Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series
DDECORATED ONE-DIMENSIONAL COBORDISMS AND TENSORENVELOPES OF NONCOMMUTATIVE RECOGNIZABLE POWERSERIES
MIKHAIL KHOVANOV
Abstract.
The paper explores the relation between noncommutative power series and topo-logical theories of one-dimensional cobordisms decorated by labelled zero-dimensional sub-manifolds. These topological theories give rise to several types of tensor envelopes of noncom-mutative recognizable power series, including the categories built from the syntactic algebraand syntactic ideals of the series and the analogue of the Deligne category.
Contents
1. Introduction 12. Decorated one-dimensional cobordisms and their evaluations 62.1. Categories C and C ′ α ̃C of decorated cobordisms with inner boundary 213.2. Tensor envelopes of ̃C Introduction
In the universal construction approach to low-dimensional topological theories [BHMV,Kh1, RW1] one starts with an evaluation of closed n -dimensional objects M taking valuesin a ground commutative ring or a field and then defines state spaces A ( N ) for ( n − ) -dimensional objects N via the bilinear pairing on n -dimensional objects M with a givenboundary, ∂M ≅ N , by coupling two such objects M , M along the boundary and evaluatingthe resulting closed object M ∪ N M . The n -dimensional objects may be manifolds, manifoldswith decorations, embedded manifolds or foams, or one of many other variations of theseexamples. The universal pairing theory of Freedman, Kitaev, Nayak, Slingerland, Walkerand Wang [FKNSWW], further developed by Calegari, Freedman, Walker and others [CFW,W], is closely related to the universal construction. Some other examples of the universal Date : October 12, 2020. a r X i v : . [ m a t h . QA ] O c t MIKHAIL KHOVANOV construction for n = A ( N ) that one assigns to ( n − ) -dimensional objects in universal constructions usually donot satisfy the Atiyah tensor product axiom A ( N ⊔ N ) ≅ A ( N ) ⊗ A ( N ) , see [A]. Instead,there are maps A ( N ) ⊗ A ( N ) A ( N ⊔ N ) , which one can think of as a sort of a lax tensor structure.In this note we explain that the universal construction approach is interesting even indimension one. Studying the universal construction for one-manifolds decorated by dotslabelled by elements of a finite set S , we recover the notion of noncommutative recognizable(equivalently, rational) power series in the alphabet S as developed by Sch¨utzenberger [Sch],Fliess [F], Eilenberg [E1, E2], Conway [Co], Reutenauer, Carlyle and Paz, and others. A fullset of references and introductions to this theory can be found in the textbooks by Berstel andReutenauer [BR2], Salomaa and Soittola [SS], ´Esik and Kuich [EK], Kuich and Salomaa [KuS],also see [Sa2, HMS]. For short introductions to noncommutative rational power series we referto Reutenauer [Re3, Re4, Re5].Theory of noncommutative recognizable power series has its roots in the theory of rationallanguages and finite state automata [Co, E1, E2, EK], and can be viewed as a linearization ofthe latter [BR2, HMS]. We briefly review the basics of noncommutative rational (recognizable)power series in Section 2.2 and Proposition 2.1 stated there. Part of the motivation for thistheory comes from an earlier theorem of Kleene that rational languages are precisely thoserecognizable by FSA (finite state automata). A language L is a subset of S ∗ (the set of wordsin the letters of the alphabet S ) and gives rise to series α ( L ) with the coefficient of w oneif w ∈ L and zero otherwise. Coefficients of α ( L ) belong to the Boolean semiring B = { , } with 1 + =
1. Kleene’s theorem and the theory of finite state automata can be found inmany textbooks on the field, see for instance [Ch] and foundational work of Conway [Co]and Eilenberg [E1, E2]. Alternatively, we refer to Underwood [U, Chapter 2] for a briefintroduction to finite state automata, regular languages, and their relation to bialgebras.We consider various flavours of the category of S -decorated one-dimensional cobordisms. S -labelled dots placed along a one-dimensional cobordism can also be thought of as codimensionone defects on it.In the first example, the category C of oriented one-dimensional cobordisms with labelsfrom S is considered in Section 2. We work over a ground field k for simplicity, but theconstruction extends to an arbitrary commutative ring R and at least parts of it extend tocommutative semirings, mirroring the theory of recognizable power series over semirings.To build an evaluation one needs a number (an element of the ground field k ) associatedto each circle carrying a collection of S -labelled dots. This collection is determined by a finitesequence w of elements of S up to a cyclic order. Consequently, to build various evaluationcategories, we need to assign a number α ( w ) ∈ k to each such sequence or word w ∈ S ∗ , subjectto the condition α ( uv ) = α ( vu ) for all words u, v .An evaluation of this type is encapsulated by a formal expression(1) Z α = ∑ w ∈ S ∗ α ( w ) w, α = { α ( w )} w ∈ S ∗ , ECORATED 1D COBORDISMS 3 known as a noncommutative power series Z α , an element of the vector space k ⟪ S ⟫ dual tothe free associative algebra k ⟨ S ⟩ generated by elements of S : k ⟪ S ⟫ ∶= k ⟨ S ⟩ ∗ = Hom k ( k ⟨ S ⟩ , k ) . Recognizable noncommutative power series are singled out by the condition that their syntactic algebra A α , see Section 2.2, is finite-dimensional. The syntactic algebra [Re1] is thequotient of k ⟨ S ⟩ by the largest two-sided ideal I α of k ⟨ S ⟩ that lies in the hyperplane ker ( α ) ,when α is considered as a linear map k ⟨ S ⟩ —→ k .Property α ( uv ) = α ( vu ) for all words u, v ∈ S ∗ describes a particular type of series that werefer to as symmetric series. Reutenauer [Re1] calls such series central .In Section 2 we show that for recognizable symmetric series α there is a satisfactory theoryof tensor envelopes [Kn], that is, tensor categories associated to α , that mirrors the theoryof the Deligne categories associated to symmetric groups and of negligible quotients of thesecategories [D, CO, EGNO]. Similar theories have recently been introduced for evaluations oftwo-dimensional cobordisms in [Kh2, KS3], two-dimensional cobordisms with corners [KQR],and two-dimensional cobordisms with dots (codimension two defects) [KKO]. One can alsocompare our construction with tensor envelopes of the ”one-sided inverse” algebras and Leav-itt path algebras considered in [KT] for categorifications of rings of fractions and with thediagrammatic categorification of the polynomial ring in [KS1].There are several categories and functors between them associated to rational symmetricnoncommutative series α , defined throughout Section 2.4 and summarized in Section 2.5 anddiagram (23) there. Various skein and quotient categories that one obtains extend the notionof the syntactic algebra A α of α (the quotient of noncommutative polynomials k ⟨ S ⟩ by thelargest two-sided ideal contained in ker α , see above) and can be thought of as forming varioustensor and Karoubi closures of the latter. The theory of syntactic algebras of noncommutativerecognizable (or rational) power series was introduced and developed by Reutenauer [Re1].Syntactic algebra A α appears as the endomorphism algebra of the generating object (+) inseveral categories associated to α .In Section 3 we go beyond the restriction that noncommutative power series be symmetricby enlarging our category of cobordisms. We consider category ̃C of S -decorated cobordisms M that may have endpoints strictly ”inside” the cobordism, that it, not on the top or bottomboundary ∂ M and ∂ M . We call these inner or floating endpoints. Such floating endpointsappear in diagrammatical calculi in [KS1, KT], for instance. Cobordisms of this type betweenempty 0-manifolds (closed or floating cobordisms) have connected components that are either S -decorated oriented intervals or circles. A multiplicative evaluation on such cobordismsassigns an element α ● ( w ) ∈ k to an oriented interval with word w written along it via labelleddots, and an element α ○ ( v ) ∈ k to an oriented circle, with word v , well-defined up to cyclicrotation, written along it.Consequently, the analogue of noncommutative power series in this case is a pair(2) α = ( α ● , α ○ ) where α ● is a noncommutative power series and α ○ is a symmetric noncommutative powerseries. There does not have to be any relation between α ● and α ○ .Pair α of series as above allows to evaluate S -decorated floating intervals (via α ● ) andfloating circles (via α ○ ). In Section 3 to the evaluation data α = ( α ● , α ○ ) we assign several MIKHAIL KHOVANOV tensor categories similar to those for the symmetric series. The resulting categories have thebest behaviour when both α ● and α ○ are recognizable series, and we specialize to this caseearly. We say that α is recognizable if both α ● and α ○ are recognizable.We follow the path familiar from Section 2 and papers [KS3, KQR] and assign severalcategories and functors between them to each recognizable pair α , including the followingcategories: ● The category
V ̃C α of viewable cobordisms, where any closed (floating) component isreduced via evaluation α . ● The skein category
S ̃C α , where, additionally, elements of two-sided and one-sidedsyntactic ideals I α and I (cid:96)α ● , I rα ● evaluate to zero when placed in the middle of thestrand or by its floating endpoint, respectively. ● The category ̃C α , the quotient of either V ̃C α or S ̃C α by the ideal of negligible mor-phisms. ● Additive Karoubi closure
D ̃C α of S ̃C α , which is the analogue of the Deligne category. ● Additive Karoubi closure
D ̃C α of ̃C α , equivalent to the quotient of D ̃C α by the idealof negligible morphisms.These four categories have finite-dimensional hom spaces (again, assuming α is recogniz-able), see diagram (31) and Section 3. They can be thought of as various tensor envelopes of α and the syntactic algebra A α .Categories built out of a single symmetric recognizable series in Section 2 can be considereda special case of this construction, given by setting the first series α ● to zero. Setting thesecond series α ○ to zero, instead, results in another specialization of the theory, with alldecorated circles evaluating to zero, while decorated intervals evaluating to coefficients of α ● ,see the remark at the end of Section 3.In this paper we use rational and recognizable interchangeably to refer to noncommuta-tive power series over a field with the syntactic ideal of finite codimension. Coincidence ofrational and recognizable power series with coefficients in an arbitrary semiring is a result ofSch¨utzenberger [Sch], see also [BR2, SS, EK, KuS] for more details and references. For moregeneral monoids, beyond the free monoid on a finite set S , the sets of recognizable and rationalseries may differ, see [DG, Sa1] and references therein. The difference between rational andrecognizable series is also visible in examples in [KQR], where a recognizable series in two ormore commuting variables needs to be rational with denominators restricted to polynomialsin single generating variables.The theory of recognizable noncommutative power series makes sense over non-commutativesemirings [BR2, SS]. One can look to generalize the theory of tensor envelopes of such seriesfrom series over a field or a commutative ring to series over a semiring. Note that closedcobordisms would then evaluate to elements of the ground semiring. Components of a closedcobordism ”commute”, in the sense of sliding past each other, as elements of the commutativemonoid of endomorphisms of the unit object of the tensor category of cobordisms, the emptyzero-manifold. For this reason, it is natural to restrict to commutative semirings in thisfuller extension of the theory of tensor envelopes of noncommutative power series. We do notconsider the general case of a ground commutative semiring K in this paper, though, limitingourselves to a ground field, but it may be interesting to develop. The case when K = B is the ECORATED 1D COBORDISMS 5 boolean semiring, gives, in particular, the notion of tensor envelopes of a rational language L or, equivalently, tensor envelopes of a finite state automaton. To get the definition, run theconstructions of Section 3 with B in place of field k and the pair α = ( α ● , ) of series with thezero symmetric series α ○ = α ● the series of a regular language L . To test whether thisnotion is useful, one may study examples of quotients ̃C α of the skein category S ̃C α for such α . In the follow-up paper, we will consider one-dimensional cobordisms with more generaldecorations, by edges and vertices of an oriented graph (or a quiver) Γ. The graph Γ may befinite or infinite. Dots on a cobordisms are labelled by oriented edges of Γ. Intervals of thecobordisms separated by dots along a connected component are labelled by vertices of Γ. A dotlabelled by an edge s ∶ a → b is surrounded by intervals labelled by vertices a and b , respectively,in the order that matches the orientation of the corresponding connected component. Suchdecorations are possible for both interval and circle connected components of a cobordism.There are suitable monoidal categories C( Γ ) and ̃C( Γ ) of Γ-decorated cobordisms generalizingcategories C and ̃C in this paper. Cobordisms with floating endpoints are allowed in ̃C( Γ ) butnot in C( Γ ) . Objects of C( Γ ) and ̃C( Γ ) are finite sequences of vertices of Γ or, equivalently,finite sequences of objects of category S ( Γ ) , see next.To Γ one assigns the small category S ( Γ ) of paths in Γ, with vertices of Γ being theobjects of S ( Γ ) and paths in Γ – morphisms, with concatenation of paths as the composition.Traveling along a connected component of a Γ-decorated cobordism one encounters a path inΓ, that is, a morphism in S ( Γ ) . If the component is a circle, the path, in addition, must beclosed, that is, start and end at the same vertex of Γ.An evaluation α , in the case of ̃C( Γ ) , where floating endpoints are allowed, consists of twomaps: ● Map α ● from the set of morphisms in S ( Γ ) (paths in Γ) to the ground field k or, moregenerally, a commutative ring or a semiring, ● Map α ○ from the set of circular morphisms , that is, closed paths in Γ without a choiceof the basepoint to k .The pair α = ( α ● , α ○ ) is the analogue of the pair in (2), generalizing the special case consideredin this paper where Γ has a single vertex and oriented loops from the vertex to itself areenumerated by elements of S .One can then define the analogues of all the categories in the diagram (31), including V ̃C α , S ̃C α , ̃C α , in this case. In particular, the category ̃C( Γ ) α is the quotient of the k -linearization k ̃C( Γ ) by the two-sided ideal of negligible morphisms, defined via the trace given by evaluation α . The pair α is called recognizable or locally-recognizable (when Γ is infinite) if the ”gligiblequotient” category ̃C( Γ ) α has finite-dimensional hom spaces. Note that the boundary pointsand floating endpoints of Γ-decorated cobordisms are labelled by objects of S ( Γ ) , that is, byvertices of Γ, the label inherited from the label of the adjacent edge of the cobordism.These constructions can be further generalized to cobordisms between finite sets of bound-ary points given by graphs and Γ-decorated graphs rather then by Γ-decorated one-manifolds.Cobordisms given by graphs can still be viewed as one-dimensional cobordisms between zero-dimensional objects (finite sets of points, possibly decorated by vertices of Γ and orientations,as necessary). MIKHAIL KHOVANOV
A natural open problem is to extend the universal construction, for decorated cobordisms(or cobordisms with defects), beyond dimension one. Parts of this extension are visible in ● [KKO], where two-dimensional cobordisms are decorated by dots labelled by elementsof a commutative monoid or a commutative algebra, with non-trivial interactionsbetween these dots and topology of cobordisms coming from the handle cobordismequated to a nontrivial element of the monoid or algebra. ● [KQR], where side boundaries of two-dimensional cobordisms with corners may becolored by elements of a finite set. ● Foam theory [Kh1, B, EST, RWd, RW1], see more references in [KK], where ratherparticular evaluations of two-dimensional decorated CW-complexes with generic sin-gularities embedded in R (foams) are used as an intermediate step to build homologytheories of links that categorify various one-variable specializations of the HOMFLYPTpolynomial. Soergel bimodules, singular Soergel bimodules, and some other structuresin representation theory admit a foam description as well [Wd, RW2, KRW]. ● Evaluation theory for two-dimensional cobordisms and evaluations of overlappingfoams [Kh2], pointing towards further connections to arithmetic topology, representa-tion theory, and the Heegaard-Floer theory. ● [KL], which considers evaluations in the two-dimensional planar case with one-dimensionaldefects.Freedman et. al [FKNSWW] mention possible decorations on low-dimensional cobordisms fortheir universal pairings.Studying evaluations not just for n -manifolds but for decorated n -manifolds, n -manifoldsand their foam analogues embedded in R n + , and other such refinements should ease one’sway into understanding recognizable evaluations in dimension n +
1. This program makessense at least in dimensions n = , , Acknowledgments.
The author is grateful to Kirill Bogdanov, Mee Seong Im, andVladimir Retakh for interesting discussions and to Victor Shuvalov for help with creating thefigures for the paper. The author was partially supported by the NSF grant DMS-1807425while working on this paper.2.
Decorated one-dimensional cobordisms and their evaluations
Categories C and C ′ . Fix a finite set S of cardinality r ≥
0, which we often write as S = { s , s , . . . , s r } .Consider the category C = C S of S -decorated compact oriented one-dimensional cobordisms.Its objects are oriented one-dimensional manifolds N , that is, finite sets with a sign assignment + or − to each element (signed finite sets). A morphism from N to N is an oriented one-dimensional manifold M decorated by finitely many dots labelled by elements of S , with ∂M = N ⊔ (− N ) , see Figure 2.1.1 for an example, which also sets the orientation conventionfor the boundary.Dots can move along a connected component of M where they are placed but withoutcrossing through other dots or moving to a boundary point. Two morphisms are equal if theyare diffeomorphic rel boundary and keeping track of dots and their labels.Each component c of M is either an oriented circle or an oriented interval. Going along c in the direction of its orientation, one can read off the labels of marked points. When c is an ECORATED 1D COBORDISMS 7 s s s s s s s − + − + ++ − − + + Figure 2.1.1.
A morphism from (− + − + +) to (+ − − + +) in C . It has twoclosed (or floating) components, one undecorated, the other decorated by s s .The two U-turns are decorated by s s and s , respectively. There are threethrough arcs: two undecorated, one decorated by s s . Whether a crossing isover- or under-crossing is irrelevant.interval, the sequence of labels is an invariant of c . When c is a circle, the sequence of labelsis an invariant up to a cyclic rotation of the sequence. A component may carry no dots; thecorresponding sequence is empty then.Composition of morphisms is given by their concatenation.To reduce to fewer objects, we take the objects to be sequences of signs (cid:15) = ( (cid:15) , . . . , (cid:15) n ) , (cid:15) i ∈ {+ , −} . To (cid:15) we associate an ordered signed zero-manifold with one point for each termin the sequence, with the signs given by (cid:15) , . . . , (cid:15) n . We may alternatively write (cid:15) i = − + or − .Permutation cobordisms show that permuting signs in a sequence (cid:15) leads to an isomorphicobject, and that isomorphism classes of objects are parametrized by pairs of non-negativeintegers n = ( n , n ) , counting the number of plus and minus signs.When restricting to a skeleton category (one object for each isomorphism class), we thusreduce objects to pairs n = ( n , n ) , where n is the number of plus points and n is thenumber of minus points. In the sequence of signs that n represents, we put plus signs first,and can also write n = (+ n − n ) .Denote by C = C S the category of S -decorated cobordisms with objects—finite sign se-quences (cid:15) as above. The skeleton category of S -decorated cobordisms, with objects n , isdenoted C ′ . Category C is slighter larger than the equivalent category C ′ .These categories are rigid symmetric tensor, with the tensor product in C given on mor-phisms by placing their diagrams next to each other. On objects, the tensor product isthe concatenation of sequences. In C ′ when forming the tensor product of morphisms, wegroup plus points together and minus points together. Tensor product on objects is given by ( n , n ) ⊗ ( m , m ) = ( n + m , n + m ) .In both categories C and C ′ object (+) has object (−) as its dual, with the duality morphismsin C shown in Figure 2.1.2. MIKHAIL KHOVANOV + − ε + − − + ε − + + − δ + − − + δ − + Figure 2.1.2.
Duality morphisms for + and − in C . ++1 + −− − ++ ss + −− ss − + ++ + P ++ − ++ − P − + Figure 2.1.3.
Generating morphisms in C : identity 1 + of (+) , identity 1 − of (−) , morphism s + ∶ (+) —→ (+) for s ∈ S , morphism s − ∶ (−) —→ (−) , permutation morphismm P ++ . Other permutation morphisms, such as P −+ ,can be obtained as compositions of these morphisms and those in Figure 2.1.2. = = = = == s s s s = s s = = = Figure 2.1.4.
Some relations in C . They hold for any choice of orientations.Orientation of the LHS of each equation determines the orientation of the RHSand vice versa. A set of these relations with some restrictions on orientationscan be taken for a defining set of relations. Commutativity relations on gener-ators in horizontally separated regions are not shown, since they are built intothe axioms of a tensor category.The empty sequence ∅ is the unit object of the tensor category C . The pair ∶= ( , ) isthe unit object of C ′ . Generating morphisms in C ′ are shown in Figures 2.1.2 and 2.1.3.Some defining relations in C are shown in Figure 2.1.4. We do not list a full set of definingrelations and will not need it. These relations can be hidden in the definition of C , wheremorphisms are declared equal if the corresponding decorated cobordisms are diffeomorphicrel boundary. ECORATED 1D COBORDISMS 9
A morphism M from (cid:15) to (cid:15) ′ in C consists of some number of oriented circles and orientedintervals. Boundaries of oriented intervals and their orientations match entries of (cid:15) and (cid:15) ′ inpairs.Denote by ∣ (cid:15) ∣ the difference of the number of plus and minus signs in (cid:15) and call it the weightof the sequence. For instance, ∣(+ − − + + + +−)∣ = − =
2. A morphism from (cid:15) to (cid:15) ′ existsiff the two sequences have the same weight, ∣ (cid:15) ∣ = ∣ (cid:15) ′ ∣ . The weight is additive under the tensorproduct of objects (concatenation of sequences). Denote by ∣∣ (cid:15) ∣∣ the length of the sequence (cid:15) .Connected components c of a morphism M are circles and intervals (arcs). Circles are closedcomponents, also called floating components. Arcs have boundary and, borrowing terminologyfrom [KS2], separate into U-turns and through arcs. A U-turn has both endpoints on the sameside of a morphism (either on the source one-manifold or on the target, while a through archas one endpoint on the source and one on the target, also see types (1)-(3) of components inFigure 3.1.3. The endpoints of a U-turn have opposite signs, while the endpoints of a througharc carry the same sign, see Figures 2.1.2, 2.1.3.By analogy with [KS3, KQR], we can also call arcs viewable or visible components, sincethey have endpoints on the boundary of the cobordism (either top or bottom or both), andcall circles floating components [KS1, KQR], since they are disjoint from the boundary of thecobordism.Denote by S ∗ the set of finite sequences of elements of S , including the empty sequence ∅ (also see Section 2.2). Going along an arc c of a cobordism M gives us a word w ( c ) ∈ S ∗ .Going along a circle c in x gives a word w ( c ) well defined up to a cyclic rotation or conjugation of words, w w ∼ w w .In this paper we encounter sequences (cid:15) of signs, which are objects of C , and sequences w ∈ S ∗ ,which are sequences of labels encountered along connected components of a cobordism, thelatter a morphism in C .2.2. Noncommutative power series.
For simplicity we work over a ground field k , although the theory of noncommutativepower series and rational and recognizable series makes sense over an arbitrary semiring R ,not necessarily commutative [BR2, SS]. For definitive treatments we refer the reader tobooks [BR2, SS] and to [Re3, Re4, Re5] for quick introductions and reviews.Let S ∗ = ∅ ⊔ S ⊔ S ⊔ . . . be the set of sequences w = t . . . t n of elements of a finite set S = { s , . . . , s r } . We call elements of S letters and elements of S ∗ words or sequences in S .The empty word ∅ is allowed. A noncommutative power series α over k is any function(3) α ∶ S ∗ —→ k , α ( w ) ∈ k , w ∈ S ∗ . We formally write this series as(4) Z α = ∑ w ∈ S ∗ α w w, α = ( α w ) w ∈ S ∗ , α w = α ( w ) ∈ k , using either α w or α ( w ) to denote the value of α on a noncommutative monomial or word w . Denote by k ⟪ S ⟫ the k -vector space of noncommutative power series and by k ⟨ S ⟩ the freenoncommutative k -algebra on generators in S (the algebra of noncommutative polynomials).Given two series α, β , their product is the series αβ that on w evaluates to(5) αβ ( w ) = ∑ w = w w α ( w ) β ( w ) , the sum over all decompositions of w . There are (cid:96) ( w ) + (cid:96) ( w ) is thelength of w . This product turns k ⟪ S ⟫ into a k -algebra, noncommutative if S has more thanone element. The inclusion k ⟨ S ⟩ ⊂ k ⟪ S ⟫ is a ring homomorphism.We say that series α ∈ k ⟪ S ⟫ is recognizable iff there is a homomorphism ψ ∶ k ⟨ S ⟩ —→ End ( k n ) of the free algebra into the algebra of n × n matrices, a vector and a dual vector λ, µ T ∈ k n such that(6) α ( w ) = µ ψ ( w ) λ for all words w . That is, the number α ( w ) is the product of the 1 × n matrix µ , n × n matrix ψ ( w ) and n × λ . Denote by k ⟪ S ⟫ rec the set of all recognizable series.Vector space k ⟪ S ⟫ is a k ⟨ S ⟩ -bimodule with f ⊗ g ∈ k ⟨ S ⟩ ⊗ k ⟨ S ⟩ op acting on α ∈ k ⟪ S ⟫ by ( f ⊗ g )( α )( w ) = α ( gwf ) , w ∈ S ∗ . We write f αg ∶= ( f ⊗ g )( α ) . This action gives rise to the left, right, and two-sided ideals I (cid:96)α , I rα and I α in k ⟨ S ⟩ : ● Left ideal I (cid:96)α consists of all f ∈ k ⟨ S ⟩ such that f α = , that is, all f such that α ( wf ) = w ∈ S ∗ . It is the largest left ideal contained in the hyperplane ker ( α ) ⊂ k ⟨ S ⟩ . ● Right ideal I rα consists of all g ∈ k ⟨ S ⟩ such that αg =
0, that is, all g such that α ( gw ) = w ∈ S ∗ . It is the largest right ideal contained in ker ( α ) . ● Ideal I α consists of all f ∈ k ⟨ S ⟩ such that α ( wf v ) = w, v ∈ S ∗ . It is the largesttwo-sided ideal contained in the hyperplane ker ( α ) .Ideal I (cid:96)α has finite codimension in k ⟨ S ⟩ iff the series α is recognizable. Given triple as in (6),ideal I (cid:96)α contains the finite codimension subspace { x ∈ k ⟨ S ⟩∣ ψ ( x ) λ = } . Vice versa, if I (cid:96)α hasfinite codimension, it is straightforward to produce the data in (6) by taking k n ≅ k ⟨ S ⟩/ I (cid:96)α , λ = µ = α . Given a triple as in (5), ker α contains two-sided ideal of finite codimension { f ∈ k ⟨ S ⟩∣ ψ ( f ) = ∈ End ( k n )} . Vice versa, if I α has finite codimension, ideals I (cid:96)α ⊃ I α and I rα ⊃ I α have finite codimension too.Consequently, if one of I (cid:96)α , I rα , I α have finite codimension in k ⟨ S ⟩ , the other two have finitecodimension as well.Two-sided ideal I α of k ⟨ S ⟩ is called the syntactic ideal of α . Denote by(7) A α ∶= k ⟨ S ⟩/ I α the quotient algebra, the syntactic algebra of α , see [Re1]. It is defined for any α, but wemostly restrict to considering it for recognizable α , when A α is finite-dimensional. We alsocall I (cid:96)α and I rα the left and right syntactic ideals of α .Algebra k ⟨ S ⟩ acts on k ⟪ S ⟫ on the left and on the right, and k ⟨ S ⟩ -bimodule generated by α (a subbimodule of k ⟪ S ⟫ ) is naturally isomorphic to the syntactic algebra A α , the latterequipped with k ⟨ S ⟩ -bimodule structure via left and right multiplications:(8) A α ≅ k ⟨ S ⟩ ⊗ k ⟨ S ⟩ op ( α ) . The quotient k ⟨ S ⟩/ I (cid:96)α is naturally a faithful left A α -module via the left multiplicationaction. Likewise, k ⟨ S ⟩/ I rα is a faithful right A α -module via the right multiplication action. ECORATED 1D COBORDISMS 11
We see that α ∈ k ⟪ S ⟫ is recognizable iff the cyclic k ⟨ S ⟩ ⊗ k ⟨ S ⟩ op -module generated by α in k ⟪ S ⟫ is finite-dimensional, or, equivalently,dim k A α < ∞ . The
Hankel matrix M α of α is the infinite square matrix with rows and columns enumeratedby elements of S ∗ with the ( w , w ) -entry α ( w w ) .Given any series α with α (∅) = proper series ), we can form the Kleene plus series α + as the formal sum(9) α + = α + α + . . . , where α n = αα . . . α is the product of n copies of α . The term α n evaluates to 0 on any wordof length less than n . Consequently, a given word evaluates nontrivially only on finitely manyterms in the sum, and α + makes sense as an element of k ⟪ S ⟫ . The series 1 + α + is the inverseof the series 1 − α in the ring k ⟪ S ⟫ .A series is finite if it contains finitely many terms. Finite series are those in the ring ofnoncommutative polynomials k ⟨ S ⟩ ⊂ k ⟪ S ⟫ .Denote by k ⟪ S ⟫ rat the smallest subset of series that ● Contains all finite series. ● Closed under the product and finite k -linear combinations of series. ● Contains α + for any proper series α in the subset.Series in k ⟪ S ⟫ rat are called rational series. Proposition 2.1.
The following properties of series α are equivalent. (1) α is rational. (2) α is recognizable. (3) The Hankel matrix M α of α has finite rank. (4) The syntactic ideal I α has finite codimension in k ⟨ S ⟩ . (5) The left ideal I (cid:96)α has finite codimension in k ⟨ S ⟩ . (6) The right ideal I rα has finite codimension in k ⟨ S ⟩ . (7) α can be computed by a weighted finite automaton. Equivalence of (2), (4), (5), (6) is explained above.For a proof of all equivalences see Sections 1 and 2 of [BR2], Salomaa-Soitola [SS], orreferences there to the original work of Sch¨utzenberger [Sch], Fliess [F], Eilenberg [E2] andothers. Most of these equivalences hold in much greater generality than over a field, in manycases over an arbitrary semiring. The Hankel matrix of noncommutative series was introducedby Fliess [F].The notion of weighted finite automaton linearizes the concept of finite state automatonand, over a field k , is equivalent to the triple ( λ, ψ, µ ) as in (6), see [BR2, Section 1.6], forinstance. ◻ We have k ⟪ S ⟫ rec = k ⟪ S ⟫ rat , since rational and recognizable series coincide.Assume that α is recognizable. The trace form α on the finite-dimensional algebra A α hasthe following nondegeneracy property:(10) for any a ∈ A α , a /= b, c ∈ A α such that α ( bac ) /= This is a much weaker condition than the usual Frobenius condition on a linear form β ona finite-dimensional algebra B :(11) for any a ∈ B, a /= b such that β ( ab ) /= β equips B with the structure of a Frobenius algebra.Given any finite-dimensional algebra B with a linear form α ∶ B —→ k , the condition that(12) for any a ∈ B, a /= b, c ∈ B such that α ( bac ) /= ( ) being the only two-sided ideal in ker ( α ) . Let us call a pair ( B, α ) with this property a syntactic pair . A finite set of generators b , . . . , b m of B gives riseto a surjective homomorphism(13) ρ ∶ k ⟨ S ⟩ —→ B, ρ ( s i ) = b i , S = { s , . . . , s m } from the free algebra k ⟨ S ⟩ to B and induced noncommutative power series in the set ofvariables S , also denoted α . This gives a bijection between recognizable power series in S andisomorphism classes of syntactic pairs ( B, α ) as above with a choice of generators ( b , . . . , b m ) of B .An algebra is called syntactic if it admits a presentation (7) for some S and α . E xamples:(1) Any Frobenius algebra B with a non-degenerate form β gives a syntactic pair ( B, β ) .(2) Take the matrix algebra B = M n ( k ) and define α ( x ) = x , to pick the first diagonalcoefficient of the matrix x . In this example the form α satisfies the weaker property(10), so that ( B, α ) is a syntactic pair, but α is not a Frobenius trace. Algebra B is Frobenius for a different linear form on it (for example, for the usual trace onmatrices).(3) Take the path algebra B of the quiver with two vertices 0 , ( ) con-necting them, with the multiplication given by concatenation of paths: ( )( ) =( ) , ( )( ) = ( ) , etc. Algebra B has a basis {( ) , ( ) , ( )} . Take any linear form α with α (( )) /=
0. Then ( B, α ) is a syntactic algebra with this linear form. It canbe generated by two elements. B is neither Frobenius nor quasi-Frobenius.(4) A finite-dimensional commutative algebra is Frobenius iff it is syntactic.See Reutenauer [Re1] and Perrin [Pe] for more results on syntactic algebras and the latteralso for another brief introduction to the subject.2.3. Evaluations and symmetric series.
We say that series α ∈ k ⟪ S ⟫ is symmetric if α ( w w ) = α ( w w ) for any w , w ∈ S ∗ . Transformation w w ↦ w w is also called conjuga-tion , so one can say that α is conjugation invariant. An evaluation α is symmetric iff it onlydepends on a sequence up to cyclic order.We use the word ”symmetric” to define such series, since the word ”cyclic” is already taken,see [BR1, KaR, Re2] and [BR2, Section 12.2]. A series α is called cyclic if, in addition tothe conjugation invariance condition, it satisfies α ( w n ) = α ( w ) for any non-empty w . Thus,a cyclic series is symmetric but most symmetric series are not cyclic. Reutenauer [Re1] uses central instead of our symmetric .Denote the set of symmetric series by k ⟪ S ⟫ s and by k ⟪ S ⟫ s,rec the set of recognizablesymmetric series. ECORATED 1D COBORDISMS 13
A series α ∈ k ⟪ S ⟫ can be averaged out to a series av ( α ) given by(14) av ( α )( w ) = ∑ uv = w,v /=∅ α ( vu ) , if w /= ∅ , av ( α )(∅) = α (∅) . That is, take the sum over all possible ways to split w into the product uv and evaluate α on vu . Series av ( α ) is symmetric. Only one of the two degenerate splittings ∅ w and w ∅ is usedto avoid having α ( w ) twice in the sum. Proposition 2.2. av ( α ) ∈ k ⟪ S ⟫ s,rec if α ∈ k ⟪ S ⟫ rec . In other words, averaging out a recognizable series produces a symmetric recognizableseries. This result is proved in Rota [Ro], see also [Re2]. It gives a large supply of symmetricrecognizable series. ◻ Symmetric series with semisimple syntactic algebra A α are studied in [Re1, Pe].2.4. Tensor envelopes of series α . (1) Category k C . We fix a base field k and form the k -linearization k C of C . Category k C has the same objects as C , that is, finite sequences (cid:15) of plus and minus signs. Morphismsin k C are finite linear combinations of morphisms in C , with the composition rules extended k -bilinearly from those of C . (2) Category VC α of viewable cobordisms. Next, choose a symmetric power series α ∈ k ⟪ S ⟫ s .Define the category VC α as the quotient of k C by the relations that a circle ̂ w ) with a sequence w written on it evaluates to α ( w ) . Since ̂ w w = ̂ w w , we need the condition that α issymmetric to define this quotient.Another way to define VC α is to say that it has the same objects as C : sequences (cid:15) ofelements of S . A morphism in VC α from (cid:15) to (cid:15) ′ is a finite k -linear combination of viewablecobordisms in C from (cid:15) to (cid:15) ′ . Recall that a cobordism is viewable if it has no floating connectedcomponents, that is, components homeomorphic to circles.Composition of morphisms in VC α is given by concatenating cobordisms and removing eachclosed circle ̂ w from the composition simultaneously with multiplying the remaining diagramby α ( w ) .The hom space Hom VC α ( (cid:15), (cid:15) ′ ) has a basis given by a choice of orientation-respecting match-ing of the elements in the pair of sequences (cid:15), (cid:15) ′ together with a choice of word in S for eachpair in the matching. An orientation-respecting matching consists of a bijection betweenpluses and minuses in the sequence (− (cid:15) ) (cid:15) ′ , which is the concatenation of − (cid:15) and (cid:15) ′ , with thesequence − (cid:15) given by reversing the signs of (cid:15) . An example in Figure 2.4.1 shows a basis ele-ment in one such hom space, with (cid:15) = (− − + + + + −) and (cid:15) ′ = (+ − − + +) . Note that the sizeof hom spaces in VC α does not depend on α , only the composition of morphisms does.Each word w = t . . . t m , t i ∈ S , i = , . . . , m defines a cobordism cob ( w ) given by puttingletters t , . . . , t m along the interval, with the orientation going towards decreasing the index,see Figure 2.4.2 left. Extended by linearity, this assignment is an algebra isomorphism(15) k ⟨ S ⟩ —→ End VC α ((+)) from the algebra of noncommutative polynomials to the algebra of endomorphisms of thesequence (+) in VC α . s s s s s s s s s s − − + + + + − + − − + + Figure 2.4.1.
A basis element in the hom space in VC α , with S ={ s , s , s , s } . Floating components (circles) are absent. ++ t t ... t m ++ w cob( w ) = = ++ w i ++ u cob( u ) = k X i =1 a i = Figure 2.4.2.
Left: cobordism cob ( w ) for a word w = t . . . t m ∈ S ∗ isgiven by placing dots labelled by letters of word w along the oriented interval.Alternatively, cob ( w ) can be denoted by a box labelled w on an interval. Right:a linear combination cob ( u ) of such cobordisms and its shorthand box notation.To a noncommutative polynomial u = k ∑ i = a i w i ∈ k ⟨ S ⟩ , a i ∈ k , w i ∈ S ∗ we assign the endomorphismcob ( u ) = k ∑ i = a i cob ( w i ) ∈ End VC α ((+)) of the sequence (+) given by the linear combination of words w i written on an upward orientedinterval, see Figure 2.4.2 right. It can be compactly denoted by a box on a strand with u written in it.Monomials in k ⟨ S ⟩ and their linear combinations can be placed along any component ofa cobordism. Taking the union over all viewable cobordisms with a given boundary (onecobordism for each diffeomorphism class rel boundary) and then over all ways of placingmonomials in k ⟨ S ⟩ along each component of the cobordism gives a basis in the hom space inthe category VC α between two objects. Recall that objects of VC α are sequences of + and − . ECORATED 1D COBORDISMS 15
The hom space between the objects (cid:15) , (cid:15) ′ is non-zero if the objects have the same weight, ∣ (cid:15) ∣ = ∣ (cid:15) ′ ∣ . Assuming the latter, the hom space Hom ( (cid:15), (cid:15) ′ ) is infinite-dimensional unless (cid:15) = (cid:15) ′ = ∅ is the empty sequence or if the set S of labels is empty. In the latter case k ⟨ S ⟩ ≅ k is theground field. Another special case is when S consists of a single element, S = { s } , for then k ⟨ S ⟩ ≅ k [ s ] is commutative. The endomorphism algebra of (+) in the category VC α is k ⟨ S ⟩ ,see (15).Category VC α is a k -linear pre-additive category. (3) The skein category SC α . Consider the syntactic ideal I α ⊂ k ⟨ S ⟩ associated to thesymmetric series α ∈ k ⟪ S ⟫ s . This ideal has finite codimension iff α ∈ k ⟪ S ⟫ s,rec , that is, if α is, in addition, a recognizable series. Denote by(16) A α ∶= k ⟨ S ⟩/ I α the quotient algebra (the syntactic algebra) of the algebra of noncommutative polynomials bythe syntactic ideal. Algebra A α is finite-dimensional iff α is recognizable. In the latter case,let(17) d α = dim k ( A α ) = codim k ( I α ) be the dimension of the syntactic algebra.We quotient the category VC α of viewable cobordisms by the relation that elements of I α are zero along any component of a cobordism. Namely, an element of I α is a finite linearcombination(18) u = k ∑ i = a i w i , a i ∈ k , w i ∈ S ∗ of words in the alphabet S . Element cob ( u ) , see Figure 2.4.2, can be inserted along anycomponent of a cobordism x . We impose the condition that any such insertion results inthe zero morphism in SC α between the corresponding sequences (cid:15), (cid:15) ′ . Equivalently, we canset cob ( u ) ∈ End ((+)) to zero for all u ∈ I α and take the monoidal closure of the relationscob ( u ) = u , which is equivalent to the previous condition. Alternatively, we canchoose generators { u j } , j ∈ J , for the 2-sided ideal I α , impose relation cob ( u j ) = , j ∈ J andtake their monoidal closure.Note that relations cob ( u ) = u ∈ I α are compatible with the evaluation of closedcomponents (circles). Namely, for any v ∈ k ⟨ S ⟩ , the closures ̂ uv and ̂ vu define the sameelement in End k C (∅) , namely the circle that carries the box uv or vu , and α ( uv ) = α ( vu ) = SC α the resulting quotient category. It has the same objects as VC α and ad-ditional relations cob ( u ) = u ∈ I α placed anywhere along one-dimensional S -decoratedcobordisms that span hom spaces in VC α .Since relations in the syntactic ideal are imposed along each connected component of acobordism, an element along a component can be reduced accordingly. Choose a set ofelements B α ⊂ k ⟨ S ⟩ that descend to a basis of A α (if needed, one can choose monomials in S ). Modulo I α , an element of k ⟨ S ⟩ can be reduced to a linear combination of elements of B α .Accordingly, we can reduce a morphism in SC α to a linear combination of viewable morphisms uv uv c uv vu c vu == α α ( uv ) = 0 Figure 2.4.3. α (̂ uv ) = α (̂ vu ) = u ∈ I α , v ∈ k ⟨ S ⟩ since uv, vu ∈ I α .such that along each component an element of B α is placed. Call these morphisms basic anddenote the set of basic morphisms from (cid:15) to (cid:15) ′ by B α ( (cid:15), (cid:15) ′ ) . Recall that a morphism from (cid:15) to (cid:15) ′ exists in C if the two sequences have the same weight,that is, the difference between the number of plus and minus signs in them: ∣ (cid:15) ∣ = ∣ (cid:15) ′ ∣ . Inthe latter case, the number of viewable morphisms (i.e., without circle components) is thenumber of ways to pair up elements of (cid:15) and (cid:15) ′ in an orientation-respecting way. Reverse thesigns in one of the sequences, say in (cid:15) , and concatenate with the other to get (cid:15) ′ (− (cid:15) ) . Thissequence has the same number n of plus and minus signs, equal to half the length of thesequence: 2 n = ∣∣ (cid:15) ∣∣ + ∣∣ (cid:15) ′ ∣∣ . Isomorphism classes of viewable cobordisms from (cid:15) to (cid:15) ′ are in aone-to-one correspondence with bijections between plus and minus signs in (cid:15) ′ (− (cid:15) ) . There are n ! such bijections. For each bijection, there are d nα ways to assign an element of B α to eachcomponent of a cobordism. The following proposition and corollary result. Proposition 2.3.
The set of basic morphisms B α ( (cid:15), (cid:15) ′ ) is a basis of the hom space Hom SC α ( (cid:15), (cid:15) ′ ) . Corollary 1.
Dimensions of hom spaces in SC α are given by: (19) dim Hom SC α ( (cid:15), (cid:15) ′ ) = ⎧⎪⎪⎨⎪⎪⎩ n ! d nα if ∣ (cid:15) ∣ = ∣ (cid:15) ′ ∣ , n = ∣∣ (cid:15) ∣∣ + ∣∣ (cid:15) ′ ∣∣ , otherwise . In particular, hom spaces in the category SC α are finite-dimensional.For the endomorphism algebra of the sequence (+) we have (compare with (15))(20) End SC α ((+)) ≅ A α , and the endomorphism algebra of (+) has dimension d α . Skein category SC α is similar tothe oriented Brauer category [R], but with lines decorated by elements of S , leading to manychoices for evaluations of floating components, one for each sequence in S ∗ up to the cyclicequivalence. (4) Negligible morphisms and gligible quotient C α . The trace tr α ( x ) of a cobordism x from (cid:15) to (cid:15) is an element of k given by closing x via ∣∣ (cid:15) ∣∣ suitably oriented arcs connecting n top with n bottom points of x into a floating cobordism ̂ x and applying α , tr α ( x ) ∶= α (̂ x ) . This operation is depicted in Figure 2.4.4. The trace is extended to all endomorphisms of (cid:15) in k ̃C by linearity. It is well-defined on trace of endomorphisms of objects (cid:15) in categories V ̃C α and S ̃C α as well. ECORATED 1D COBORDISMS 17 x x b x = α α ( b x ) ∈ k Figure 2.4.4.
The trace map: closing endomorphism x of (cid:15) into ̂ x and ap-plying α . In this example (cid:15) = (+ + − + −) .The trace is symmetric, that is tr α ( yx ) = tr α ( xy ) for a morphism x from (cid:15) to (cid:15) ′ and y from (cid:15) ′ to (cid:15) . The ideal J α ⊂ SC α is defined as follows.A morphism y ∈ Hom ( (cid:15), (cid:15) ′ ) is called negligible and belongs to the ideal J α if tr α ( zy ) = z ∈ Hom ( (cid:15) ′ , (cid:15) ) . Negligible morphisms constitute a two-sided ideal in thepre-additive category SC α . We call J α the ideal of negligible morphisms , relative to the traceform tr α . Define the quotient category C α ∶= SC α / J α . The quotient category C α has finite-dimensional hom spaces, as does SC α (recall that α isrecognizable). The trace is nondegenerate on C α and defines perfect bilinear pairingsHom ( (cid:15), (cid:15) ′ ) ⊗ Hom ( (cid:15) ′ , (cid:15) ) —→ k on its hom spaces. We may call C α the gligible quotient of SC α , having modded out by theideal of negligible morphisms. State spaces of recognizable series α . Recall that in the category VC α objects are signsequences (cid:15) and morphisms are finite linear combinations of viewable cobordisms. The spaceof homs V (cid:15) ∶= Hom VC α (∅ , (cid:15) ) , has a basis of all viewable cobordisms (no floating components) M with ∂M = (cid:15) . This spacecarries a symmetric bilinear form, given on pairs of basis elements (viewable cobordisms) by ( x, y ) (cid:15) ∶= α ( yx ) ∈ k , where y is the reflection of y about a horizontal line combined with the orientation reversalon y , and yx is the closed cobordism which is the composition of y and x .Define A α ( (cid:15) ) as the quotient of V (cid:15) by the kernel of this bilinear form. Then there is acanonical isomorphism A α ( (cid:15) ) ≅ Hom C α (∅ , (cid:15) ) as well as isomorphisms A α ((− (cid:15) ) ⊔ (cid:15) ′ ) ≅ Hom C α ( , (− (cid:15) ) ⊔ (cid:15) ′ ) ≅ Hom C α ( (cid:15), (cid:15) ′ ) given by moving the bottom boundary (cid:15) of a cobordism to the top via a cobordism with ∣∣ (cid:15) ∣∣ parallel arcs. Here the sequence (− (cid:15) ) ⊔ (cid:15) ′ is the concatenation of − (cid:15) and (cid:15) ′ .Note that (cid:15) must be balanced for A α ( (cid:15) ) to be nonzero, that is, (cid:15) must have the samenumber n of pluses and minuses. We can then define(21) A α ( n ) ∶= A α ((+ n − n )) . Spaces A α ( n ) come with a lot of structure, including multiplication maps A α ( n ) ⊗ A α ( m ) —→ A α ( n + m ) . We have A α ( ) ≅ k and A α ( ) ≅ A α . Vector space A α ( n ) carries an action of the symmetricgroup product S n × S n by the permutation cobordisms, as well as an action of the tensor powerof the syntactic algebra A ⊗ nα ⊗( A opα ) ⊗ n , with one copy of A α ≅ End C α ((+)) or A opα ≅ End C α ((−)) acting at each sign of + n − n . More generally, a version of the oriented walled Brauer algebrawith strands carrying S -labelled dots and closed decorated circles evaluating via α acts on A α ( n ) and, more generally, on Hom C α ( (cid:15), + n − n ) for any sign sequence (cid:15) . This generalizedwalled Brauer algebra Br n,α is straightforward to define. It is associated to any recognizableseries α , finite-dimensional, and isomorphic to the endomorphism algebra End SC α ((+ n − n )) of the object (+ n − n ) in the skein category SC α . The action of Br n,α on Hom C α ( (cid:15), (+ n − n )) descends to the action of its quotient algebra End C α ((+ n − n )) on the same space.Multiplication maps turn the direct sum(22) A ∗ α ∶= ⊕ n ≥ A α ( n ) into a graded associative k -algebra, with compatible actions of S n on A α ( n ) over all n , making A ∗ α into what Sam and Snowden call a twisted commutative algebra or tca in [SSn, Definition7.2.1]. A twisted commutative algebra in that sense may be very from being commutative:for instance, the free associative algebra (the tensor algebra of a vector space) has the obvioustca structure [SSn, Example 7.2.2]. More generally, given an n -dimensional topological theory α as defined in [Kh2], perhaps for manifolds with defects, etc. and an ( n − ) -manifold N ,the direct sum A ∗ ( N ) ∶= ⊕ n ≥ α (⊔ n N ) of state spaces of disjoint unions of n copies of N , over all n , is naturally a tca in the senseof [SSn]. (5) The Deligne category DC α and its gligible quotient DC α . The skein category SC α isa rigid symmetric monoidal k -linear category with signed sequences (cid:15) as objects and finite-dimensional hom spaces. We form the additive Karoubi closure DC α ∶= Kar (SC ⊕ α ) by allowing formal finite direct sums of objects in SC , extending morphisms correspondingly,and then adding idempotents to get a Karoubi-closed category. Category DC α plays the roleof the Deligne category in our construction.The trace tr α extends to DC α and defines a 2-sided ideal DJ α ⊂ DC α of negligible morphismsrelative to tr α . Define the gligible quotient category by DC α ∶= DC α /DJ α . ECORATED 1D COBORDISMS 19
This category is equivalent to the additive Karoubi envelope of C α . It is a Karoubi-closedrigid symmetric category with non-degenerate bilinear forms on its hom spaces.2.5. Summary of categories and functors.
Here is the summary of the categories that have been introduced. ● C : the category of S -decorated one-dimensional cobordisms. Its objects are sequences (cid:15) of plus and minus signs and morphisms are one-manifolds with boundary decorated by S -labelled dots. That is, the morphisms are S -decorated one-manifolds with boundary. ● k C : this category has the same objects as C ; its morphisms are formal finite k -linearcombinations of morphisms in C . ● VC α : in this quotient category of k C we reduce morphisms to linear combinationsof viewable cobordisms. Floating connected components (circles, possibly carrying S -dots) are removed by evaluating them via α . ● SC α : to define this category, specialize to rational α and add skein relations by mod-ding out by elements of the ideal I α in k [ S ] , along each component of the cobordism.Hom spaces in this category are finite-dimensional. ● C α : the quotient of SC α by the ideal J α of negligible morphisms. This category is alsoequivalent (even isomorphic) to the quotients of k C and VC α by the correspondingideals of negligible morphisms in them. The trace pairing in C α between Hom ( n, m ) and Hom ( m, n ) is perfect. ● DC α is the analogue of the Deligne category obtained from SC α by allowing finitedirect sums of objects and then adding idempotents as objects to get a Karoubi-closedcategory. ● DC α : the quotient of DC α by the two-sided ideal of negligible morphisms. Thiscategory is equivalent to the additive Karoubi closure of C α and sits in the bottomright corner of the commutative square below.We arrange these categories and functors, for recognizable α , into the following diagram:(23) C ———→ k C ———→ VC α ———→ SC α ———→ DC α (cid:215)(cid:215)(cid:215)(cid:214) (cid:215)(cid:215)(cid:215)(cid:214)C α ———→ DC α All seven categories are rigid symmetric monoidal. All but the leftmost category C are k -linear. Except for the two categories on the far right, the objects of each category aresequences (cid:15) of plus and minus signs. The four categories on the right all have finite-dimensionalhom spaces. The two categories on the far right are additive and Karoubi-closed. The fourcategories in the middle of the diagram are pre-additive but not additive.The arrows show functors between these categories considered in the paper. The squareis commutative. An analogous diagram of functors and categories can be found in [KS3] forthe category of oriented 2D cobordisms in place of C and in [KQR] for suitable categories oforiented 2D cobordisms with side boundary and corners.For convenience, one- or two-word summaries of these categories are provided below, in thediagram essentially identical to that in [KQR, Section 3.4]: (24) S -dottedcobordisms (cid:47) (cid:47) k -linear (cid:47) (cid:47) viewable (cid:47) (cid:47) skein (cid:15) (cid:15) (cid:47) (cid:47) Deligne (Karoubian) (cid:15) (cid:15) gligible (cid:47) (cid:47) gligible and KaroubianIt is possible to go directly from k C to C α by modding out by the ideal of negligiblemorphisms in the former category. It is convenient to arrive at this quotient in several steps,introducing categories VC α and SC α on the way.If α is not recognizable, we can still define categories VC α , C α and DC α , but then, forinstance, one can potentially get two non-equivalent categories in place of DC α by followingalong the two different paths in the square above. To justify considering these categoriesfor some non-recognizable α one would want to find interesting examples where the gligiblequotient category C α has additional relations beyond those in SC α , that is, beyond the relationsthat elements of the syntactic ideal I α are zero in End (+) in SC α and C α .2.6. Examples and variations of the construction.
An involution.
Categories C and k C carry contravariant involution that reflects the cobor-dism about the middle, reversing its source and target objects, and reverses the orientation ofthe cobordism. This involution takes the object (cid:15) to − (cid:15) , that is, reverses the sign (orientation)of boundary zero-manifolds as well. To match this involution to evaluation α , assume that k comes with an involution, also denoted , and α satisfies α ( w ) = α ( w ) , where w = t n . . . t is the word w = t . . . t n in reverse. Then there are induced contravariant involutions on allthe categories associated to α and displayed in diagram (23), and one can, for instance, studysuch unitary 1D topological theories, with the set of defects S , for k = C and the complexinvolution. Examples. (1) If the set S = ∅ is empty, there are no decorations and the series α is given by its valueon the empty sequence, that is, by its constant term, and we can write α = λ ∈ k for thatvalue. A circle cobordism evaluates to λ . The skein category SC λ is isomorphic to the viewablecategory VC λ and to the oriented Brauer category B λ for for the parameter λ . Category C λ is then the quotient of B λ by the ideal of negligible morphisms, while DC λ is the additiveKaroubi closure of B λ , etc. Note that these categories depend both on the field k and λ ∈ k .(2) If S = { s } is a one-element set, the series α is a one-variable series, with the generatingfunction(25) Z α ( T ) = ∑ n ≥ α n T n , α n = α ( s n ) .α is recognizable iff Z α ( T ) is a rational function, with Z α ( T ) = P ( T )/ Q ( T ) for some polyno-mials P ( T ) , Q ( T ) .This example is similar to the ones in [Kh2, KS3], where the topological theory is 2-dimensional but there are no defects. The analogue of the Hankel matrix measuring bilinearpairing on connected cobordisms with the boundary S (such cobordisms are determined by ECORATED 1D COBORDISMS 21 the genus g ), see [Kh2], is the Hankel matrix for evaluations of x m x n where x n is an arc with n dots, viewed as a cobordism from ∅ to (+−) . Cobordism x m from (+−) to ∅ is an arc with m dots. Closed cobordism x m x n is a circle with n + m dots, and once again the Hankel matrix H with the ( n, m ) -entry α n + m results, as in [Kh2]. In both cases the state space (of (+−) ,respectively of S ) is the quotient of R N by the null space of H .The theories diverge beyond this example, but there is another connection between thetwo, slightly different from the one above due to an additional shift in the dots versus handlescorrespondence between 1D and 2D cobordisms. Namely, the state space of (+ k − k ) in theone-dimensional theory with the series α maps to the state space for the union ⊔ k S of k circles in the two-dimensional theory for the series α ′ = ( a, α , α , . . . ) for any a ∈ k . In termsof generating functions, Z α ′ = a + Z α T . On the topological side, an arc with n dots is mappedto a an annulus with n handles, while a circle carrying n dots is mapped to to the torus withadditional n handles (thus a surface of genus n + n to n + α . When the set S has more than one element,recognizable power series still admit an analogue of the partial fraction decomposition, see [FH]and references therein, which should lead to decompositions of associated tensor categories. Unoriented cobordisms.
There is an obvious unoriented version of the category C , whereone-dimensional cobordisms are unoriented and the objects, in the skeletal category case,are numbers n ∈ Z + , counting the number of top and bottom endpoints of the cobordism.Evaluation α must be -invariant, that is, to satisfy α ( w ) = α ( w ) , for any word w , in additionto being symmetric, as earlier: α ( w w ) = α ( w w ) , for any words w , w . The dihedral group D n acts on the set S n of words of length n in the alphabet S , and the function α ∶ S ∗ —→ k ,when restricted to these words, must be D n -invariant. Such series can be called d-symmetric ,for instance.The theory then goes through and one can define the viewable category VC α , the skeincategory SC α , the gligible quotient C α , and so on. The interesting case, as before, is when α is recognizable , that is, when the category C α has finite-dimensional hom spaces. A d-symmetricseries α is recognizable iff it is recognizable as noncommutative series iff the syntactic ideal I α has finite codimension in k ⟨ S ⟩ .If the set S is empty, cobordisms do not carry any dots (defects), and the category SC α isthe unoriented Brauer category Br unλ for the parameter λ = α ( ) ∈ k , while C α is the gligiblequotient of B unλ . 3. Cobordisms with inner (floating) boundary
Category ̃C of decorated cobordisms with inner boundary. To connect decorated one-dimensional cobordisms with noncommutative rational powerseries that are not necessarily symmetric we enlarge the category C by allowing cobordisms M that may have additional boundary points ( floating boundary points) strictly inside thecobordism, not being part of the top ∂ M or bottom ∂ M boundary of N . s s s s s s s s s s s s s s − − + − + + − − + + − + Figure 3.1.1.
A morphism from (− − + − ++) to (− − + + −+) . Over- andundercrossing and intersections are ”virtual” and should be ignored. Hollowdots are not labels and show inner (floating) endpoints of the cobordism.Define the category ̃C of S -labelled cobordism with floating (or inner) boundary to havethe same objects as C , that is, finite sequences (cid:15) of plus and minus signs. A morphism in ̃C from (cid:15) to (cid:15) ′ , see Figure 3.1.1 for an example, is a compact oriented S -decorated one-manifold M with(26) ∂M = (cid:15) ′ ⊔ (− (cid:15) ) ⊔ ∂ in M, where ∂ in M is the inner or floating boundary of M that is disjoint from the top boundary,given by (cid:15) ′ and from the bottom boundary, given by − (cid:15) . In (26) we interpret a sign sequence (cid:15) ′ as a zero-dimensional oriented manifold, with oriented connected components describedby elements of the sequence. The sequence − (cid:15) opposite to (cid:15) corresponds to the orientationreversal of 0D manifold (cid:15) . An S -decoration is a collection of points (dots) labelled by elementsof the set S inside M (not on the boundary ∂M ). Labelled points can move along a connectedcomponent but not cross through each other.Morphisms are such decorated 1D cobordisms, possibly with inner endpoints (inner bound-ary points) considered up to rel boundary diffeomorphisms. Figure 3.1.1 shows an exampleof a morphism from (cid:15) = (− − + − ++) to (cid:15) ′ = (− − + + −+) .Composition of morphisms in ̃C is given by concatenation of cobordisms. The category ̃C contains C as the subcategory with the same objects as ̃C and morphisms – morphisms of ̃C that have no inner (floating) boundary points.Connected components of a cobordism in ̃C split into viewable and floating types. Fig-ure 3.1.1 cobordism has three floating components: one circle and two intervals. The samecobordism has eight viewable components: four of them have both endpoints on top or bottomboundary, while the other four have one floating endpoint. Floating components terminologywas introduced in [KS1].Going along a component c in the direction of its orientation we read off the labels of dots.If the component is an arc, the result is a sequence sec ( c ) ∈ S ∗ , a word in the alphabet S . Ifthe component is a circle, the sequence sec ( c ) is defined up to cyclic rotation. Our conventionis to write the sequence from right to left as we follow the orientation. For instance, in ECORATED 1D COBORDISMS 23 s i s i s i s i k s i s j s j s j m Figure 3.1.2.
Left: three interval floating components, with sequences (∅) , ( s i ) , and ( s i s i . . . s i k ) . Right: three circle components, with sequences (∅) , ( s i ) and ( s j m . . . s j s j ) up to cyclic rotation. Figure 3.1.3.
Five types of viewable components, left to right: an intervalconnecting (1) a top and a bottom point, (2) two top points, (3) two bottompoints; a interval with an inner boundary point and a (4) top endpoint, (5)bottom endpoint. Labels of dots and orientations of lines are not shown. Inthe subcategory C viewable components are of types (1)-(3) only.Figure 3.1.6 left the sequence is s s s , while in Figure 3.1.6 right the sequence is s s s .Orientation reversal of a component corresponds to reversing the sequence.The sequences for components of Figure 3.1.1 cobordism are: ● The empty sequence (∅) and ( s ) for the two floating arc components. ● Sequence ( s s s ) , up to cyclic rotation, for the unique floating circle component. ● Sequences ( s s ) and ( s ) for the two connected components that connect a topendpoint and a bottom endpoint. ● The empty sequence (∅) for the unique component that connects two top endpoints. ● Sequence ( s s ) for the unique component connecting two bottom endpoints. ● Sequences ( s s ) and ( s s ) for arc components with one top and one inner boundarypoint. ● Sequences (∅) and ( s ) for arc components with one bottom and one inner boundarypoint.A floating component of a cobordism x in ̃C is either an interval or a circle, see Figure 3.1.2.A viewable component has one of the five types shown in Figure 3.1.3, with some numberof dots (perhaps none) on it.Monoidal category ̃C has generators shown in Figures 2.1.3 and 2.1.4 and common with itssubcategory C and two additional generators shows in Figure 3.1.4 left. These are ars withone floating and one top endpoint. Applying U-turns to them results in arcs with one floatingand one bottom endpoint, see Figure 3.1.4 right. Some additional defining relations in ̃C areshown in Figure 3.1.5, see also Figure 2.1.4 for defining relations in the subcategory C , whichalso give a subset of defining relations in ̃C . We will not need a full set of defining relationsfor ̃C in this paper. + − + += − − = Figure 3.1.4.
Left: additional generating morphisms for monoidal category ̃C beyond the generating morphisms common for ̃C and C shown in Figures 2.1.3and 2.1.4. = = = = Figure 3.1.5.
Some additional relations in ̃C . s s s s s s = s s s = Figure 3.1.6.
Orientation matters: evaluations α ( s s s ) and α ( s s s ) aredifferent, in general. Reversal of a sequence corresponds to orientation reversalof the corresponding floating arc or circle.3.2. Tensor envelopes of ̃C . Floating (closed) cobordisms in ̃C (endomorphisms of the empty zero-manifold (∅) ) areunions of floating intervals and circles. A floating interval carries a sequence w ∈ S ∗ , a floatingcircle carries a sequence v well-defined up to cyclic rotation, v v ≡ v v . Consequently amultiplicative evaluation of floating cobordisms in ̃C , as explained in the introduction, consistsof a pair of series(27) α = ( α ● , α ○ ) , where α ● ∈ k ⟪ S ⟫ is a noncommutative series and α ○ ∈ k ⟪ S ⟫ s is a symmetric series.A multiplicative evaluation on closed cobordisms in ̃C assigns α ● ( w ) ∈ k to an orientedinterval with word w written along it via labelled dots, see Figure 3.2.1. Element α ○ ( v ) ∈ k isassigned to an oriented circle with word v , well-defined up to a cyclic rotation, written alongit. ECORATED 1D COBORDISMS 25 w w . . . w n c • ( w ) α • ( w ) w w w w n c ◦ ( w ) α ◦ ( w ) Figure 3.2.1.
Evaluation α ● ( w ) of the floating interval c ● ( w ) and evaluation α ○ ( w ) of the circle c ○ ( w ) in V ̃C , for a word w = w . . . w n .We now proceed along a familiar route, as in [KS3, KQR] and Section 2, to build varioustensor envelopes of a pair α = ( α ● , α ○ ) .(1) Pre-linearization category k ̃C . Category k ̃C has the same objects as ̃C , and the mor-phisms are finite k -linear combinations of morphisms in ̃C . This is a naive linearization or pre-linearization of ̃C .(2) Viewable cobordisms category
V ̃C α . To form category
V ̃C α , we mod out tensor category k ̃C by relations that evaluate floating (closed) cobordisms to elements of the ground fieldvia α . Namely, a floating oriented interval with a sequence w ∈ S ∗ on it, denoted c ● ( w ) ,evaluates to α ● ( w ) ∈ k . A floating oriented w -decorated circle c ○ ( w ) evaluates to α ○ ( w ) ,see Figure 3.2.1. Recall that α ○ is symmetric and α ○ ( v v ) = α ○ ( v v ) for any words v v ,matching circle rotation, c ○ ( v v ) = c ○ ( v v ) . Since all floating components of a cobordism reduce to elements in k , the vector space ofhoms from (cid:15) to (cid:15) ′ in V ̃C α has a basis of viewable cobordisms from (cid:15) to (cid:15) ′ with any sequenceswritten on its connected components.Denote by I( (cid:15), (cid:15) ′ ) the set of diffeomorphism classes of viewable cobordisms (without dotdecorations) from (cid:15) to (cid:15) ′ . A viewable cobordism has no circles and all its connected compo-nents are intervals. Such a cobordism C may have some number of viewable components oftypes (4) and (5), see Figure 3.1.3. Each such component has one floating boundary point andone boundary point among elements of (cid:15) ⊔ (cid:15) ′ . Other connected components (of types (1)-(3))give an orientation-respecting matching of the remaining elements of (cid:15) and (cid:15) ′ .To specify an element of I( (cid:15), (cid:15) ′ ) we select a subset I ′ of elements in the sequence (− (cid:15) )⊔ (cid:15) ′ sothat the remaining sequence is balanced , that is, has the same number of pluses and minuses.We then choose a bijection b between pluses and minuses of (− (cid:15) ) ⊔ (cid:15) ∖ I ′ . Such pairs ( I ′ , b ) arein a bijection with isomorphism classes of viewable undecorated cobordisms between (cid:15) and (cid:15) ′ , that is, elements of I( (cid:15), (cid:15) ′ ) . Figure 3.2.2 shows elements of the set I(+ , +−) . Figure 3.2.3shows elements of the set I(+ , + + −) .To allow S -decorations, we consider the set I S ( (cid:15), (cid:15) ′ ) which consists of a pair: an elementof I( (cid:15), (cid:15) ′ ) and a choice of word w ( c ) in S ∗ for each component c of I( (cid:15), (cid:15) ′ ) . To such a pairwe assign an S -decorated viewable cobordism given by the element of I( (cid:15), (cid:15) ′ ) and words w ( c ) written on components c of the cobordism. Some words may be empty (have length zero).Denote the cobordism associated with t ∈ I S ( (cid:15), (cid:15) ′ ) by C ( t ) . Proposition 3.1.
Viewable cobordisms C ( t ) , over all t in I S ( (cid:15), (cid:15) ′ ) , constitute a basis in thehom space Hom ( (cid:15), (cid:15) ′ ) in the category V ̃C α . + − + + − + + − + Figure 3.2.2.
Three elements of the set
I(+ , +−) . + − ++ + − ++ + − ++ + − ++ + − ++ + − ++ + − ++ Figure 3.2.3.
Seven elements of the set
I(+ , + + −) Composing cobordisms from these bases sets results in cobordisms that, in general, havefloating components. These components are evaluated via α , viewable components are kept,and the composition of two basis elements is a basis element in a suitable hom space, scaledby an element of k .As earlier, to each word w ∈ S ∗ we associate the upward interval with w written on it, thatis, cobordism cob ( w ) from (+) to (+) , see Figure 2.4.2 left. Each element u of k ⟨ S ⟩ gives riseto a linear combination cob ( u ) of these cobordisms, see Figure 2.4.2 right. The resulting mapcob ∶ k ⟨ S ⟩ —→ End
V ̃C α ((+)) is an injective homomorphism from the free algebra k ⟨ S ⟩ to the ring of endomorphisms of (+) in category V ̃C α (in the smaller category VC α considered in Section 2 this map is anisomorphism). One-sided inverse homomorphism to cob is given by the surjectioncob ′ ∶ End
V ̃C α ((+)) —→ k ⟨ S ⟩ that sends any cobordism with floating endpoints to zero. The latter cobordisms span a two-sided ideal in End V ̃C α ((+)) , with the quotient isomorphic to k ⟨ S ⟩ . This ideal is naturallyisomorphic to k ⟨ S ⟩ ⊗ k ⟨ S ⟩ when viewed as a k ⟨ S ⟩ -bimodule. Multiplication in this ideal isgiven by ( x ⊗ x )( y ⊗ y ) = α ● ( x y ) x ⊗ y . Composition cob ′ ○ cob = Id k ⟨ S ⟩ .(3) Skein category
S ̃C α . This category has finite-dimensional hom spaces when α is recog-nizable, and we restrict to that case. We say that α = ( α ● , α ○ ) is recognizable if both series α ● and α ○ are recognizable. Series α ● and α ○ has syntactic ideals I α ● , I α ○ ⊂ k ⟨ S ⟩ , respectively.Recognizability means that both ideals have finite codimension in k ⟨ S ⟩ . Equivalently, the ECORATED 1D COBORDISMS 27 u = 0 uv v α • uv α ◦ Figure 3.2.4.
Left: endomorphism cob ( u ) of (+) is set to zero in S ̃C α for u ∈ I α . Middle: α ● ( v uv ) = v , v ∈ k ⟨ S ⟩ since u ∈ I α ⊂ I α ● . Right: α ○ ( vu ) = v ∈ k ⟨ S ⟩ since u ∈ I α ⊂ I α ○ .two-sided ideal(28) I α ∶= I α ● ∩ I α ○ ⊂ k ⟨ S ⟩ has finite codimension in k ⟨ S ⟩ . Denote by A α ∶= k ⟨ S ⟩/ I α the syntactic algebra of the pair α .Starting with the category V ̃C α , we add tensor relations cob ( u ) = u ∈ I α , seeFigure 3.2.4 left. These relations are consistent with the evaluation α of floating components.Consistency is due to restricting to u in the syntactic ideal, which is contained in both ideals I α ● and I α ○ . Elements of the first ideal evaluate to zero when placed anywhere on a floatinginterval, see Figure 3.2.4 middle. Elements of the second ideal evaluate to zero when placedon a circle, see Figure 3.2.4 right.Recall that in addition to two-sided syntactic ideals I α ● , I α ○ and their intersection I α = I α ● ∩ I α ○ there are one-sided syntactic ideals I (cid:96)α ● and I rα ● . Here I (cid:96)α ● = { x ∈ k ⟨ S ⟩∣ α ● ( yx ) = ∀ y ∈ k ⟨ S ⟩} and I rα ● = { x ∈ k ⟨ S ⟩∣ α ● ( xy ) = ∀ y ∈ k ⟨ S ⟩} are left and right ideals in k ⟨ S ⟩ , respectively.For u ∈ k ⟨ S ⟩ denote by cob + ( u ) the element of Hom (∅ , (+)) given by putting u on aninterval at its ”out” floating endpoint, see Figure 3.2.5 left. Define cob − ( u ) likewise, seeFigure 3.2.5 right. We add relations that cob + ( u ) = u ∈ I (cid:96)α ● and cob − ( v ) = v ∈ I rα ● .This finishes our definition of category S ̃C α .Note that ideals I α , I (cid:96)α ● , I rα ● have finite codimensions in k ⟨ S ⟩ , and each of these ideals isfinitely-generated. In particular, one can restrict to adding finitely many relations to V ̃C α toget the ”skein” category S ̃C α .Due to consistency of these relations with the evaluation α on floating components wecan describe a basis in the hom spaces in the category S ̃C α , as follows. Choose subsets B α , B (cid:96)α ● , B rα ● ⊂ k ⟨ S ⟩ that descend to bases of A α , k ⟨ S ⟩/ I (cid:96)α ● and I rα ● / k ⟨ S ⟩ , respectively.Recall the basis I S ( (cid:15), (cid:15) ′ ) of the hom space from (cid:15) to (cid:15) ′ in V ̃C α constructed earlier. It consistsof a floating cobordism x from (cid:15) to (cid:15) ′ with various monomials written on components of thecobordism (all components are viewable). Define the set B α ( (cid:15), (cid:15) ′ ) to also consists of floatingcobordisms from (cid:15) to (cid:15) ′ , but now we write an element of one of the three sets B α , B (cid:96)α ● , B rα ● oneach component of c , depending on its type: + u = 0 uv α • − v = 0 vu uv = α • Figure 3.2.5.
Left: Element cob + ( u ) of Hom (∅ , (+)) is set to zero in S ̃C α for u ∈ I (cid:96)α ● . This relation is compatible with evaluations of floating diagrams, since α ● ( vu ) = ∀ v ∈ k ⟨ S ⟩ , second left. Right: Element cob − ( v ) of Hom (∅ , (−)) isset to zero in S ̃C α for v ∈ I rα ● . This relation is also compatible with evaluationsof floating diagrams, since α ● ( vu ) = ∀ u ∈ k ⟨ S ⟩ for such v , see the last equationon the right. c ++ c c ++ Figure 3.2.6.
There are two types of cobordisms from (+) to (+) in ̃C . Mir-roring that decomposition, basis B α ((+) , (+)) consists of elements of c ∈ B α placed on through strand and pairs of elements c ∈ B (cid:96)α ● and c ∈ B rα ● placedon the two strands with floating endpoints. ● If a component has no floating endpoints, thus connects two boundary points (at thetop or bottom boundary, or both), put an element of B α along it. ● If a component has a floating endpoint and is oriented away from this endpoint, putan element of B (cid:96)α ● along this component. ● If a component has a floating endpoint and is oriented towards it, put an element of B rα ● along this component.An undecorated viewable cobordism x with n , n , n components of these three types, re-spectively, admits ∣ B α ∣ n ∣ B (cid:96)α ● ∣ n ∣ B rα ● ∣ n possible decorations. The set B α ( (cid:15), (cid:15) ′ ) is the union ofthese decorated cobordisms, where we start with any viewable undecorated cobordism x from (cid:15) to (cid:15) ′ and decorate it in all possible such ways. The set B α ( (cid:15), (cid:15) ′ ) is finite.For example, B α ((+) , (+)) has cardinality ∣ B α ∣ + ∣ B (cid:96)α ● ∣ ⋅ ∣ B rα ● ∣ and consists of diagrams oftwo types, see Figure 3.2.6. ECORATED 1D COBORDISMS 29
Proposition 3.2.
The set B α ( (cid:15), (cid:15) ′ ) is a basis of the hom space Hom S ̃C α ( (cid:15), (cid:15) ′ ) in the category S ̃C α . This construction gives a basis in hom spaces of
S ̃C α for non-recognizable α as well, butthen A α and hom spaces are infinite-dimensional. Recall that we restrict to consideringrecognizable α for most of this section.The endomorphism ring of (+) in S ̃C α contains a two-sided ideal isomorphic to the tensorproduct k ⟨ S ⟩/ I (cid:96)α ● ⊗ I rα ● / k ⟨ S ⟩ , with the quotient algebra isomorphic to A α , so there is an exactsequence of k ⟨ S ⟩ -bimodules(29) 0 —→ k ⟨ S ⟩/ I (cid:96)α ● ⊗ I rα ● / k ⟨ S ⟩ —→ End
S ̃C α ((+)) —→ A α —→ . The quotient map onto A α admits a section, and A α is naturally a subalgebra of End S ̃C α ((+)) .This decomposition corresponds to two types of endomorphisms of (+) in ̃C (without floatingendpoints versus having two floating endpoints) and corresponding bases in endomorphismsof (+) in S ̃C α , see Figure 3.2.6.(4) Gligible quotient category ̃C α . The trace tr α ( x ) of a cobordism x from (cid:15) to (cid:15) is definedin the same way as for cobordisms in the smaller category C α , by closing x into a floatingcobordism ̂ x , see Figure 2.4.4 and evaluating via α : tr α ( x ) ∶= α (̂ x ) . This operation extends to a k -linear trace on k ̃C α that descends to a trace on V ̃C α and S ̃C α :End k ̃C ( (cid:15) ) —→ End
V ̃C α ( (cid:15) ) —→ End
S ̃C α ( (cid:15) ) tr α —→ k . The trace is symmetric. The two-sided ideal ̃ J α ⊂ S ̃C α of negligible morphisms is definedas usual, see Section 2.4 for the definition of negligible ideal in the subcategory SC α of S ̃C α .Define the quotient category ̃C α ∶= S ̃C α / J α . The quotient category ̃C α has finite-dimensional hom spaces, as does S ̃C α , since α is recog-nizable. The trace is nondegenerate on ̃C α and defines perfect bilinear pairingsHom ( (cid:15), (cid:15) ′ ) ⊗ Hom ( (cid:15) ′ , (cid:15) ) —→ k on its hom spaces. We call ̃C α the gligible quotient of S ̃C α , having modded out by the idealof negligible morphisms.Up to an isomorphism, the state space(30) A α ( (cid:15) ) ∶= Hom ̃C α (∅ , (cid:15) ) depends only on the number of pluses and minuses in (cid:15) and A α ( (cid:15) ) ≅ A α (+ n − m ) , where n and m is the number of pluses and minuses in (cid:15) . Summing A α (+ n − m ) over n, m ≥ S n × S m action on the homogeneous ( n, m ) component withthe properties similar to that of a tca algebra [SSn]. (5) The Deligne category and its gligible quotient. From the skein category
S ̃C α we canpass to its additive Karoubi closure D ̃C α ∶= Kar (S ̃C ⊕ α ) , which is the analogue of the Deligne category. The quotient of D ̃C α by the ideal D ̃J α ofnegligible morphisms, D ̃C α ∶= D ̃C α /D ̃J α , is equivalent to the additive Karoubi closure of ̃C α . Summary:
To summarize, the following categories are assigned to a recognizable pair α asin (27): ● The category
V ̃C α of viewable cobordisms with the α -evaluation of floating (or closed)components. ● The skein category
S ̃C α where closed (floating) S -decorated intervals and circles areevaluated via α and elements of the syntactic ideal I α evaluate to zero when placedalong any interval in a cobordism. Furthemore, elements of left and right syntacticideals I (cid:96)α ● and I rα ● evaluate to zero when placed at the beginning or end of an intervalwith the corresponding endpoint floating. ● The quotient ̃C α of S ̃C α by the two-sided ideal of negligible morphisms. We alsocall ̃C α the gligible category or the gligible quotient. Hom spaces in ̃C α come withnondegenerate bilinear formsHom ( (cid:15), (cid:15) ′ ) ⊗ Hom ( (cid:15) ′ , (cid:15) ) —→ k , where (cid:15), (cid:15) ′ are objects ̃C α , sequences of pluses and minuses describing oriented zero-manifolds that are the source and the target of decorated one-cobordisms. The uni-versal construction, in this case, assigns the vector space Hom ̃C α (∅ , (cid:15) ) of morphismsfrom the empty zero-manifold ∅ to (cid:15) to the oriented zero-manifold (cid:15) . ● Additive Karoubi closure
D ̃C α of S ̃C α , analogous to the Deligne category. The quotientof D ̃C α by the ideal of negligible morphisms is denoted D ̃C α .We arrange these categories and functors, for recognizable α = ( α ● , α ○ ) , into the followingdiagram, with a commutative square on the right:(31) ̃C ———→ k ̃C ———→ V ̃C α ———→ S ̃C α ———→ D ̃C α (cid:215)(cid:215)(cid:215)(cid:214) (cid:215)(cid:215)(cid:215)(cid:214)̃C α ———→ D ̃C α Properties of the categories in the analogous diagram (23) in Section 2.5, as explained in theparagraph following (23), hold for the categories in (31) as well.The category ̃C and categories built out of it require a pair of series α = ( α ● , α ○ ) forevaluation. When working with the subcategory C of cobordisms without floating endpoints,only circles appear as connected components of floating cobordisms, and series α ○ is neededfor evaluation. Instead of working with C , one can use ̃C but set the connected component α ● =
0, so that α = ( , α ○ ) . Then in the viewable category V ̃C α any cobordism that contains afloating interval evaluates to zero. Syntactic ideals I (cid:96)α ● , I rα ● = k ⟨ S ⟩ , and in the skein category S ̃C α any cobordism containing an interval (including viewable intervals, with one boundaryand one floating endpoint) evaluates to 0. This results in equivalences of categories(32) S ̃C ( ,α ○ ) ≅ SC α ○ , ̃C ( ,α ○ ) ≅ C α ○ , D ̃C ( ,α ○ ) ≅ DC α ○ , D ̃C ( ,α ○ ) ≅ DC α ○ , ECORATED 1D COBORDISMS 31 that, furthermore, respect commutative squares of categories in diagrams (23) and (31).Thus, the construction in Section 2 of the skein category SC α , the gligible quotient category C α and other categories defined there and associated to symmetric series α can be considereda special case of the construction of the present section, specializing to the pair ( , α ) withthe first recognizable series in the pair being zero.Alternatively, one can set α ○ to zero and consider a pair α = ( α ● , ) . In the viewablecobordism category V ̃C α for this α a circle (necessarily floating, and with any decoration)evaluates to 0. Only the ideals I α ● , I (cid:96)α ● , I rα ● (two-sided, left and right, respectively) are usedin the definition of the skein category S ̃C α . A decorated U -turn as in Figure 3.1.3, case (2)or (3), may be non-zero in ̃C α , since it may be coupled to two intervals on the other side witha non-zero evaluation.When α = ( α ● , ) , another approach is to restrict possible cobordisms and disallow U -turnsas cobordisms. Then a cobordism c must have no critical points under the natural projectiononto the interval [ , ] under which ∂ i c projects onto i , for i = ,
1. When components ofcobordisms are unoriented, such restricted cobordisms appear in [KS1] in a categorification ofthe polynomial ring (without dot decorations) and in [KT] in a categorification of Z [ / ] andpotential categorifications of Z [ / n ] as monoidal envelopes of certain Leavitt path algebras andthe ”one-sided inverse” algebra k ⟨ a, b ⟩/( ab − ) . The latter cobordisms carry dot gecorations,corresponding to the generators of these algebras.When S = ∅ , the evaluation again reduces to two numbers (evaluations of the orientedinterval and oriented circle), and the skein category S ̃C is the oriented partial Brauer category,see the remark below. When S = { s } has cardinality one, recognizable series α is encoded bytwo rational functions Z α ● ( T ) , Z α ○ ( T ) in a single variable T . Remark:
Instead of power series α = ( α ● , α ○ ) in noncommuting variables one can insteadstart with an associative k -algebra B and two k -linear maps(33) α ● ∶ B —→ k , α ○ ∶ B —→ k such that α ○ is symmetric, α ○ ( ab ) = α ○ ( ba ) , a, b ∈ B . Two-sided syntactic ideals I α ● , I α ○ ⊂ B ,their intersection I α ∶= I α ● ∩ I α ○ , and one-sided syntactic ideals I rα ● , I (cid:96)α ● are defined in the sameway as for noncommutative series.The nondegenerate case is that of I α = B , but the arbitrary casecan be reduced to it by passing to the quotient B / I α . Recognizable case corresponds to finite-dimensional B . Analogues of all categories in (31) can be defined for such pair of traces on afinite-dimensional B . Defects on cobordisms are now labelled by elements of B rather thanby elements of S . The difference from noncommutative recognizable power series is that onedoes not pick any particular set of generators S of B , working with the entire B instead, butthe resulting categories, starting with the category S ̃C α in (31), are equivalent to the onesbuilt from a noncommutative power series once a set S of generators of B is chosen. Remark:
It is straightforward to modify the constructions of this section to the case ofunoriented one-manifolds with floating endpoints and S -decorated dots. If, in addition, S = ∅ ,there are no dots and floating cobordisms reduce to unions of intervals and circles. Theevaluation is then a pair ( α ● ( ) , α ○ ( )) of elements of k and the unoriented skein category S ̃C α is the partial Brauer category [MM], also known as the rook-Brauer category [HdM]. References [A] M. F. Atiyah,
Topological quantum field theory , Publ. Math. IHES, tome (1988), 175-186.[BR1] J. Berstel and C. Reutenauer, Zeta functions of formal languages , Transactions of the AMS ,no. 2 (1990), 533-546.[BR2] J. Berstel and C. Reutenauer,
Noncommutative rational series with applications , Encyclopedia ofMath. Appl. , Cambridge U Press, 2011.[B] C. Blanchet,
An oriented model for Khovanov homology , JKTR
02, (2010), 291–312,arXiv:1405.7246.[BHMV] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel,
Topological quantum field theories derivedfrom the Kauffman bracket , Topology Positivity of the universal pairing in 3 dimensions ,Journal of the AMS A course in formal languages, automata and groups , Universitext series, Springer 2009.[Co] J. H. Conway,
Regular algebra and finite machines , Chapman and Hill 1971, republished by Dover,2012.[CO] J. Comes and V. Ostrik,
On blocks of Deligne’s category
Rep ( S t ) , Adv. in Math. (2011),1331-1377, arXiv:0910.5695 .[D] P. Deligne, La cat´egorie des repr´esentations du groupe sym´etrique S t , lorsque t n’est pas en entiernaturel , in Proceedings of the Int. Colloquium on Alg. Groups and Homogeneous Spaces, Tata Inst.Fund. Res. Studies Math. Mumbai 2007, 209-273.[DG] M. Droste and P. Gastin, On recognizable and rational power series in partially commuting variables ,in Degano P., Gorrieri R., Marchetti-Spaccamela A. (eds) Automata, Languages and Programming,ICALP 1997, Lect. Notes Comp. Sci. , Springer 1997.[EST] M. Ehrig, C. Stroppel and D. Tubbenhauer,
Generic gl -foams, webs, and arc algebras, arXiv:1601.08010.[E1] S. Eilenberg, Automata, languages and machines , vol. A, Academic Press, 1974.[E2] S. Eilenberg,
Automata, languages and machines , vol. B, Academic Press, 1976.[EK] Z. ´Esik and W. Kuich,
Modern automata theory
Tensor categories , Math. Surveys andMonographs , AMS 2015.[F] M. Fliess,
Matrices de Hankel , J. Math. Pures Appl (9), (1974), 197-222.[FH] M. V. Foursov and C. Hespel, About the decomposition of rational series in noncommutativevariables into simple series , in Proc. 6th Int. Conf. on Informatics in Control, Automation andRobotics - Signal Processing, Systems Modeling and Control (2009), 214-220.[FKNSWW] M. H. Freedman, A. Kitaev, C. Nayak, J. K. Slingerland, K. Walker and Z. Wang,
Universalmanifold pairings and positivity , Geometry and Topology (2005), 2303-2317.[HdM] T. Halverson and E. delMas, Representations of the rook-Brauer algebra , Comm. in Algebra , is. 1(2014), 423-443, arXiv:1206.4576.[HMS] J. W. Helton, T. Mai and R. Speicher, Applications of realizations (aka linearizations) to freeprobability , Journal of Functional Analysis (2018), 1-79.[KaR] C. Kassel and C. Reutenauer,
Algebraicity of the zeta function associated to a matrix over a freegroup algebra , Algebra and Number Theory :2 (2014), 497-511.[Kh1] M. Khovanov, sl(3) link homology , Alg. Geom. Top. (2004), 1045–1081, arXiv:0304375.[Kh2] M. Khovanov, Universal construction of topological theories in two dimensions , arXiv:2007.03361.[KK] M. Khovanov and N. Kitchloo,
A deformation of Robert-Wagner foam evaluation and link homology ,arXiv:2004.14197.[KL] M. Khovanov and R. Laugwitz, In preparation.[KKO] M. Khovanov, Y. Kononov, and V. Ostrik, In preparation.[KRW] M. Khovanov, L.-H. Robert and E. Wagner, In preparation.[KQR] M. Khovanov, Y. Qi, and L. Rozansky,
Evaluating thin flat surfaces , arXiv:2009.01384.[KS1] M. Khovanov and R. Sazdanovic,
Categorification of the polynomial ring , Fundamenta Math. ,no.3 (2015), 251-280, arXiv:1101.0293.
ECORATED 1D COBORDISMS 33 [KS2] M. Khovanov and R. Sazdanovic,
Diagrammatic categorification of the Chebyshev polynomials of thesecond kind , to appear in Journal of Pure Appl. Algebra, arXiv:2003.11664.[KS3] M. Khovanov and R. Sazdanovic,
Bilinear pairings on two-dimensional cobordisms andgeneralizations of the Deligne category , arXiv:2007.11640.[KT] M. Khovanov and Y. Tian,
How to categorify the ring of integers localized at two , QuantumTopology , is. 4 (2019), 723–775, arXiv:1702.07466.[Kn] F. Knop, Tensor envelopes of regular categories , Advances in Mathematics (2007), 571-617.[KuS] W. Kuich and A. Salomaa,
Semiring, automata, languages , EATCS Monographs of Theor. Comp. Sci. , Springer 1986.[MM] P. Martin and V. Mazorchuk, On the representation theory of partial Brauer algebras , Quarterly J.Math. , no. 1 (2014), 225-247, arXiv:1205.0464.[Pe] D. Perrin, Completely reducible sets , Int. Journal of Algebra and Computation , 4 (2013), 915-942.[R] A. Reynolds, Representations of the oriented Brauer category , PhD Thesis 2015, University ofOregon.[Re1] C. Reutenauer,
S´eries formelles et alg´ebres syntactiques , Journal of Algebra (1980), 448-483.[Re2] C. Reutenauer, Cyclic derivation of noncommutative algebraic power series , Journal of Algebra (1983), 32-39.[Re3] C. Reutenauer, Noncommuting variables: I. Finite automata, II. Rational generating series, III.Realization of bilinear systems , in Encyclopedia of Systems and Control, (M. Singh, editor),Pergamon Press, 1988, 3268-3279.[Re4] C. Reutenauer,
A survey of noncommutative rational series , DIMACS Series in Discrete Math. Theor.Comp. Sci. (1996), 159-169.[Re5] C. Reutenauer, Michel Fliess and non-commutative formal power series , International Journal ofControl , no. 3 (2008), 338-343.[Ro] G.-C. Rota, A cyclic derivative in noncommutative algebra , Journal of Algebra (1980), 54-75.[RW1] L.-H. Robert and E. Wagner, A closed formula for the evaluation of sl N -foams , to appear inQuantum Topology, arXiv:1702.04140.[RW2] L.-H. Robert and E. Wagner, Symmetric Khovanov–Rozansky link homologies , Journal de l’´EcolePolytechnique — Math´ematiques, Tome (2020) , 573-651, https://jep.centre-mersenne.org/item/JEP_2020__7__573_0/ .[RWd] D. E. V. Rose, P. Wedrich, Deformations of colored sl(N) link homologies via foams , Geom. Topol. (2016), 3431–3517, arXiv:1501.02567.[SS] A. Salomaa and M. Soittola, Automata-theoretic aspects of formal power series , Texts andMonographs in Computer Science, Springer, 1978.[SSn] S. Sam and A. Snowden,
Gr¨obner methods for representations of combinatorial categories , J. Amer.Math. Soc. (2017), 159-203, arXiv:1409.1670 .[Sa1] J. Sakarovitch, Easy multiplications I. The realm of Kleene’s theorem , Information and Computation (1987), 173-197.[Sa2] J. Sakarovitch, Rational and recognizable power series , Chapter 4 in Handbook of WeightedAutomata (eds. M. Droste, W. Kuich, H. Vogler), 105-174, Springer, 2009.[Sch] M. P. Sch¨utzenberger,
On the definition of a family of automata , Information and Control (1961),245–270.[U] R. G. Underwood, Fundamentals of Hopf algebras , Universitext series, Springer 2015.[W] K. Walker,
Universal manifold pairings in dimension 3 , Celebratio Mathematica, MichaelH. Freedman, https://celebratio.org/Freedman_MH/article/93/ , (2012).[Wd] P. Wedrich,
Exponential growth of colored HOMFLY-PT homology , Adv. in Math. , (2019),471–525, arXiv:1602.02769.
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