additional virtual Reidemeister relations, shown in Figure 7, which describe howvirtual crossings interact with each other and with classical crossings. Figure 7.
The "virtual" and "mixed" Reidemeister moves, as planar algebra relations.The relations are imposed in all possible (consistent) strand orientations. This connection is further strengthened by the equivalence of circuit algebras and wheeled props in Theorem 5.5, as wheeledprops play a role in formality theorems for Lie bialgebras [Mer16]. The direct comparison between circuit algebras and thesewheeled props is part of the authors’ motivation for this project.
It is a basic fact of virtual knot theory called the detour move (see for example [DK05, Section 2]) thatany purely virtual part of a strand of a virtual tangle diagram (i.e. a strand that only intersects others invirtual crossings) can be re-routed in any other purely virtual way . This suggests that the virtual crossing,morally speaking, isn’t a “true generator” of virtual tangles, but merely a structural, diagrammatic artifact.This motivates the description of virtual tangles as a circuit algebra [BD17, Section 3]: the generators aresimply two crossings { ! , " } , and the relations are the ordinary Reidemeister moves { R , R , R } : vT = CA (cid:10) ! , " | R , R , R (cid:11) . In this description, “virtual crossings” only exist in the pictorial representation of wiring diagrams, however,as wiring diagrams are fundamentally combinatorial objects (Remark 2.3 and Lemma 5.3), it is understoodthat they don’t hold any mathematical meaning. The detour move is so tautological that it doesn’t evenmake sense as a statement in a circuit algebra context.This also gives an illustrative example of the relationship between circuit algebras and planar algebras:virtual tangles naturally form a circuit algebra, and all circuit algebras are also planar algebras by Lemma 3.1and Proposition 3.3, hence virtual tangles also form a planar algebra. However, we see above that the circuitalgebra description in terms of generators and relations is simpler and in the authors’ opinion more elegant.Classical tangles, on the other hand, naturally form a planar algebra, with a simple, elegant description, anddo not admit a (reasonable) circuit algebra structure.4.
Wheeled Props
A (linear) prop is a strict symmetric tensor category in which the monoid of objects is freely generatedby a single object. Props are often used to “encode” a class of algebraic structures. For example, there existsa prop LieB with the property that strict symmetric monoidal functors from
LieB to the category of vectorspaces are in one-to-one correspondence with Lie bialgebras [Mer16, Definition 2.1].Wheeled props are an extension of props which “encode” algebraic structures with a notion of trace .They arise naturally in geometry, deformation theory and theoretical physics. For example, in the Batalin-Vilkovisky quantization formalism, formal germs of SP-manifolds are in one-to-one correspondence withrepresentations of a certain wheeled prop [MMS09, Theorem 3.4.3]. More detailed examples of wheeledprops are given in Section 6.2.In this section we give a monadic definition for wheeled props (Definition 4.15 and 4.16) using the notionof oriented graphs and the operation of graph substitution . This is the most intuitive route – in the authors’opinion – to describing the relationship between wheeled props and circuit algebras in Section 5. An alternate,axiomatic, definition of wheeled props is presented in Section 6.4.1.
Oriented graphs.
For this purpose, a graph is a graph with open edges : an edge may be adjacentto one, two or no vertices. Graphs may have free-floating loops with no vertices, as in Figure 8. Thereare several definitions of such graphs in the literature; in Definitions 4.1 and 4.3 we adapt a combinatorialdefinition for generalized graphs from [YJ15], which gives a direct comparison to wiring diagrams.For a user-friendly preview before the technical definition, a graph consists of a collection F of flags , or half edges , along with an (ordered) partition on F . The sets in the partition are the vertices of the graph,with the exception of one set, set aside for flags that are part of a free-floating edge or loop, and hence notincident to a vertex. This set in the partition is called the exceptional cell , denoted by v (cid:15) .An involution on flags ι : F → F glues the flags together to form edges . Two flags a, a (cid:48) ∈ v (cid:15) with theproperty that ι ( a ) = a (cid:48) form a free-floating loop detached from any vertex. In addition, there is a fixed pointfree involution π on the ι -fixed points of v (cid:15) , which assembles these flags into free-floating edges not attachedto a vertex. To summarise: Definition 4.1.
Fix a countable alphabet I . A labelled graph G is a finite set F = F ( G ) , called the set of flags , or half-edges , together with • a finite ordered partition F = ( (cid:96) α ∈ V ( G ) v α ) (cid:116) v (cid:15) , • an involution ι : F → F with the property ι ( v (cid:15) ) = v (cid:15) , • a fixed point free involution π on the ι -fixed points in v (cid:15) , and Some authors prefer to capitalise the word PROP to emphasise that prop refers to “PROduct and Permutation category." Also referred to as Borisov-Manin graphs. Sometimes defined as a quadruple of vertices, flags, attachment map and aninvolution. Our generalization allows for vertex-less loops.
IRCUIT ALGEBRAS ARE WHEELED PROPS 11
Figure 8.
A labelled graph with flags partitioned into vertices: v = { , , , , } and v = { , , , } with exceptional cell v (cid:15) = { , , , } . • a labelling function λ : (cid:96) v α ∈ V ( G ) v α → I , injective on each v α , • an injective boundary labelling function β : ∂G → I , where ∂G ⊆ F is the set of ι -fixed flags.The vertices v α , and the exceptional cell v (cid:15) , are subsets of flags, and as such they can be empty. If a vertex v α is empty, it is an isolated vertex with no incident flags. If v (cid:15) is empty, then the graph has no free-floatingedges or loops. The set V ( G ) is called the vertex set. In this paper, we assume for simplicity that verticesare numbered, i.e. there is a bijection V ( G ) → { , , ..., r } . From now on we will refer to labelled graphssimply as graphs . Example 4.2.
The graph in Figure 8 has flags F = { , , , , , , , , , , , , } partitioned into vertices: v = { , , , , } and v = { , , , } with exceptional cell v (cid:15) = { , , , } .The involution ι acts on F with ι (5) = 7 , ι (4) = 6 and ι (12) = 13 , the last pair making up the floating circle.The ι -fixed points are , , , , , and . The fixed point free involution π acts on the ι -fixed pointsin v (cid:15) and sets π (10) = 11 . The labelling λ : v ∪ v → I labels each flag k for k = 1 , ..., by a k . In thisexample, ∂G = { , , , , , , } , and the boundary labelling assigns the label β ( k ) = i k to each k ∈ ∂G .Note that the flags in the floating loop remain un-labelled. Definition 4.3. An oriented graph is a graph G with an orientation function δ : F → {− , } , where F = (cid:96) α ∈ V ( G ) v α (cid:116) { ϕ ∈ v (cid:15) : ι ( ϕ ) = ϕ } , such that δ ( ιx ) = − δ ( x ) whenever ιx (cid:54) = x , and δ ( πx ) = − δ ( x ) whenever π is defined. We writein ( v α ) = λ ( δ | v α ) − (1) , and out ( v α ) = λ ( δ | v α ) − ( − , in ( G ) = β ( δ | ∂G ) − (1) , and out ( G ) = β ( δ | ∂G ) − ( − , where δ | v α and δ | ∂G are the restrictions of δ to a vertex v α and the boundary ∂G , respectively. For an orientedgraph we only require the labelling function λ to be injective on the sets ( δ | v α ) − (1) and ( δ | v α ) − ( − asopposed to all of v α . Similarly, the boundary labelling β is only required to be injective on ( δ | ∂G ) − (1) and ( δ | ∂G ) − ( − .In pictures we indicate the direction function by drawing an arrow from a negative flag a to its positivepair ι ( a ) ; or from a positive flag a to its negative pair π ( a ) for free edges. Free loops do not have directions.See Figure 9.For non-exceptional ι -fixed flags – that is, half-edges attached to a vertex – negative flags are drawn asoutgoing and positive flags are drawn as incoming. As an example, the oriented corolla in Figure 9a hasincoming, or positive, flags with (boundary) labels i , i and i , and outgoing (negative) flags labelled j and j . Figure 9b shows another example of a directed graph. Boundary labels are shown with incoming flagslabelled i α and outgoing flags labelled j β for α = 1 , , and β = 1 , . Vertex labels (i.e. the λ labelling) Outgoing edges are also commonly called outputs, and incoming edges are called inputs. We use the outgoing/incomingterminology to avoid confusion with circuit algebra inputs and outputs. (a)
A corolla C ( I ; J ) . (b) A directed graph with inputs { i , i , i } and outputs { j , j } . The dashed red circle indicates the boundary of G , ∂G . Figure 9.
Two depictions of oriented graphs with incoming labels { i , i , i } and outgoinglabels { j , j } . Figure 10.
An example of a graph substitution which creates a free-floating loop.are suppressed. Note that any complete edge has a well-defined beginning and end. The free loop has nodirection.The labels of the incoming and outgoing flags adjacent to a vertex v α are called the neighbourhood of thevertex : nbh ( v ) = ( in ( v ); out ( v )) = ( λ ( δ | − v )(1); λ ( δ | − v )( − . By a slight abuse of notation (using ∂G to denote the set of boundary flags or their labels, depending oncontext) for oriented graphs we also write: ∂G = ( in ( G ); out ( G )) = ( β ( δ | − ∂G )(1); β ( δ | − ∂G )( − . Definition 4.4.
We say that two oriented labelled graphs G and G are isomorphic if there is a bijection offlags φ : F ( G ) → F ( G ) , which preserves the ordered partitions, the involutions, the orientation function,and the labelling.The operation of graph substitution for oriented graphs parallels the composition of wiring diagrams ina circuit algebra. Intuitively, graph substitution “glues” a graph H v into a vertex v of a graph G in such away that ∂H v = nbh ( v ) . The result is a new graph G ( H v ) . While the intuition is clear, writing down theresulting graph in terms of involutions is tedious due to the possible creation of floating loops: see Figure 10for an example. Below we give an intuitive definition and for full details refer the reader to [YJ15, Chapter5]. Sometimes the vertex is said to be labelled by the input and output flags contained in the neighbourhood, and the graphis labelled by its boundary flags. The notion also exists for non-oriented graphs, by simply omitting the condition that orientations match.
IRCUIT ALGEBRAS ARE WHEELED PROPS 13
Figure 11.
An example of a graph substitution which permutes the labels of a graph.
Definition 4.5.
Let G be an oriented graph, v ∈ V ( G ) a vertex, and H v an oriented graph with ∂H v =( in ( H v ); out ( H v )) = ( in ( v ); out ( v )) = nbh ( v ) . Define the graph substitution G ( H v ) as the graph obtained by • replacing the vertex v ∈ V ( G ) with the graph H v , and • identifying each leg (boundary flag) of H v with the flag of v with the same label and same orientation.The boundary of the graph G ( H v ) is identified with the boundary of G . Moreover, there is a canonicalidentification of vertex sets V ( G ( H v )) = ( V ( G ) \ { v } ) (cid:116) V ( H v ) . The ordering (numbering) of the vertices of G ( H v ) is as follows: first follow the ordering of V ( G ) before v , then the ordering of V ( H v ) , then the orderingof V ( G ) after v .Graph substitution is associative and unital , see Theorem 5.32 and Lemma 5.31 in [YJ15]. The unit forsubstitution into a given vertex v of a graph G is a corolla : a single vertex with the same number of incomingand outgoing legs as v , with the same labelling. An example of a corolla is shown in Figure 9a.Associativity implies that graph substitution can be carried out en masse , given a substitutable graph H v for each of the vertices of a graph G . This operation is denoted G ( { H v } v ∈ V ( G ) ) , or G ( { H v } ) for short. Theboundary flags of G ( { H v } ) are identified with the boundary flags of G , and there is a canonical identification(4.1) V ( G ( { H v } )) = (cid:97) v ∈ V ( G ) V ( H v ) . Example 4.6.
Graph substitution captures important structural changes in graphs. A key example is the relabelling substitution: one can change the boundary labelling of a graph G arbitrarily, by substituting G into a corolla whose vertex labels agree with the boundary labels of G , but whose boundary labels are thedesired new labels. The example shown in Figure 11 is a graph substitution which implements a permutationof boundary labels.4.2. Wheeled props.
In this section, wheeled props are defined as algebras over a monad as in [MMS09,Definition 2.1.7]. We first include some short reminders of the necessary categorical notions.An endofunctor T : C → C is called a monad if it comes equipped with two natural transformations µ : T → T and η : Id C → T called multiplication and unit which satisfy the associativity (4.2) and unitconditions of a monoid (4.3).(4.2) T T T T TT T T
T µµT µµ (4.3)
T T T T T ηηT = µµ Given a monad ( T, µ, η ) on a category C , a T -algebra is a pair ( X, γ ) where X is an object in C togetherwith a structure map γ : T ( X ) → X in C which commutes with the monad multiplication (4.4) and withthe unit (4.5). A morphism φ : X → Y between T -algebras is a morphism in C which is compatible with the T -action (4.6). The category of all T -algebras in C is denoted Alg T ( C ) . (4.4) T ( T ( X )) T ( X ) T ( X ) X T ( γ ) µ X γγ (4.5) X T ( X ) X η X id γ (4.6) T ( X ) T ( Y ) X Y γ X T ( φ ) γ Y φ Example 4.7.
A standard example of a monad is the “monad for monoids” T : Set → Set given by
T X = (cid:96) n ≥ X n , that is, words in X . Monadic multiplication is concatenation of words. The unit is givenby the inclusion η X : X = X (cid:44) → T X . An algebra over this monad is a choice of set M together withan associative and unital multiplication M × M → M . In other words, an algebra over this monad is anassociative monoid in Set .The monad “encoding” wheeled props arises from graph substitution. In short, it is an endofunctor oncategories of equivariant vector space valued diagrams indexed by a category of graphs . We explain thissentence in detail over the next few pages; the first step is to introduce the category of S -bimodules in Vect . Definition 4.8.
Given a countable alphabet I , an S -bimodule E is a family of vector spaces { E [ I ; J ] } I ; J ,where I and J run over finite subsets of I . Each E [ I ; J ] is equipped with commuting left and right actionsof the symmetric groups S I and S J , respectively. A morphism f : E → E (cid:48) of S -bimodules is an S I × S J -equivariant family of linear maps f I,J : E [ I ; J ] → E (cid:48) [ I ; J ] . The category of S -bimodules is denoted by Vect S .In the context of this paper S -bimodules are used to decorate vertices of graphs: if E is an S -bimoduleand v is a vertex of a graph G with ( in ( v ); out ( v )) = ( I ; J ) , then v can be decorated by an element of E [ I ; J ] and, to decorate the entire graph, these elements are tensor multiplied together in the order of the verticesof G : Definition 4.9.
Given an S -bimodule E = { E [ I ; J ] } and an isomorphism class of directed graphs [ G ] , wedefine the E -decorated graph as the tensor product E [ G ] = (cid:79) v ∈ V ( G ) E [ in ( v ); out ( v )] . If G has no vertices, then E [ G ] = k , the tensor unit.A counter-intuitive aspect of this Definition 4.9 is that the E -decorated graph is a vector space which doesnot “remember” the isomorphism class of G . The following lemma makes this observation explicit, by notingthat E [ G ] only depends on the input and output sets of the vertices of G : Lemma 4.10.
Assume that G and G (cid:48) are directed graphs with V ( G ) = { v , ..., v n } and V ( G (cid:48) ) = { w , ..., w n } ,and assume furthermore that ( in ( v i ); out ( v i )) = ( in ( w i ); out ( w i )) , for all i = 1 , ..., n . Then there is a canon-ical isomorphism E [ G ] ∼ = E [ G (cid:48) ] .Proof. Immediate from the definition. (cid:3)
The next lemma establishes the relationship between decorations and graph substitution:
Lemma 4.11.
Given an S -bimodule E , an oriented graph G and a collection { H v } v ∈ V ( G ) so that the graphsubstitution G ( { H v } ) is defined, there is a natural isomorphism E [ G ( { H v } )] ∼ = (cid:78) v ∈ V ( G ) E [ H v ] .Proof. This is a direct consequence of Formula (4.1). (cid:3)
Next, we define a category of graphs : Definition 4.12.
Let G ( I ; J ) denote the category whose objects are strict isomorphism classes (as in Def-inition 4.4) of graphs [G] with ∂G = ( in ( G ); out ( G )) = ( I ; J ) . Morphisms in G ( I ; J ) are bijections of flagswhich preserve the vertex sets, involutions, directions, and labels. Morphisms may permute the vertex order– that is, permute ordering of the partition, while fixing the exceptional cell. The category G is the coproduct (cid:96) G ( I ; J ) , where I and J run over finite subsets of I . IRCUIT ALGEBRAS ARE WHEELED PROPS 15
Note that morphisms in G ( I ; J ) are “isomorphisms” of graphs in a slightly looser sense than the strictisomorphisms of Definition 4.4.To summarise, the category Vect S is indexed by graphs, that is, graph decoration defines the object-levelof a bifunctor Vect S × G → Vect , were G is the category of graphs.We’re now ready to define the monad of graph substitution : this is an endofunctor F : Vect S Vect S . At the level of objects, F is defined by sending an S -bimodule E to an ( I ; J ) -indexed collection of vectorspaces, with symmetric group actions to be defined afterwards. As a vector space, FE [ I ; J ] := colim [ G ] ∈G ( I ; J ) E [ G ] , where E [ G ] is the graph decoration as in Definition 4.9. The colimit here is essentially a direct sum of vectorspaces; for convenience, we briefly recall the definition in this context:The colimit colim [ G ] ∈G ( I ; J ) E [ G ] is a vector space, equipped with, for every [ G ] ∈ G ( I ; J ) , a structuremap E [ G ] → colim [ G ] ∈G ( I ; J ) E [ G ] . Each morphism [ G ] → [ G (cid:48) ] of G ( I ; J ) induces a tensor factor permutingisomorphism E [ G ] → E [ G (cid:48) ] , and the structure maps are compatible (form commutative triangles) with eachof these isomorphisms. The colimit has the universal property that given any vector space V , with maps f [ G ] : E [ G ] → V for all [ G ] ∈ G ( I ; J ) , all the maps f [ G ] factor through the colimit via the structure maps,and the colimit is the unique vector space, up to a unique isomorphism, with this property.We say that FE [ I ; J ] is the space of E -decorations of (isomorphism classes of) graphs G with boundarylabels ( I ; J ) .To make F an endofunctor, it needs to take its values in Vect S , that is, we need to define commuting left S I and right S J actions on FE [ I ; J ] . Note that S I and S J have natural commuting actions on the indexingcategory G ( I ; J ) , by permuting the incoming and outgoing boundary labels of a graph G ∈ G ( I ; J ) . This canbe accomplished by substituting into the appropriate label permuting corolla, as in Figure 11. For σ ∈ S I , τ ∈ S J , let σGτ := C σ,τ ( G ) denote the graph with permuted boundary labels, where C σ,τ is the labelpermuting corolla. Then, by Lemma 4.10, there is a canonical isomorphism E [ G ] = E [ σGτ ] . The action ofthe pair ( σ, τ ) on FE [ I ; J ] sends the summand E [ G ] to E [ σGτ ] via this canonical isomorphism. Hence, F isindeed an endofunctor on Vect S .For the monad structure on F , we need to define the monadic multiplication µ : F → F . Informally, F E [ I ; J ] is “the space of graphs whose vertices are decorated by E -decorated graphs”, so one can picture amonadic multiplication µ : F E [ I ; J ] → FE [ I ; J ] defined by substituting all the “decorating graphs” into theappropriate vertices. This is possible as decorations “commute” with graph substitution by Lemma 4.11.The next proposition makes this paragraph precise. Proposition 4.13.
Graph substitution induces a natural transformation µ : F F . Proof.
Note that, for each graph G in G ( I ; J ) , graph substitution (Definition 4.5) describes a functor S : (cid:89) v ∈ V ( G ) G ( in ( v ); out ( v )) → G ( I ; J ) which sends a family of graphs { H v } v ∈ V ( G ) to G ( { H v } ) at the level of objects.A morphism (cid:81) v ∈ V ( G ) ϕ v : { H v } → { H (cid:48) v } is a permutation of the vertex sets of each H v . Via thecanonical bijection V ( G ( { H v } )) = (cid:70) v ∈ V ( G ) V ( H v ) , the disjoint union of the permutations ϕ v induces avertex permutation on V ( G ) , which is, in turn, a morphism in G ( I ; J ) . This defines S at the level ofmorphisms, and it is clear that S is a functor.By definition, F E [ I ; J ] = colim [ G ] ∈G ( I ; J ) ( F E )[ G ] . To define a natural transformation µ : F → F , it issufficient to define, for each G ∈ G ( I ; J ) , a map m : ( F E )[ G ] → FE [ I ; J ] . Then the universal property of thecolimit gives rise to the map µ :(4.7) ( F E )[ G ] F E [ I ; J ] FE [ I ; J ] . m µ To define the map m , observe the following: ( F E )[ G ] = (cid:79) v ∈ V ( G ) ( FE )[ in ( v ); out ( v )] = (cid:79) v ∈ V ( G ) colim [ H v ] ∈G ( in ( v ); out ( v )) E [ H v ] (4.8) ∼ = colim G ( v ) (cid:79) v ∈ V ( G ) E [ H v ] (cid:39) colim G ( v ) E [ G ( { H v } )] , (4.9)where we have written G ( v ) = (cid:81) v ∈ V ( G ) G ( in ( v ); out ( v )) to shorten notation. The equalities in (4.8) follow fromthe definition of the functor F . The first isomorphism is the fact that tensor products commute with colimitsin Vect , and the second isomorphism is by Lemma 4.11.Since [ G ( { H v } )] ∈ G ( I ; J ) , the graph substitution functor S induces a map ˜ S , which composes with theisomorphism above to define m : ( F E )[ G ] colim { H v }∈G ( v ) E [ G ( { H v } )] colim [ K ] ∈G ( I ; J ) E [ K ] FE [ I ; J ] . ∼ = m ˜ S ∼ = (cid:3) To define the monad unit, note that if C ( I ; J ) is the corolla whose vertex labels agree with its boundarylabels ( I ; J ) then E [ C ( I ; J ) ] = E [ I ; J ] . Definition 4.14.
The map E [ C ( I ; J ) ] colim G ( I ; J ) E [ G ] defines a natural transformation η E : { E [ I ; J ] } { FE [ I ; J ] } . The following proposition is used to define wheeled props in [MMS09, Section 2]:
Proposition 4.15.
The endofunctor F together with the natural transformations µ : F → F and η :Id Vect S → F is a monad on the category Vect S .Proof. The natural transformations µ (Definition 4.13) and η (Definition 4.14) are associative and unitalsince graph substitution is associative and unital. (cid:3) Definition 4.16.
A linear wheeled prop E is an algebra over the monad F in the category of S -bimodules Vect S . The category of linear wheeled props is the category of F -algebras and is denoted by wProp .In other words, a wheeled prop is an S -bimodule E together with an action γ : F E → E . Note that,given an S -bimodule E , the free wheeled prop generated by E is the S -bimodule F E with structure map themonadic multiplication µ : F E → F E . We give examples and an alternative description of wheeled props inSection 6. Remark 4.17.
The reader may have noticed that circuit algebras are described as algebras over an operadand wheeled props are described as algebras over a monad. In general, one can associate, to any operad O , amonad M O with the property that O -algebras are M O -algebras. It is not the case, however, that all monadscome from operads. For full details see [Lei04, Appendix C].5. Equivalence
In this section we prove that there is an equivalence of categories between the category of circuit algebrasand the category of linear wheeled props. The key observation is that wiring diagrams (Definition 2.1)are in bijection with oriented graphs (Definition 4.3), and under this bijection, wiring digram compositioncorresponds with graph substitution. This correspondence leads to the equivalence of categories proven inTheorem 5.5.
IRCUIT ALGEBRAS ARE WHEELED PROPS 17
Graphs and Wiring Diagrams.
The goal of this subsection is to define a (structure respecting)correspondence
Φ : { iso. classes of oriented labelled graphs } → { wiring diagrams } . The idea behind Φ is that the vertices of graphs can be viewed as input circles of wiring diagrams, asillustrated in Figure 12. The technical details take more work. Figure 12.
The correspondence between graphs and wiring diagrams.Recall that the data of an oriented graph G consists of a set of flags F ( G ) , an ordered partition F ( G ) =( (cid:116) α ∈ V ( G ) v α ) (cid:116) v (cid:15) , an involution ι , a fixed point free involution π on the ι -fixed points of v (cid:15) , an orientationmap F → {− , } , and labelling functions λ : (cid:96) α ∈ V ( G ) v α → I on the vertices, and β : ∂G → I on theboundary.By Proposition 2.3, a wiring diagram is a triple D = ( A , p, l ) where A is the set of finite sets of labels { A in / out , ..., A in / out r } , p is a perfect matching (bijection) between the finite sets (cid:70) ri =0 A out i and (cid:70) ri =0 A in i , and l ∈ Z ≥ is a non-negative integer (the number of circles). Construction 5.1.
We construct a correspondence Φ which assigns to an isomorphism class of orientedlabelled graphs [ G ] – represented by a graph G – a wiring diagram D [ G ] = ( A , p, l ) .(1) We define the number l = { ϕ ∈ v (cid:15) : ι ( ϕ ) (cid:54) = ϕ } , where the sign denotes the cardinality of theset that follows it. Simply put, l is the number of free loops in G .(2) The vertices V ( G ) = { , , ..., r } give rise to A = { A in / out , A in / out , ..., A in / out r } by setting- in ( G ) = A out , out ( G ) = A in . (Note: the switch of in/out here is intentional and is due toopposite conventions.)- in ( v i ) = A in i , out ( v i ) = A out i for i = 1 , ..., r. (3) The bijection p is built as follows:- For each label a ∈ A out / in i , ≤ i ≤ r , there is a unique negative/positive (respectively) flag λ − ( a ) ∈ v i . If λ − ( a ) is not fixed by ι , and ιλ − ( a ) ∈ v j then p ( a ) := λιλ − ( a ) ∈ A in / out j . If ι ( λ − ( a )) = λ − ( a ) then p ( a ) := βλ − ( a ) ∈ A in / out .- For each label a ∈ A out / in , there is a unique positive, respectively negative (once again in/outconventions are opposite here) flag β − ( a ) ∈ ∂G . If β − ( a ) ∈ v i for some i , then define p ( a ) := λβ − ( a ) ∈ A in / out i . If β − ( a ) ∈ v (cid:15) , then define p ( a ) := βπβ − ( a ) ∈ A in / out .Since labelled graph isomorphisms preserve the ordered partitions, involutions, orientation and labellings(Definition 4.4), the wiring diagram D [ G ] does not depend on the representative of the isomorphism class [ G ] . Hence, Φ is well-defined.Next, we construct an inverse map Ψ : { Wiring diagrams } → {
Iso. classes of oriented labelled graphs } , which, intuitively, turns input circles into vertices and removes the output circle. Construction 5.2.
Given a wiring diagram D = ( { A out / in , ..., A out / in r } , p, l ) , we define an isomorphism classof graphs Ψ( D ) = [ G D ] as follows. (1) The set of flags F ( G D ) := { ( a, i, out / in ) : a ∈ A out / in i for ≤ i ≤ r } ∪{ ( a, , out / in ) : a ∈ A out / in and p ( a ) ∈ A in / out } ∪ { c j } lj =1 . Here, “ ( a, i, out / in ) ” stands for “ ( a, i, out ) and ( a, i, in ) ”.(2) The flags F ( G ) are partitioned into vertices (cid:83) ri =1 v i ∪ v (cid:15) with v i := { ( a, i, out / in ) : a ∈ A out / in i } for ≤ i ≤ r, and v (cid:15) = { ( a, , out / in ) : a ∈ A out / in , and p ( a ) ∈ A in / out } ∪ { c j } lj =1 . (3) The involutions ι and π are set as follows:- On the flags in the vertices v i , i = 1 , ..., r , ι ( a, i, out / in ) = (cid:40) ( p ( a ) , j, in / out ) , where p ( a ) ∈ A in / out j , j (cid:54) = 0( a, i, out / in ) , where p ( a ) ∈ A in / out . This defines the internal edges of the graph G D , and the boundary edges connected to a vertex.- If a ∈ A out / in such that p ( a ) ∈ A in / out , then ( a, , out / in ) is an ι -fixed point, and π ( a, , out / in ) =( p ( a ) , , in / out ) . This gives free-floating edges in the graph G D .- It remains to describe the free floating loops on G D . For c j , where j is odd, ι ( c j ) = c j +1 andwhere j is even, ι ( c j ) = c j − .(4) The labelling functions λ and β are as follows. For each ( a, i, out / in ) ∈ v i ( i = 1 , ..., r ) , we set λ ( a, i, out / in ) = a . Furthermore, if ( a, i, out / in ) ∈ v i is an ι -fixed point, then set β ( a, i, out / in ) = p ( a ) . For the ι -fixed points of type ( a, , out / in ) , define β ( a, , out / in ) = a .(5) The direction function is defined by δ ( a, i, in ) = 1 and δ ( a, i, out ) = − where i = 1 , ..., r ; while δ ( a, , in ) = − and δ ( a, , out ) = 1 . Keep in mind that free floating loop flags don’t have signs – intechnical notation, F = F \ { c j } lj =1 . Lemma 5.3.
The maps Φ and Ψ are inverse maps of sets, therefore set bijections.Proof. It is straightforward to check that Φ ◦ Ψ is the identity map on wiring diagrams. For a graph G ,the composition Ψ(Φ( G )) renames the flags of G, but retains the labelling, therefore it does not change theisomorphism class of G . (cid:3) Lemma 5.4.
The map Ψ translates wiring diagram composition to graph substitution: Ψ( D ◦ i D ) = Ψ( D ) (cid:0) Ψ( D ) i (cid:1) . Proof.
A straightforward verification. (cid:3)
A concise way to summarize the above is that the operad structure of wiring diagrams induces the monadstructure on graphs, via the map Φ . Formally, this implies that they admit the same algebras, a statementwe unpack in the final proof below.5.2. Equivalence of Categories.
We are now ready to prove the main result of this paper:
Theorem 5.5.
There is an equivalence of categories
CA wProp (cid:101) Φ (cid:101) Ψ between the category of circuit algebras and the category of linear wheeled props. Lemma 5.6.
Every circuit algebra has an underlying S -bimodule.Proof. This is simply stating that the collection of vector spaces that comprise a circuit algebra V = { V [ T out ; T in ] } each carry natural symmetric group actions by S T out on the left and S T in on the right, where T out , T in ⊆ I . Indeed, this is the case, induced by the action of label permuting wiring diagrams as explainedin Example 2.6. (cid:3) IRCUIT ALGEBRAS ARE WHEELED PROPS 19
Proof of Theorem 5.5.
First, we show that every a circuit algebra V admits the structure of a wheeled prop (cid:101) Φ( V ) = V . By Lemma 5.6, V has an underlying S -bimodule V = { V [ I ; J ] } I,J ⊆I . To describe a wheeledprop structure on V , we need to exhibit a structure map γ : F V → V .Recall from Section 4.2 that FV [ I ; J ] = colim [ G ] ∈G ( I ; J ) V [ G ] . In other words, FV [ I ; J ] is linearly spannedby isomorphism classes of labelled directed graphs [ G ] with ∂G = ( I ; J ) , where the vertices v i ∈ V ( G ) aredecorated with the vector spaces V [ in ( v i ); out ( v i )] .The map Φ from Construction 5.1 assigns to [ G ] a wiring diagram D [ G ] . By definition of the circuitalgebra structure on V , D [ G ] induces a linear map F D [ G ] : (cid:78) ri =1 V [ in ( v i ); out ( v i )] → V [ I ; J ] .The maps F D [ G ] are natural in [ G ] : any map [ G ] → [ G (cid:48) ] in G ( I ; J ) corresponds to a permutation of vertexorder, which is respected by assignment of linear maps in the circuit algebra (Axiom (2) in Definition 2.4)and thus we define the structure map γ using the universal property of the colimit, for each pair of label sets I, J ⊆ I : V [ G ] FV [ I, J ] = colim [ G ] ∈G ( I,J ) V [ G ] V [ I ; J ] F D [ G ] γ Thus, V is a wheeled prop. It is clear from its construction that a circuit algebra map V → W is automaticallyalso wheeled prop map V → W ( (cid:101) Φ is natural in V ), and thus we have defined a functor (cid:101) Φ : CA → wProp .In the other direction, given a linear wheeled prop W we construct a circuit algebra (cid:101) Ψ( W ) = W . First, W has an underlying S -bimodule, which in particular is a collection of vector spaces W [ I ; J ] , where I and J run over finite subsets of I . It remains to construct the action maps F D : W [ A out ; A in ] ⊗ ... ⊗ W [ A out r ; A in r ] → W [ A in ; A out ] for each wiring diagram D = ( A , p, l ) . The map Ψ assigns to D an isomorphism class of graphs [ G D ] = Ψ( D ) .We define the action F D as the restriction of γ : colim [ G ] ∈G ( I,J ) W [ G ] → W [ I ; J ] to the component W [ G D ] .The composition axiom – Axiom (1) – of Definition 2.4 holds by Lemma 5.4 and axiom (4.4) of an algebraover a monad, as µ captures wiring diagram composition, and γ captures the assignment of linear maps towiring diagrams. The equivariance – Axiom (2) – holds by the naturality of Ψ in G D : input set permutations D → D (cid:48) correspond to morphisms G D → G D (cid:48) in G . The assignment (cid:101) Ψ : wProp → CA is natural in W andthus (cid:101) Ψ defines a functor. The fact that (cid:101) Φ and (cid:101) Ψ are inverse functors follows from Lemma 5.3. (cid:3) Wheeled props as tensor categories
In [Mac65, Chapter V], Mac Lane introduced props as strict symmetric tensor categories whose monoidof objects has a single generator: that is, a symmetric tensor category equipped with a distinguished object x such that every object is a tensor power x ⊗ n , for some n ≥ . Hence, morphisms in a prop are of theform f : x ⊗ n → x ⊗ m . Diagrammatically, such a morphism is illustrated by an ( n, m ) -corolla whose vertexis decorated by f (as on the left in Figure 13).Composition of morphisms, also called vertical composition , is modelled diagrammatically by attachingsome of the outputs of one corolla to inputs of another, resulting in directed graphs. Directed graphs arecomposed the same way. The tensor product of the prop is realised by taking disjoint unions of graphs, andis called horizontal composition . For examples of both compositions see Figure 13.In other words, props are categories in which morphisms are directed graphs, where every edge “carries”a copy of the generator x . Note that these graphs have no floating loops or closed cycles; a floating edgedenotes the identity id : x → x . For more details on this point of view, and examples of props, we suggestthe survey article [Mar08, Section 8].A wheeled prop is a prop where every object has a dual. This gives rise to a family of linear “trace” or“contraction” maps t ji : x ∗⊗ m ⊗ x ⊗ n x ∗⊗ m − ⊗ x ⊗ n − . Diagrammatically, contractions are represented by connecting a chosen output of a graph (the j th copy of x ) to a chosen input of the same graph (the i th copy of x ∗ ), as in Figure 14. Figure 13.
The tensor product of morphisms f and hg where f : x → x ⊗ and hg is thecomposite of morphisms g : x → x ⊗ and h : x ⊗ → x ⊗ in a prop P . Stacking morphismsnext to each other like this is called horizontal composition . The composition hg is called vertical composition . Figure 14.
The horizontal product t f ∗ hg .We note that in the literature a strict symmetric tensor category with duals is also called a rigid sym-metric tensor category. Saying that such a tensor category has a single generating object is equivalent tosaying that the monoid of objects has a single generator x . [Del90, JSV96]In this section, we present an axiomatic definition of a wheeled prop in line with this categorical view,which is equivalent to the monadic definition given in Definition 4.16. In light of Theorem 5.5, one canequivalently interpret the following as a set of axioms satisfied by circuit algebras. In Section 6.2, we presenttwo prominent examples of wheeled props. Finally, in Section 6.3 we show that every wheeled prop is,in particular, a strict pivotal category. This gives a fully faithful embedding into the category of pivotalcategories, parallel to that between circuit algebras and planar algebras in Proposition 3.3.6.1. Axiomatic definition.Definition 6.1.
Let I denote a fixed countable alphabet. A wheeled prop E := (cid:16) E , ∗ , t ji (cid:17) consists of:(1) an S -bimodule E = { E [ I ; J ] } ;(2) a horizontal composition ∗ : E [ I ; J ] ⊗ E [ K ; L ] E [ I ∪ K ; J ∪ L ] , where I ∩ K = ∅ and J ∩ L = ∅ ;(3) a linear map ∅ : k → E [ ∅ ; ∅ ] called the empty unit ;(4) a contraction operation E [ I ; J ] E [ I \ { i } ; J \ { j } ] , t ji for every pair i ∈ I and j ∈ J ; Rigid symmetric tensor categories are also called compact closed categories.
IRCUIT ALGEBRAS ARE WHEELED PROPS 21 (5) a linear map i : k → E [ { i } ; { i } ] , for every i ∈ I , called the unit .This data satisfies a list of axioms that we will present in detail shortly. In particular, the horizontalcomposition and contractions commute with each other and are associative, equivariant and unital. Remark 6.2.
Wheeled props are, in particular, examples of props ([MMS09, Example 2.1.1]). To definevertical composition of morphisms such as the composition of h and g in Figure 13, one combines horizontalcompositions and contractions, as in Figure 15.In fact, for i ∈ I, l ∈ L , and I ∩ K = ∅ and J ∩ L = ∅ , the horizontal composition and contractionoperations combine to give an additional dioperadic composition denoted i ◦ l , which joins the l th output ofone graph to the i th input of another: E [ I ; J ] ⊗ E [ K ; L ] E [ I \ { i } ∪ K ; J ∪ L \ { l } ] E [ I ∪ K ; J ∪ L ] i ◦ l ∗ t li . Vertical composition can then be obtained by a horizontal composition followed by iterated contractions.As an example, the morphism hg from Figure 14 is obtained by first taking a horizontal composition of h ∈ x ∗⊗ ⊗ x ⊗ and g ∈ x ∗⊗ ⊗ x ⊗ and then applying contractions t j i and t j i . See Figure 15. Figure 15.
The vertical composition hg depicted on the right is horizontal compositionfollowed by two contractions: t j i t j i ( g ∗ h ) .The following is a comprehensive list of axioms satisfied by the horizontal composition, contractionsand the units. Throughout, by an abuse of notation we write “ σI ” to indicate that the set I has been“permuted” by σ ∈ S I . Of course, I is a set and σ is a bijection so as sets σI = I , however this notation isuseful in practical examples with a naturally ordered alphabet, e.g. when I = Z ≥ . For example, the pair ( σ, τ ) ∈ S I × S J acts on the vector space E [ I ; J ] and we write ( σ, τ ) : E [ I ; J ] → E [ σI ; Jτ ] . H1:
The horizontal composition is associative in the sense that the following square commutes: E [ I ; J ] ⊗ E [ K ; L ] ⊗ E [ M ; N ] E [ I ∪ K ; J ∪ L ] ⊗ E [ M ; N ] E [ I ; J ] ⊗ E [ K ∪ M ; L ∪ N ] E [ I ∪ K ∪ M ; J ∪ L ∪ N ] . ∗⊗ idid ⊗∗ ∗∗ The i ◦ l notation means “identify output l with input i .” H2:
The horizontal composition is bi-equivariant . Explicitly, for any two pairs of disjoint finite subsets
I, K and
J, L of I , the following square commutes: E [ I ; J ] ⊗ E [ K ; L ] E [ I ∪ K ; J ∪ L ] E [ σ I ; Jτ ] ⊗ E [ σ K ; Lτ ] E [ σ I ∪ σ K ; Jτ ∪ Lτ ] . ∗ ( σ ; τ ) ⊗ ( σ ; τ ) ( σ ∪ σ ; τ ∪ τ ) ∗ The notation σ ∪ σ refers to the element of S I ∪ K which acts as σ on I and σ on K . Similarly, τ ∪ τ ∈ S J ∪ L acts by τ on J and τ on L . H3:
Horizontal composition is symmetric . Let β denote the block permutation β = (12) (cid:104) I, K (cid:105) ∈ S I ∪ K ,which swaps the blocks I and K of I ∪ K . In the same notation, γ = (12) (cid:104) J, L (cid:105) ∈ S J ∪ L swaps the blocks J and L in J ∪ L . Then, for any two pairs of disjoint subsets I, K and
J, L of I , the following diagramcommutes: E [ I ; J ] ⊗ E [ K ; L ] E [ I ∪ K ; J ∪ L ] E [ K ; L ] ⊗ E [ I ; J ] E [ K ∪ I ; L ∪ J ] . ∗⊗ -swap ( β ; γ ) ∗ H4:
The empty unit ∅ is a two-sided unit for the horizontal composition: E [ I ; J ] k ⊗ E [ I ; J ] E [ I ; J ] ⊗ k E [ ∅ ; ∅ ] ⊗ E [ I ; J ] E [ I ; J ] ⊗ E [ ∅ ; ∅ ] E [ I ; J ] . ∼ = ∼ = =1 ∅ ⊗ id id ⊗ ∅ ∗ ∗ Remark 6.3.
A wheeled prop E is, in particular, a prop, and thus, a symmetric tensor category with asingle generating object. The axioms H1 – H4 are the axioms that govern the symmetric tensor product on E . The next set of axioms shows that contractions in E are also bi-equivariant, and commute with each otherand the horizontal composition, providing the remainder of the rigid symmetric tensor category structure on E . The unit i is the unit for vertical composition – i.e. composition of morphisms in the tensor category –which arises as a horizontal composition followed by a contraction. Remark 6.4.
In order to precisely state the bi-equivariance axiom for contractions, one needs to establishthat any pair of relabelling bijections ( f, g ) : ( I × J ) → ( I (cid:48) × J (cid:48) ) induce natural isomorphisms of the vectorspaces R f ; g : E [ I ; J ] E [ I (cid:48) ; J (cid:48) ] . We leave it as an exercise to the reader to construct these relabellingisomorphisms from the wheeled prop structure. C1:
Contraction is bi-equivariant : for any pair of non-empty label sets
I, J ⊆ I , the following diagramcommutes: E [ I ; J ] E [ I \ { i } ; J \ { j } ] E [ σI ; Jτ ] E [ σI \ { i } ; Jτ \ { j } ] t ji ( σ ; τ ) R σ | I \{ i } ; τ | J \{ j } t τ − j ) σ ( i ) Here σ ∈ S I , τ ∈ S J , and σ | I \{ i } and τ | J \{ j } are restrictions of the permutations – note that in generalthese are no longer permutations, but relabellings, and R σ | I \{ i } ; τ | J \{ j } is the induced isomorphism as inRemark 6.4. C2:
Contraction maps commute: given any labelling sets
I, J ⊆ I with | I | ≥ , | J | ≥ , i (cid:54) = k ∈ I and j (cid:54) = l ∈ J , then the operations t ji and t lk commute: IRCUIT ALGEBRAS ARE WHEELED PROPS 23 E [ I ; J ] E [ I \ { i } ; J \ { j } ] E [ I \ { k } ; J \ { l } ] E [ I \ { i, k } ; J \ { j, l } ] . t ji t lk t lk t ji HC1:
Horizontal composition and contraction maps commute with one another: for any pairs of disjointsubsets
I, K and
J, L of I , and any chosen i ∈ I , j ∈ J , k ∈ K and l ∈ L , the following two squares commute. E [ I ; J ] ⊗ E [ K ; L ] E [ I ∪ K ; J ∪ L ] E [ I \ { i } ; J \ { j } ] ⊗ E [ K ; L ] E [ I \ { i } ∪ K ; J \ { j } ∪ L ] ∗ t ji ⊗ id t ji ∗ E [ I ; J ] ⊗ E [ K ; L ] E [ I ∪ K ; J ∪ L ] E [ I ; J ] ⊗ E [ K \ { k } ; L \ { l } ] E [ I ∪ K \ { k } ; J ∪ L \ { l } ] . ∗ id ⊗ t lk t lk ∗ HC2:
The units i are the units for the dioperadic compositions, which are themselves compositionsof horizontal compositions and contractions as defined in Remark 6.2. Specifically, for every pair of sets I, J ⊆ I and labels i ∈ I, j ∈ J , the following diagrams commute. k ⊗ E [ I ; J ] E [ I ; J ] E [ { i } ; { i } ] ⊗ E [ I ; J ] i ⊗ id ∼ = t ii ◦∗ E [ I ; J ] ⊗ k E [ I ; J ] E [ I ; J ] ⊗ E [ { j } ; { j } ] id ⊗ j ∼ = t jj ◦∗ In the context of this definition, morphisms
E E (cid:48) f of wheeled props are tensor functors whichrespect the contractions.The axiomatic Definition 6.1 is equivalent to Definition 4.16. The key to understanding this is thetranslation between graph substitution, and the horizontal composition and contraction operations. Observethat any connected graph G – which is not a free floating loop or edge – can be constructed from iteratedsubstitution of elementary directed graphs : graphs which have either one or two vertices, and one or zerointernal edges (edges where both flags are part of a vertex). An E -decorated graph with one vertex representsa composition of contraction operations, and a graph with two vertices a dioperadic composition – itself acombination of horizontal composition with contractions – with possibly additional contractions. Figure 14shows an example of an elementary graph with one vertex, as well as a graph with two vertices obtainedfrom two elementary graphs.Thus, a graph G without floating loops and edges represents a sequence of iterated contractions andhorizontal compositions. The empty unit is represented by the empty graph; the unit by a floating edge,and the floating loop represents a contraction applied to the unit. It is non-trivial to check that the axiomsabove are equivalent to the algebra structure over the monad of graph substitutions. In the literature thisis often called an equivalence of the unbiased definition (monadic) and the biased definition (axiomatic). Afull proof of this equivalence can be found, for example, in [YJ15, 11.9.3, Corollary 11.35].6.2. Examples.
Wheeled props arise in the literature in a range of contexts, for example as natural wheeledextensions of the associative and commutative operads in [MMS09], and have applications in geometry andphysics. We recommend the survey article [Mer10b] for full details.In this section we present an example that in the authors’ opinion illuminates some of the structureencoded in a wheeled prop: namely, a wheeled prop whose category of algebras is the category of semisimpleLie algebras. An algebra over a wheeled prop W is a morphism from W to an endomorphism wheeled prop End( E ) , defined in Example 6.6. The term “algebra” is somewhat confusing: based on the definition, algebrasover a wheeled prop may be more intuitively named representations of the wheeled prop. There is a corresponding statement for circuit algebras, stating that all wiring diagrams are generated via compositionsfrom “elementary” wiring diagrams, which realise disjoint unions, dioperadic compositions and contractions.
Remark 6.5.
In fact, the statement we prove below is stronger than simply describing the wheeled propfor semisimple Lie algebras. In [Kap99], Kapranov constructs a prop from any operad P , by adjoining amodule of P -algebra forms to the prop generated by P . We will show below that for P = Lie (the operadfor Lie algebras), considering Kapranov’s prop as a wheeled prop, a finite dimensional algebra over it thatsatisfies non-degeneracy conditions for the P -algebra forms is a semisimple Lie algebra. We take no creditfor originality of this construction – Kapranov’s construction preceded the definition of wheeled prop in[MMS09] by several years.The contraction operations in a wheeled prop can be seen as a generalized trace operation. We begin withthe definition of endomorphism wheeled props, which makes this precise, as there the contraction maps arethe standard trace maps of linear algebra. Given k -vector spaces U , V , and W , a linear map f : V ⊗ U → W ⊗ U is given by f ( v i ⊗ u j ) = Σ k,m α kmij w k ⊗ u m , where v i , u j and w k run over a chosen basis for V , U and W ,respectively. Recall that the trace of f with respect to U is given by t uu f ( v i ) = Σ j,k α kjij w k . If V and W areone dimensional, this formula reduces to the trace of the matrix of the linear map f : U → U . Example 6.6.
For simplicity, set the alphabet I = Z ≥ to be non-negative integers, and use label sets n = { , , ..., n } . Fix a finite dimensional k -vector space E and denote its linear dual by E ∗ . We define afamily of vector spaces, for ( n, m ) ∈ Z ≥ : End( E )[ n ; m ] := Hom k ( E ⊗ n , E ⊗ m ) ∼ = ( E ∗ ) ⊗ n ⊗ E ⊗ m . Linear maps can be pre- and post-composed with actions of the symmetric groups S m and S n which permutethe tensor factors, making the collection End( E ) = { End( E )[ n ; m ] } into an S -bimodule.Using abbreviated notation, write ( φ ⊗ . . . ⊗ φ n ) ⊗ ( w ⊗ . . . ⊗ w m ) ∈ ( E ∗ ) ⊗ n ⊗ E ⊗ m ∼ = End( E )[ n ; m ] as φ ⊗ w for short. The horizontal composition ∗ : End( E )[ n ; m ] ⊗ End( E )[ k ; l ] End( E )[ n + k ; m + l ] is defined as concatenation ( φ ⊗ w ) ∗ ( φ (cid:48) ⊗ w (cid:48) ) := ( φ ⊗ φ (cid:48) ) ⊗ ( w ⊗ w (cid:48) ) , and extended linearly.Using the same notation, the contraction maps are defined by t ji ( φ ⊗ w ) := φ i ( w j ) · (cid:0) ( φ ⊗ · · · ⊗ φ i − ⊗ φ i +1 ⊗ · · · ⊗ φ n ) ⊗ ( w ⊗ · · · ⊗ w j − ⊗ w j +1 ⊗ · · · ⊗ w m ) (cid:1) for any ≤ i ≤ n and ≤ j ≤ m . In other words, following the standard definition of trace above, thecontraction operation t ji : End( E )[ n ; m ] → End( E )[ n − m − applied to a linear map φ ⊗ w in End( E )[ n ; m ] given by t ji ( φ ⊗ w ) is the classical trace detailed above withrespect to the i th copy of E ∗ and the j th copy of E , using the isomorphism E ∼ = E ∗ specified by the choiceof basis. See [MMS09, Example 2.1.1] for full details. Figure 16.
A graphical interpretation of trace.
IRCUIT ALGEBRAS ARE WHEELED PROPS 25
Figure 17.
The antisymmetry and Jacobi relations, to be understood locally, that is, i , i , i , j and j each denote trivalent graphs. Figure 18.
An element t ( p ) in Lie [2; 0]
Example 6.7.
The main example for this section is the wheeled prop
Lie w for semisimple Lie algebras. Weassemble the vector spaces Lie w [ n ; m ] from two ingredients: the vector spaces Lie ( n ) generated by free Liewords on n letters, and formal traces t ( p ) for a Lie word p .Let Lie ( n ) denote the k -vector space spanned by all the Lie words in the free Lie algebra generated by letters x , . . . , x n , with each letter x i appearing exactly once. Diagrammatically, such Lie words are representedby directed trivalent graphs with n inputs labelled x , . . . , x n , and output; satisfying that every trivalentvertex has two inputs and one output; and these graphs are considered modulo the antisymmetry and Jacobirelations of Figure 18. For the reader familiar with operads, these are the spaces that make up the arity n -operations of the operad Lie . The vector space
Lie ( n ) admits a natural S n action, by permuting the letters x i . The prop Lie , whosealgebras are Lie algebras, is freely generated – using horizontal and vertical compositions – by setting
Lie [ n,
1] :=
Lie ( n ) . We will not describe the prop Lie in detail, but instead adjoin formal trace operations toobtain the wheeled prop
Lie w .Denote by t ( p ) the result of identifying the single output of a Lie word p ∈ Lie ( n + 1) with its last input,as in Figure 18. Denote by Lie [ n ; 0] , for n ≥ , the vector space formally spanned by the symbols t ( p ) for p ∈ Lie ( n + 1) . The symmetric group S n acts on Lie [ n ; 0] by permuting the first n letters of p . The symbols t ( p ) satisfy the following relations:(1) For σ ∈ S n , t ( σp ) = σ t ( p ) , where σp is the action of σ on p via the standard embedding S n (cid:44) → S n +1 .(2) For p ∈ Lie ( n + ) and q ∈ Lie ( m + ) , write p ◦ i q ∈ Lie ( n + m +1) for the dioperadic composition givenby gluing the unique output of q to the i th input of p (This would be denoted p i ◦ q in Remark 6.2,we have dropped the indexing of the output of q , as it is unique). Then, t ( p ◦ i q ) = t ( p ) ◦ i q , whenever p ∈ Lie ( n + 1) , q ∈ Lie ( m + 1) and i (cid:54) = n + 1 .(3) Finally, the trace operations are cyclically symmetric: t ( p ◦ n +1 q ) = β t ( q ◦ m +1 p ) , where β is the blocktransposition of [1 , ..., n ] and [1 , ..., m ] , as shown in Figure 19. Precisely, this is the operad for Lie algebras, considered as a prop as in [BV73].
Figure 19.
An example of relation (3) for
Lie [ n ; 0] .Now we’re ready to define Lie w = { Lie w [ n ; m ] } , by first setting Lie w [ n ; m ] = (cid:77) A ∪···∪ A m ∪ B ∪···∪ B r (cid:79) i Lie ( a i ) ⊗ (cid:79) j Lie [ b j ; 0] , where the sets A ∪· · ·∪ A m ∪ B ∪· · ·∪ B r run over all partitions of the set n , the numbers a i = | A i | , b j = | B j | denote their cardinalities, and the words in Lie ( a i ) are on the letters x α for α ∈ A i . Diagrammatically, thesespaces are spanned by graphs whose connected components are the Lie graphs in Lie ( n ) or trace graphs t ( p ) .The permutation group S n acts by permuting the input labels x α of the graphs. There is also a right actionby S m , where τ ∈ S m acts as τ − on the sets A i and tensor factors Lie ( a i ) .Horizontal composition on Lie w is given by concatenation of tensor factors, that is, disjoint union ofgraphs. Contraction on Lie w is given by identifying an output of a graph in Lie w [ n ; m ] with an input. This iseither a dioperadic composition (joining two separate connected components), or a permuted trace symbol(connecting the output of a connected component to one of its own inputs). Next, we need to check thatthis structure satisfies the axioms of a wheeled prop.The horizontal composition axioms are easy to verify. Essentially, the contraction axioms follow fromthe relations (1), (2) and (3) imposed on Lie [ n ; 0] . For example, the equivariance axiom C1 is true by therelation (1) for traces.To verify the commutativity of contraction maps, C2 , one needs to analyse different cases depending onwhether the two contraction maps are of the “dioperadic composition” or the “trace” type: • For two dioperadic compositions – that is, if the two contractions involve three or four separateconnected components – the axiom clearly holds. The same is true for a dioperadic compositionwith a trace map on a separate component; and for two trace maps on separate components. • For a dioperadic composition between components, and a permuted trace on one of those components,the axiom C2 holds by the relation (2). • Given two dioperadic compositions between the same pair of Lie graph components, the first isperformed as a dioperadic composition, while the second is a permuted trace map on the resultinggraph. If each outgoing edge is connected to the last incoming edge of the other graph, then theaxiom C2 follows directly from property (3), see Figure 19 to visualize this. If the outgoing edgesare connected to other incoming edges, then one applies antisymmetry permutations to reduce thisto the earlier scenario.We leave it as an exercise to the reader to verify the remaining axioms. To summarize, the relations (1),(2) and (3) make Lie w a wheeled prop. Note that the converse is also true: the wheeled prop axioms force therelations (1), (2) and (3). In other words, Lie w can equivalently be described as the wheeled prop generatedby the Lie word [ x , x ] , denoted I ; using horizontal compositions, contractions and units; and modulo theAnti-Symmetry and Jacobi relations: Lie w = WP (cid:10) I | Anti-Symmetry , Jacobi (cid:11)
IRCUIT ALGEBRAS ARE WHEELED PROPS 27
In effect, the symbols in
Lie [ n ; 0] parametrise generalized Killing forms κ n := t ([ x , [ x , ... [ x n , x n +1 ] ... ]) . Indeed, Proposition 3.4.4 [Kap99] shows that, as vector spaces,
Lie [ n ; 0] is isomorphic to the vector spacewhose basis is given by the non-cyclic permutations of the κ n . An algebra over the wheeled prop Lie w is, by definition, a morphism of wheeled props α : Lie w → End( E ) .As such, every algebra is determined by where it sends the wheeled prop generator. The image of I picksout a bracket in End( E )[2; 1] = Hom k ( E ⊗ , E ) . This is subject to the Anti-Symmetry and Jacobi relationswhich hold in Lie w .Moreover, the elements t ( p ) ∈ Lie [ n ; 0] (which are themselves obtained from I using horizontal com-positions and contractions) are sent to the Killing forms κ n = t ([ x , [ x , ... [ x n , x n +1 ] ... ]) in End( E ⊗ n , k ) .Note that relation (3) guarantees that this is a symmetric bilinear form. If the target vector space E is afinite-dimensional vector space, then the κ n ’s are the Killing forms x ⊗ . . . ⊗ x n (cid:55)→ tr ( ad ( x ) . . . ad ( x n )) . In this finite dimensional case it follows that an algebra α : Lie w → End( E ) that takes t ([ x , [ x , x ]]) to anon-degenerate form makes E into a semisimple Lie algebra.6.3. Wheeled props and pivotal categories. A pivotal category is a particular kind of tensor categorywith a notion of dual. That is, a tensor category C , equipped with a (strict) contravariant functor of monoidalcategories ( − ) ∗ , with ( − ) ∗∗ = id C , and a family of maps (cid:15) c : c ⊗ c ∗ → I for each c ∈ C (here I is the unitobject of C ), which satisfy axioms P , P and P of [FY89, Definition 1.3] (a more general version can befound in [JS93]).As mentioned in the introduction, the category of planar algebras is equivalent to the category of pivotalcategories with a symmetrically self-dual generator [MPS10]. It is a straightforward exercise to check that awheeled prop is, in particular, a pivotal category with a single generator; since we couldn’t find any statementof this fact in the literature we include a proof sketch here. There is no expectation however for the oppositedirection to hold, i.e. not every pivotal category is a wheeled prop. Proposition 6.8.
There exists a fully faithful embedding ρ : wProp (cid:44) → PivCat from the category of linearwheeled props to the category of linear pivotal categories with a single generator.Proof.
Let A be a linear wheeled prop with generating object x (as in Remark 6.3). We will show directlythat A is a pivotal category. Objects in A are generated by a single object x – the “colour” of all outgoingedges of a graph. The dual of x colours incoming edges, and ( x ⊗ n ) ∗ = ( x ∗ ) ⊗ n . Since the symmetric tensorproduct ⊗ on A is horizontal composition, it is clear that ( − ) ∗ is a tensor contravariant functor on A . Thetrace map (cid:15) x : x ⊗ x ∗ → x = I is given by the contraction (cid:15) x = t , and can be generalised to all objects byiterated applications of contractions, e.g. (cid:15) x ⊗ n : x ⊗ n ⊗ ( x ⊗ n ) ∗ → I .The axioms P , P and P of [FY89] now follow from the axioms of a wheeled prop. As just one example,the axiom P states that for any x ⊗ k , x ⊗ m in x ⊗ n the following diagram commutes: ( x ⊗ k ⊗ x ⊗ m ) ⊗ ( x ⊗ m ∗ ⊗ x ⊗ k ∗ ) x ⊗ k ⊗ ( x ⊗ m ⊗ ( x ⊗ m ∗ ⊗ x ⊗ k ∗ )) x ⊗ k ⊗ (( x ⊗ m ⊗ x ⊗ m ∗ ) ⊗ x ⊗ k ∗ ) x ⊗ k ⊗ ( I ⊗ x ⊗ k ∗ ) x ⊗ k ⊗ x ⊗ k ∗ ( x ⊗ k ⊗ x ⊗ m ) ⊗ ( x ⊗ k ⊗ x ⊗ m ) ∗ I One can check that this indeed holds in any wheeled prop A , by axiom H1 and iterated applications of thecontraction operations. The remaining axioms follow in a similar manner.As a morphism f : A → B between wheeled props is, in particular, a monoidal functor between rigidtensor categories, it is also a morphism between pivotal categories. It follows that the category of wProp issubcategory of PivCat and the natural inclusion gives a fully faithful embedding wProp (cid:44) → PivCat . (cid:3) Remark 6.9.
Pivotal categories are strictly more general than wheeled props and there is no claim that thefunctor ρ is an equivalence. For formal reasons, there exists an adjoint to ρ (similiar to [JSV96, Proposition CA wPropPA PivCat ∼ = ∼ = This problem remains open and will be pursued in future work. Moreover, the category of pivotal categoriesis naturally a -category and one would like to see the equivalences promoted to equivalences of -categories.It is not known if wheeled props have a natural -category structure, though it is likely, as it is alreadyknown that props admit the structure of a -monoid (See [JY09, Section 8] or [Lac04]). References [Bar05] D. Bar-Natan,
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