ezra getzler- operads revisited

8/3/2019 Ezra Getzler- Operads revisited

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arXiv:math/0

701767v1

[math.C

T]26Jan2007

Operads revisited

Ezra Getzler

Northwestern University, Evanston, Illinois, USA

[email protected]

This paper presents an approach to operads related to recent work in topolog-ical field theory (Costello [4]). The idea is to represent operads as symmetricmonoidal functors on a symmetric monoidal category T; we recall how this

works for cyclic and modular operads and dioperads in Section 1. This pointof view permits the construction of all sorts of variants on the notion of anoperad. As our main example, we present a simplicial variant of modularoperads, related to Segals definition of quantum field theory, as modifiedfor topological field theory (Getzler [10]). (This definition is only correct ina cocomplete symmetric monoidal category whose tensor product preservescolimits, but this covers most cases of interest.)

An operad in the category Set of sets may be presented as a symmetricmonoidal category tP, called the theory associated to P (Boardman and Vogt[2]). The category tP has the natural numbers as its objects; tensor productis given by addition. The morphisms of tP are built using the operad P:

t

P(m, n) =

f:{1,...,m}{1,...,n}

n

i=1 P(|f1

(i)|).

The category ofP-algebras is equivalent to the category of symmetric monoidalfunctors from tP to Set; this reduces the study of algebras over operads tothe study of symmetric monoidal functors.

The category of contravariant functors from the opposite category T of asmall category T to the category Set of sets

T = [T, Set]

is called the category of presheaves ofT. IfT is a symmetric monoidal category,then T is too (Day [5]); its tensor product is the coend

V W =A,BT

T(, A B) V(A) W(B).

A symmetric monoidal functor F : S T between symmetric monoidalcategories S and T is a functor F together with a natural equivalence
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2 Ezra Getzler

: F F = F .

The functor F is lax symmetric monoidal if is only a natural transformation.If : S T is a symmetric monoidal functor, it is not always the case

that the induced functor : T S

is a symmetric monoidal functor; in general, it is only a lax symmetricmonoidal functor, with respect to the natural transformation

(V) (W)A,BS

S(, A B) V( A) W( B) (V) (W)

(V W)

V,W

A,BSS(, A B) V( A) W( B)

A,BST((), (A B)) V( A) W( B)

is a functor

A,BST((), (A B)) V( A) W( B)

A,BST((), (A) (B)) V(A) W(B)

is symmetric monoidal

A,BST((), (A) (B)) V(A) W(B)

A,BTT((), A B) V(A) W(B)

universality of coends

(V W)A,BT

T((), A B) V(A) W(B)

The following definition introduces the main object of study of this paper.

Definition. A pattern is a symmetric monoidal functor : S T betweensmall symmetric monoidal categories S and T such that : T S is asymmetric monoidal functor (in other words, the natural transformation defined above is an equivalence).

Let be a pattern and let C be a symmetric monoidal category. A -preoperad in C is a symmetric monoidal functor from S to C, and a -operadin C is a symmetric monoidal functor from T to C. Denote by PreOp(C) andOp(C) the categories of -preoperads and -operads respectively. If C is acocomplete symmetric monoidal category, there is an adjunction

: PreOp(C) Op(C) : .

In Section 2, we prove that if is essentially surjective, C is cocomplete, andthe functor A B on C preserves colimits in each variable, then the functor

is monadic. We also prove that ifS is a free symmetric monoidal category andC is locally finitely presentable, then Op(C) is locally finitely presentable.

Patterns generalize the coloured operads of Boardman and Vogt [2], which

are the special case whereS

is the free symmetric category generated by adiscrete category (called the set of colours). Operads are themselves algebrasfor a coloured operad, whose colours are the natural numbers (cf. Berger


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Op erads revisited 3

and Moerdijk [1]), but it is more natural to think of the colour n as havingnontrivial automorphisms, namely the symmetric group Sn.

The definition of a pattern may be applied in the setting of simplicialcategories, or even more generally, enriched categories. We define and study

patterns enriched over a symmetric monoidal category V in Section 2; werecall those parts of the formalism of enriched categories which we will needin the appendix.

In Section 3, we present examples of a simplicial pattern which arises intopological field theory. The most interesting of these is related to modularoperads. Let cob be the (simplicial) groupoid of diffeomorphisms betweencompact connected oriented surfaces with boundary. Let Sg,n be a compactoriented surface of genus g with n boundary circles, and let the mapping classgroup be the group of components of the oriented diffeomorphism group:

g,n = 0(Diff+(Sg,n)).

The simplicial groupoid cob has a skeleton

g,n

Diff+(Sg,n),

and if 2g 2 + n > 0, there is a homotopy equivalence Diff+(Sg,n) = g,n.In Section 3, we define a pattern whose underlying functor is the inclusionS cob Cob. This pattern bears a similar relation to modular operadsthat braided operads (Fiedorowicz [8]) bear to symmetric operads. The map-ping class group 0,n is closely related to the ribbon braid group Bn Z.Denoting the generators of Bn Bn Z by {b1, . . . , bn1} and the gen-erators of Zn Bn Z by {t1, . . . , tn}, Moore and Seiberg show [21, Ap-pendix B.1] that 0,n is isomorphic to the quotient of Bn Z by its sub-group (b1b2 . . . bn1tn)n, b1b2 . . . bn1t2nbn1 . . . b1. In this sense, operads for

the patternCob

are a cyclic analogue of braided operads.

Acknowledgements

I thank Michael Batanin and Steve Lack for very helpful feedback on earlierdrafts of this paper.

I am grateful to a number of institutions for hosting me during the writingof this paper: RIMS, University of Nice, and KITP. I am grateful to my hostsat all of these institutions, Kyoji Saito and Kentaro Hori at RIMS, AndreHirschowitz and Carlos Simpson in Nice, and David Gross at KITP, for theirhospitality.

I am also grateful to Jean-Louis Loday for the opportunity to lecture onan earlier version of this work at Luminy. I received support from NSF Grants

DMS-0072508 and DMS-0505669.


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4 Ezra Getzler

1 Modular operads as symmetric monoidal functors

In this section, we define modular operads in terms of the symmetric monoidalcategory of dual graphs. Although we do not assume familiarity with the

original definition (Getzler and Kapranov [11]; see also Markl et al. [18]), thissection will certainly be easier to understand if this is not the readers firstbrush with the subject.

We also show how modifications of this construction, in which dual graphsare replaces by forests, or by directed graphs, yielding cyclic operads andmodular dioperads. Finally, we review the definition of algebras for modularoperads and dioperads.

Graphs

A graph consists of the following data:

i) finite sets V() and F(), the sets of vertices and flags of the graph;ii) a function p : F() V(), whose fibre p1(v) is the set of flags of the

graph meeting at the vertex v;iii) an involution : F() F(), whose fixed points are called the legs of

, and whose remaining orbits are called the edges of .

We denote by L() and E() the sets of legs and edges of , and by n(v) =|p1(v)| the number of flags meeting a vertex v.

To a graph is associated a one-dimensional cell complex, with 0-cellsV() L(), and 1-cells E() L(). The 1-cell associated to an edgee = {f, (f)}, f F(G), is attached to the 0-cells corresponding to thevertices p(f) and p((f)) (which may be equal), and the 1-cell associated toa leg f L(G) F(G) is attached to the 0-cells corresponding to the vertex

p(f) and the leg f itself.The edges of a graph define an equivalence relation on its vertices V();

the components of the graph are the equivalence classes with respect to thisrelation. Denote the set of components by 0(). The Euler characteristic ofa component C of a graph is e(C) = |V(C)| |E(C)|. Denote by n(C) thenumber of legs of a component C.

Dual graphs

A dual graph is a graph together with a function g : V() N. The naturalnumber g(v) is called the genus of the vertex v.

The genus g(C) of a component C of a dual graph is defined by theformula

g(C) =

vC

g(v) + 1 e(C).

The genus of a component is a non-negative integer, which equals 0 if andonly if C is a tree and g(v) = 0 for all vertices v of C.


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A stable graph is a dual graph such that for all vertices v V(), theinteger 2g(v) 2 + n(v) is positive. In other words, if g(v) = 0, then n(v) isat least 3, while if g(v) = 1, then n(v) is nonzero. If is a stable graph, then2g(C) 2 + |L(C)| > 0 for all components C of .

The symmetric monoidal category G

The objects of the category G are the dual graphs whose set of edges E()is empty. Equivalently, an object of G is a pair of finite sets L and V andfunctions p : L V, g : V N.

A morphism ofG with source (L1, V1, p1, g1) and target (L2, V2, p2, g2) is apair consisting of a dual graph with F() = L1, V() = V1, and g() = g1,together with isomorphisms : L2 L() and : V2 0() such that

p = p2 and g = g2.The definition of the composition of two morphisms = 2 1 in G is

straightforward. We have dual graphs 1 and 2, with

F(1) = L1, V(1) = V1, g(1) = g1,

F(2) = L2, V(2) = V2, g(2) = g2,

together with isomorphisms

1 : L2 L(1), 1 : V2 0(1),

2 : L3 L(2), 2 : V3 0(2).

Since the source of = 2 1 equals the source of 1, we see thatp : F() V() is identified with p : L1 V1. The involution of L1 isdefined as follows: if f F() = L1 lies in an edge of 1, then (f) = 1(f),while if f is a leg of 1, then (f) = 1(2(

11 (f))). The isomorphisms

and of are simply the isomorphisms 2 and 2 of 2.In other words, 2 1 is obtained from 1 by gluing those pairs of legs of

1 together which correspond to edges of 2. It is clear that composition inG is associative.

The tensor product on objects of G extends to morphisms, making G intoa symmetric monoidal category.

A morphism in G is invertible if the underlying stable graph has no edges,in other words, if it is simply an isomorphism between two objects of G.Denote by H the groupoid consisting of all invertible morphisms of G, andby : H G the inclusion. The groupoid H is the free symmetric monoidalfunctor generated by the groupoid h consisting of all morphisms of G withconnected domain.

The category G has a small skeleton; for example, the full subcategory ofG in which the sets of flags and vertices of the objects are subsets of the set ofnatural numbers. The tensor product ofG takes us outside this category, but

it is not hard to define an equivalent tensor product for this small skeleton.We will tacitly replace G by this skeleton, since some of our constructions willrequire that G be small.


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6 Ezra Getzler

Modular operads as symmetric monoidal functors

The following definition of modular operads may be found in Costello [4]. LetC be a symmetric monoidal category which is cocomplete, and such that the

functor A B preserves colimits in each variable. (This last condition is auto-matic ifC is a closed symmetric monoidal category.) A modular preoperadin C is a symmetric monoidal functor from H to C, and a modular operadin C is a symmetric monoidal functor from G to C.

Let A be a small symmetric monoidal category, and let B be symmetricmonoidal category. Denote by [A, B] the category of functors and naturalequivalences from A to B, and by A, B the category of symmetric monoidalfunctors and monoidal natural equivalences.

With this notation, the categories of modular operads, respectively pre-operads, in a symmetric monoidal category C are Mod(C) = G, C andPreMod(C) = H, C.Theorem. i) There is an adjunction

: PreMod(C)Mod(C) : ,

where is restriction along : H G , and is the coend

P =AH

G(A, ) P(A).

ii) The functor : Mod(C) PreMod(C)

is monadic. That is, there is an equivalence of categories

Mod(C) PreMod(C)T,

where T is the monad .

iii) The category Mod(C) is locally finitely presentable if C is locally finitelypresentable.

In the original definition of modular operads (Getzler and Kapranov [11]),there was an additional stability condition, which may be phrased in the fol-lowing terms. Denote by G+ the subcategory ofG consisting of stable graphs,and let H+ = H G+. Then a stable modular preoperad in C is a symmet-ric monoidal functor from H+ to C, and a stable modular operad in C is asymmetric monoidal functor from G+ to C. These categories of stable modularpreoperads and operads are equivalent to the categories of stable S-modulesand modular operads of loc. cit.

Algebras for modular operads

To any object M of a closed symmetric monoidal category C is associated themonoid End(M) = [M, M]: an A-module is a morphism of monoids : A


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End(M). The analogue of this construction for modular operads is called aP-algebra.

A bilinear form with domain M in a symmetric monoidal category C is amorphism t : M M such that t = t. Associated to a bilinear form

(M, t) is a modular operad End(M, t), defined on a connected object of Gto be

End(M, t)() = ML().

IfP is a modular operad, a P-algebra M is an object M of C, a bilinear formt : M M , and a morphism of modular operads : P End(M, t).

This modular operad has an underlying stable modular operad End+(M, t),defined by restriction to the stable graphs in G.

Cyclic operads

A variant of the above definition of modular operads is obtained by taking thesubcategory of forests G0 in the category G: a forest is a graph each component

of which is simply connected. Symmetric monoidal functors on G0 are cyclicoperads.Denote the cyclic operad underlying End(M, t) by End0(M, t): thus

End0(M, t)() = ML().

If P is a cyclic operad, a P-algebra M is bilinear form t with domain M anda morphism of cyclic operads : P End0(M, t).

There is a functor P P0, which associates to a modular operad itsunderlying cyclic operad: this is the restriction functor from G, C to G0, C.

There is also a stable variant of cyclic operads, in which G0 is replacedby its stable subcategory G0+, defined by restricting to forests in which eachvertex meets at least three flags.

Dioperads and modular dioperads

Another variant of the definition of modular operads is obtained by replac-ing the graphs in the definition of modular operads by digraphs (directedgraphs):

A digraph is a graph together with a partition

F() = F+() F()

of the flags into outgoing and incoming flags, such that each edge has oneoutgoing and one incoming flag. Each edge of a digraph has an orientation,running towards the outgoing flag. The set of legs of a digraph are partitioned

into the outgoing and incoming legs: L() = L() F(). Denote thenumber of outgoing and incoming legs by n().


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8 Ezra Getzler

A dual digraph is a digraph together with a function g : V() N.Imitating the construction of the symmetric monoidal category of dual graphsG, we may construct a symmetric monoidal category of dual digraphs D. Amodular dioperad is a symmetric monoidal functor on D. (These are the

wheeled props studied in a recent preprint of Merkulov [20].)IfM are objects ofC and t : M+ M is a pairing, we may construct

a modular dioperad End(M, t), defined on a connected object of D tobe

End(M)() = M

L+()+ M

L() . (1)

If P is a modular dioperad, a P-algebra is a pairing t : M+ M and amorphism of modular dioperads : P End(M, t).

A directed forest is a directed graph each component of which is simplyconnected; let D0 be the subcategory of D of consisting of directed forests. Adioperad is a symmetric monoidal functor on D0 (Gan [9]).

IfM is an object of C, we may construct a dioperad End0 (M), defined ona connected object of D0 to be

End0 (M)() = Hom

ML+(), ML()

.

When M = M is a rigid object with dual M = M+, this is a special case

of (1). IfP is a dioperad, a P-algebra M is an object M of C and a morphismof dioperads : P End0 (M).

props

MacLanes notion [17] of a prop also fits into the above framework. There is asubcategory DP of D, consisting of all dual digraphs such that each vertexhas genus 0, and has no directed circuits. A prop is a symmetric monoidalfunctor on DP. (This follows from the description of free props in Enriquez

and Etingof [7].) Note that D0 is a subcategory ofDP: thus, every prop has anunderlying dioperad. Note also that the dioperad End0 (M) is in fact a prop;if P is a prop, we may define a P-algebra M to be an object M of C and amorphism of props : P End0 (M).

2 Patterns

Patterns abstract the approach to modular operads sketched in Section 1. Thissection develops the theory of patterns enriched over a complete, cocomplete,closed symmetric monoidal category V. In fact, we are mainly interested inthe cases where V is the category Set of sets or the category sSet of simplicialsets. We refer to the appendices for a review of the needed enriched category

theory.


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10 Ezra Getzler

V1 Vn =A1,...,AnAV1(A1) Vn(An) y(A1 An).

The Yoneda functor y : A A is a symmetric monoidal V-functor. (See Day[5] and Im and Kelly [13].)

If :S

T

is a symmetric monoidal V-functor between small symmetricmonoidal V-categories, the pull-back functor F is a lax symmetric monoidalV-functor; we saw this in the unenriched case in the introduction, and theproof in the enriched case is similar.

Definition 2.2. A V-pattern is a symmetric monoidal V-functor : S Tbetween small symmetric monoidal V-categories S and T such that

: T S

is a symmetric monoidal V-functor.

Let C be a symmetric monoidal V-category which is cocomplete, andsuch that the V-functors A B preserves colimits in each variable. The V-

categories of-preoperads and -operads in C are respectively the V-categoriesPreOp(C) = S, C and Op(C) = T, C of symmetric monoidal V-functorsfrom S and T to C.

The monadicity theorem

We now construct a V-functor , which generalizes the functor taking a pre-operad to the free operad that it generates.

Proposition 2.3. Let be a V-pattern. The V-adjunction

: [S, C] [T, C] :

induces a V-adjunction between the categories of -preoperads and -operads

: PreOp(C) Op(C) : .

Proof. Let G be a symmetric monoidal V-functor from S to C. The left KanV-extension G is the V-coend

G(B) =AS

T(A,B) G(A).

For each n, there is a natural V-equivalence


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Operads revisited 11

G(B1) G(Bn) =nk=1

AkST( Ak, Bk) G(Ak)

=A1,...,AnS

k T( Ak, Bk) nk=1 G(Ak)

since preserves V-coends

=A1,...,An

Sk T( Ak, Bk) G

nk=1 Ak

since G is symmetric monoidal

=A1,...,AnS

k T( Ak, Bk) AS

S

A,nk=1 Ak

G(A)

by the Yoneda lemma

=ASA1,...,AnSn

k=1 T(Ak, Bk) S

A,nk=1 Ak

G(A)

by Fubinis theorem for V-coends

=AS

T

A,nk=1 Bk

G(A)

since is symmetric monoidal

= G n

k=1

Bk

.

This natural V-equivalence makes G into a symmetric monoidal V-functor.The unit and counit of the V-adjunction between and on PreOp(C)

and Op(C) are now induced by the unit and counit of the V-adjunctionbetween and

on [S, C] and [T, C].

For the unenriched version of the following result on reflexive V-coequalizers,see, for example, Johnstone [14, Corollary 1.2.12]; the proof in the enrichedcase is identical.

Proposition 2.4. Let C be a symmetric monoidal V-category with reflexiveV-coequalizers. If the tensor product A preserves reflexive V-coequalizers,then so does the functorS ().

Corollary 2.5. The V-categories PreOp(C) and Op(C) have reflexive co-equalizers.

Proof. The V-coequalizer R in [S, C] of a reflexive parallel pair Poof

//

g// Q in

S, C is computed pointwise: for each X Ob(S),

P(X) oof(X)

//

g(X)// Q(X) R(X)

is a reflexive V-coequalizer in C. By Proposition 2.4, R is a symmetric

monoidal V-functor; thus, R is the V-coequalizer of the reflexive pair Poof

//

g//Q

in S, C. The same argument works for T, C.


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12 Ezra Getzler

Proposition 2.6. If is an essentially surjective V-pattern, the V-functor

: Op(C) PreOp(C)

creates reflexive V-coequalizers.

Proof. The proof of Corollary 2.5 shows that the horizontal V-functors in thediagram

S, C [S, C]//

T, C

S, C

T, C [T, C]// [T, C]

[S, C]

create, and hence preserve, reflexive V-coequalizers. The V-functor

: [T, C] [S, C]

creates all V-colimits, since V-colimits are computed pointwise and is es-

sentially surjective. It follows that the V-functor : T, C S, C createsreflexive V-coequalizers.

Recall that a V-functor R : A B, with left adjoint L : B A, is V-monadic if there is an equivalence of V-categories A BT, where T is theV-monad associated to the V-adjunction

L : A B : R.

The following is a variant of Theorem II.2.1 of Dubuc [6]; reflexive V-coequalizers are substituted for contractible V-coequalizers, but otherwise,the proof is the same.

Proposition 2.7. A V-functor R : A B, with left adjoint L : B A, is

V-monadic if B has, and R creates, reflexive V-coequalizers.

Corollary 2.8. If is an essentially surjective V-pattern, then the V-functor

: Op(C) PreOp(C)

is V-monadic.

In practice, the V-patterns of interest all have the following property.

Definition 2.9. A V-pattern : S T is regular if it is essentially surjectiveand S is equivalent to a free symmetric monoidal V-category.

Denote by s a V-category such that S is equivalent to the free symmetricmonoidal V-category Ss. The V-category s may be thought of as a generalized

set of colours; the theory associated to a coloured operad is a regular patternwith discrete s.


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Theorem 2.10. If is a regular pattern and C is locally finitely presentable,then Op(C) is locally finitely presentable.

Proof. The V-category PreOp(C) is equivalent to the V-category [s, C], andhence is locally finitely presentable. By Lemma 3.8, Op

(C) is cocomplete.

The functor takes finitely presentable objects of C to finitely presentableobjects of Op(C), and hence takes finitely presentable strong generators ofPreOp(C) to finitely presentable strong generators of Op(C).

3 The simplicial patterns Cob and Cob

In this section, we construct simplicial patterns Cob = Cob[d], associatedwith gluing of oriented d-dimensional manifolds along components of theirboundary.

Definition 3.1. An object S of Cob is a compact d-dimensional manifold,with boundary S and orientation o.

In the above definition, we permit the manifold S to be disconnected. Inparticular, it may be empty. (Note that an empty manifold of dimension d hasa unique orientation.) However, the above definition should be refined in orderto produce a set of objects of Cob: one way to do this is to add to the datadefining an object ofCob an embedding into a Euclidean space RN, togetherwith a collared neigbourhood of the boundary S. We call this a decoratedobject.

We now define a simplicial set Cob(S, T) of morphisms between ob jects Sand T ofCob.

Definition 3.2. A hypersurface in an object S ofCob consists of a closed(d 1)-dimensional manifold M together with an embedding : M S.

Given a hypersurface in S, let S[] be the manifold obtained by cuttingS along the image of . The orientation o if S induces an orientation o[] ofS[].

A hypersurface in a decorated object is an open embedding of the manifoldM [1, 1] into the complement in S of the collared neighbourhoods of theboundary. This induces a collaring on the boundary of S[]. To embed S[]into RN+1, we take the product of the embedding of S into RN and thefunction 21, where 2 1 is the function on S equal to the coordinatet [1, 1] on the image of and undefined elsewhere, and (t) = sgn(t)(t).Here, Cc (1, 1) is a non-negative smooth function of compact supportequal to 1 in a neighbourhood of 0 (1, 1).

A k-simplex in the simplicial set Cob(S, T) of morphisms from S to Tconsists of the following data:

i) a closed (d 1)-manifold M;


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14 Ezra Getzler

ii) an isotopy of hypersurfaces, that is a commutative diagram

k M

k%%LLL

LLLL

k M k T

// k T

k yyrrrr

rrr

in which is an embedding;iii) a fibred diffeomorphism

k S

k%%L

LLLL

LLk S k T[]

// k T[]

kyyrrr

rrrr

compatible with the orientations on its domain and target.

It is straightforward to extend this definition to decorated objects: the onlydifference is that is a fibred open embedding of kM[1, 1] into kT.

We now makeCob

into a symmetric monoidal simplicial category. Thecomposition of k-simplices (M , , ) Cob(S, T)k and (N, , ) Cob(T, U)kis the k-simplex consisting of the embedding

k (M N)

k%%LL

LLLLL

k (M N) k U ()

// k U

kyyrrrr

rrr

and the fibred diffeomorphism

k S

k%%L

LLLL

LLk S k U[( ) ]

// k U[( ) ]

kyyrrr

rrrr

In the special case that the diffeomorphisms and are the identity, thiscomposition is obtained by taking the union of the disjoint hypersurfacesk M and k N in k U. It is clear that composition is associative,and compatible with the face and degeneracy maps between simplices.

The identity 0-simplex 1S in Cob(S, S) is associated to the empty hyper-surface in S and the identity diffeomorphism of S.

The tensor product ofCob is simple to describe: it is disjoint union. WhenSi are decorated objects of Cob, 1 i k, embedded in RNi, we embedS1 . . . Sk in Rmax(Ni)+1 by composing the embedding of Si with theinclusion RNi Rmax(Ni)+1 defined by

(t1, . . . , tNi) (t1, . . . , tNi , 0, . . . , 0, i).

Just as modular operads have a directed version, modular dioperads, thesimplicial pattern Cob[d] has a directed analogue.


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Definition 3.3. An object S ofCob is a compact d-dimensional manifold,with orientations o and o of S and its boundary S.

The boundary S of an object S of Cob is partitioned into incomingand outgoing parts S = S +S, according to whether the boundary is

positive or negatively oriented by o with respect to the orientation o of S.

Definition 3.4. A hypersurface in an object S ofCob consists of a closed(d 1)-dimensional oriented manifold M together with an embedding :M S.

Given a hypersurface in S, let S[] be the manifold obtained by cuttingS along the image of . The orientation o if S induces an orientation o[] ofS[], while the orientations ofS and M induce an orientation of its boundaryS[].

A k-simplex in the simplicial set Cob(S, T) of morphisms from S to Tconsists of the following data:

i) a closed oriented (d 1)-manifold M;

ii) an isotopy of hypersurfaces, that is a commutative diagram

k M

k%%L

LLLL

LLk M k T

// k T

kyyrrrr

rrr

in which is an embedding;iii) a fibred diffeomorphism

k S

k%%LL

LLLL

Lk S k T[]

// k T[]

kyyrrr

rrrr

compatible with the orientations on its domain and target.

We may make Cob into a symmetric monoidal simplicial category inthe same way as for Cob. The definition of decorated objects and decoratedmorphisms is also easily extended to this setting.

Let cob be the simplicial groupoid of connected oriented surfaces and theiroriented diffeomorphisms, with skeleton

g,n

Diff+(Sg,n).

The embedding cob Cob extends to an essentially surjective V-functorS cob Cob. Similarly, if cob is the simplicial groupoid of connected

oriented surfaces with oriented boundary and their oriented diffeomorphisms,the V-functor S cob Cob is essentially surjective.We can now state the main theorem of this section.


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16 Ezra Getzler

Theorem 3.5. The simplicial functors cob Cob andcob Cob induceregular simplicial patterns S cob Cob andS cob Cob

A d-dimensional modular operad, respectively dioperad, is an operad forthe simplicial pattern S cob Cob, respectively S cob Cob.

One-dimensional modular operads

When d = 1, the category cob has a skeleton with two objects, the intervalI and the circle S. The simplicial group cob(I, I) = Diff+[0, 1] is contractibleand the simplicial group cob(S, S) is homotopy equivalent to SO(2).

The simplicial set Cob(Ik, I) is empty for k = 0 and contractible for eachk > 0, and the simplicial set Cob(Ik, S) is empty for k = 0 and homotopyequivalent to SO(2) for each k > 0. Thus, a 1-dimensional modular operad ina symmetric monoidal category C consists of a homotopy associative algebraA, and a homotopy trace from A to a homotopy SO(2)-module M. If C isdiscrete, a 1-dimensional modular operad is simply a non-unital associative

algebra A in C together with an object M in C and a trace tr : A M.

One-dimensional modular dioperads

When d = 1, the category cob has a skeleton with five objects, the in-tervals I , I

+ , I

+ and I

++ , representing the 1-manifold [0, 1] with the four

different orientations of its boundary, and the circle S. The simplicial groupscob

(Ia, Ia) are contractible, and the simplicial group cob(S, S) is homo-

topy equivalent to SO(2).A 1-dimensional modular dioperad P in a discrete symmetric monoidal

category C consists of associative algebras A = P(I+) and B = P(I+ ) in C,

a (A, B)-bimodule Q = P(I ), a (B, A)-bimodule R = P(I++ ), and an object

M = P(S), together with morphisms

: Q B R A, : R A Q B, trA : A M, trB : B M.

Denote the left and right actions of A and B on Q and R by Q : A Q Q,Q : Q B Q, R : B R R and R : R A R. The above data mustin addition satisfy the following conditions:

and are morphisms of (A, A)-bimodules and (B, B)-bimodules respec-tively;

trA and trB are traces; Q ( Q) = Q (R ) : Q B R A Q Q and R ( R) =

R (Q ) : R A Q B R R; trA : Q B R M and trB : R A Q M are equal on the

isomorphic objects (Q B R) AA and (R A Q) BB .

That is, a 1-dimensional modular dioperad is the same thing as a Moritacontext (Morita [22]) (A,B,P,Q,,) with trace (trA, trB).


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Two-dimensional topological field theories

Let G be the discrete pattern introduced in Section 1 whose operads are mod-ular operads. There is a natural morphism of patterns

: Cob[2] G ;thus, application of to a modular operad gives rise to a 2-dimensional mod-ular operad. But modular operads and what we call 2-dimensional modularoperads are quite different: a modular preoperad P consists of a sequenceof Sn-modules P((g, n)), while a 2-dimensional modular preoperad is a cob-module.

An example of a 2-dimensional modular operad is the terminal one, forwhich P(S) is the unit for each surface S. A more interesting one comesfrom conformal field theory: the underlying 2-dimensional modular operadassociates to an oriented surface with boundary S the moduli space N(S)of conformal structures on S. (More accurately, N(S) is the simplicial setwhose n-simplices are the n-parameter smooth families of conformal struc-

tures parametrized by the n-simplex.) The space N(S) is contractible for allS, since it is the space of smooth sections of a fibre bundle over M with con-tractible fibres. To define the structure of a 2-dimensional modular operad onN amounts to showing that conformal structures may be glued along circles:this is done by choosing a Riemannian metric in the conformal class which isflat in a neighbourhood of the boundary, such that the boundary is geodesic,and each of its components has length 1.

A bilinear form t : M M on a cochain complex where M is non-degenerate if it induces a quasi-isomorphism between M and Hom(M, ).

Definition 3.6. Let P be a 2-dimensional modular operad. A P-algebra is amorphism of 2-dimensional modular operads

: P End(M, t),

where M is a cochain complex with non-degenerate bilinear form t : MM .

A topological conformal field theory is a C(N)-algebra.

In the theory of infinite loop spaces, one defines an E-algebra as analgebra for an operad E such that E(n) is contractible for all n. Similarly, asshown by Fiedorowicz [8], an E2-algebra is an algebra for a braided operadE such that E(n) is contractible for all n. Motivated by this, we make thefollowing definition.

Definition 3.7. A 2-dimensional topological field theory is a pair consistingof a 2-dimensional modular operad E in the category of cochain complexessuch that E(S) is quasi-isomorphic to for all surfaces S, and an E-algebra(M, t).

In particular, a topological conformal field theory is a 2-dimensional topo-logical field theory.


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18 Ezra Getzler

Appendix. Enriched categories

In this appendix, we recall some results of enriched category theory. Let Vbe a closed symmetric monoidal category; that is, V is a symmetric monoidal

category such that the tensor product functor Y from V to itself has aright adjoint for all objects Y, denoted [Y, ]: in other words,

[X Y, Z] = [X, [Y, Z]].

Throughout this paper, we assume that V is complete and cocomplete. Denoteby A A0 the continuous functor A0 = V( , A) from V to Set, where isthe unit of V.

Let V-Cat be the 2-category whose objects are V-categories, whose 1-mor-phisms are V-functors, and whose 2-morphisms are V-natural transformations.Since V is closed, V is itself a V-category.

Applying the functor ()0 : V Set to a V-category C, we obtain itsunderlying category C0; in this way, we obtain a 2-functor

()0 : V-Cat Cat.

Given a small V-category A, and a V-category B, there is a V-category[A, B], whose objects are the V-functors from A to B, and such that

[A, B](F, G) =AA

B(FA,GA).

There is an equivalence of categories V-Cat(A, B) [A, B]0.

The Yoneda embedding for V-categories

The opposite of a V-category C is the V-category C with

C(A, B) = C(B, A).

If A is a small V-category, denote by A the V-category of presheaves

A = [A, V].

By the V-Yoneda lemma, there is a full, faithful V-functor y : A A, with

y(A) = A(, A), A Ob(A).

If F : A B is a V-functor, denote by F : B A the V-functor inducedby F.


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Cocomplete V-categories

A V-category C is tensored if there is a V-functor : V C C togetherwith a V-natural equivalence of functors C(X A, B) = [X, C(A, B)] from

V

C

C to V. For example, the V-category V is itself tensored.Let A be a small V-category, let F be a V-functor from A to V, and letG be a V-functor from A to a V-category C. If B is an object of C, denote byG, B : C V the presheaf such that G, C(A) = [GA, C]. The weighted

colimit ofG, with weight F, is an objectAA

F A GA ofC such that thereis a natural isomorphism

C (F, G, ) = CAA

F A GA,

.

If C is tensored,AA

F(A) G(A) is the coequalizer of the diagram

A0,A1Ob(A)

A(A0, A1) F A1 GA0////

AOb(A)

F A GA,

where the two morphisms are induced by the action on F and coaction on Grespectively.

A V-category C is cocomplete if it has all weighted colimits, or equiva-lently, if it satisfies the following conditions:

i) the category C0 is cocomplete;ii) for each object A of C, the functor C(, A) : C0 V transforms colimits

into limits;iii) C is tensored.

In particular, V-categories A of presheaves are cocomplete.Let C be a cocomplete V-category. A V-functor F : A B between small

V-categories gives rise to a V-adjunction

F : [A, C] [B, C] : F,

that is, an adjunction in the 2-category V-Cat. The functor F is called the(pointwise) left V-Kan extension of along F; it is the V-coend

FG() =AA

A(F A, ) GA.

Cocomplete categories of V-algebras

A V-monad T on a V-category C is a monad in the full sub-2-category ofV-Catwith unique object C. If C is small, this is the same thing as a monoid in themonoidal category V-Cat(C, C).

The following is the enriched version of a result of Linton [16], and isproved in exactly the same way.


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20 Ezra Getzler

Lemma 3.8. LetT be a V-monad on a cocomplete V-categoryC such that theV-category of algebras CT has reflexive V-coequalizers. Then the V-category ofT-algebras CT is cocomplete.

Proof. We must show that CT has all weighted colimits. Let A be a smallV-category, let F be a weight, and let G : A CT be a diagram ofT-algebras.

Then the weighted colimitAA

F(A) G(A) is a reflexive coequalizer

TAA

F A TRGA

oo//

// TAA

F A RGA

.

Locally finitely presentable V-categories

In studying algebraic theories using categories, locally finite presentable cat-egories plays a basic role. When the closed symmetric monoidal categery Vis locally finitely presentable, with finitely presentable unit , these have ageneralization to enriched category theory over V, due to Kelly [15]. (There isa more general theory of locally presentable categories, where the cardinal 0

is replaced by an arbitrary regular cardinal; this extension is straightforward,but we do not present it here in order to simplify exposition.)

Examples of locally finitely presentable closed symmetric monoidal cate-gories include the categories of sets, groupoids, categories, simplicial sets, andabelian groups the finitely presentable objects are respectively finite sets,groupoids and categories, simplicial sets with a finite number of nondegener-ate simplices, and finitely presentable abelian groups. A less obvious exampleis the category of symmetric spectra of Hovey et al. [12].

An object A in a V-category C is finitely presentable if the functor

C(A, ) : C V

preserves filtered colimits. A strong generator in a V-category C is a set{Gi Ob(C)}iI such that the functor

A

iI

C(Gi, A) : C VI

reflects isomorphisms.

Definition 3.9. A locally finitely presentable V-category C is a cocom-plete V-category with a finitely presentable strong generator (i.e. the objectsmaking up the strong generator are finitely presentable).

Pseudo algebras

A 2-monad T on a 2-category C is by definition a Cat-monad on C. Denote

the composition of the 2-monadT

by m :TT

T

, and the unit by : 1 T

.Associated to a 2-monad T is the 2-category CT of pseudo T-algebras(Bunge [3] and Street [23]; see also Marmolejo [19]). A pseudo T-algebra is an


21/25


object A ofC, together with a morphism a : TA A, the composition, andinvertible 2-morphisms

TTA

TATa

77oooooo

TTA

TAmA ''OO

OOOOO

TA

A

mA

''OOOOOOOO

TA

A

a

77oooooooo

and

A

TAA 77oooooooo

A A

TA

A

a

''OOOOO

OOO

such that

TTTA

TTA

TTa

GGTTTA

TTA

mTA

//

/////

//

TTA

TA

TmA

////

////

TTA

TA

Ta

GG

TTA TATa //

TA Aa //

TTA TAmA

//

TA

A

a

//

////

///

TA

A

a

GG

equals TTTA

TTA

TTa

GGTTTA

TTA

mTA

//

/////

//TTTA TTA

TmA

//

TTA TATa //

TTA

TA

mA

GG

TTA

TA

Ta

////

////

TTA TAmA

//

TA

A

a

//

////

///

TA

A

a

GG

T

and

TA

TTATA

77oooooooTA

TTATA ''

OOOOOO

TA TA

TTA

TA

Ta

''OOOOO

OO

TTA

TAmA

77oooooo

TA Aa //

T

equals

TA TTATA

// TTA

TATa

77ooooooTTA

TAmA ''OO

OOOOO

TA

A

mA

''OOOOO

OOO

TA

A

a

77oooooooo

The lax morphisms of CT are pairs (f, ) consisting of a morphism f :A B and a 2-morphism

TA

TBTf 77ooooooo

TA

Aa ''OOOOO

OOO

TB

B

b

''OOOOO

OOO

A

Bf

77ooooooooo


22/25

22 Ezra Getzler

such that

TTA

TTB

TTf

GG

TTA

TA

mA

//

////

///

TTB

TB

mB

////

////

TA

TB

Tf

GG

TTB TBTb //

TB Bb //

TA Aa

//

TB

B

b

//

/////

//

A

B

f

GG

equals TTA

TTB

Tf

GG

TTA

TA

mA

//

////

///TTA TATa

//

TTB TBTb //

TA

TB

Tf

GG

TA

A

a

////

//

//

TA Aa

//

TB

B

b

//

/////

//

A

B

f

GG

T

and

A

TAA

77oooooooo

TA

TB

Tf

''OOOOO

OO

A

Bf ''OO

OOOOOOO

B

TBBooo

77ooo

B B

TB

B

b''OO

OOOOOO

equals

A

TAA

77ooooooooA A

TA

TBTf 77ooooooo

TA

Aa

OOOO

''OOOO

TB

B

b

''OOOOO

OOO

A

B

f

77ooooooooo

A lax morphism (f, ) is a morphism if the 2-morphism is invertible.

The 2-morphisms : (f, ) (

f, ) ofCT

are 2-morphisms : f

fsuch that

TA

TBTf 77ooooooo

TA

Aa ''OOOOO

OOO

TB

B

b

''OOOOO

OOO

A

Bf ..

A

B

f

DD

equals TA

TBTf //

TA

TB

Tf

DD

TA

Aa ''OOOOO

OOO

TB

B

a

''OOOOO

OOO

A

B

f

77ooooooooo

T

(2)

V-categories of pseudo algebras

IfA and B are pseudo T-algebras, and the underlying category ofA is small, wesaw that there is a V-category [A, B] whose objects are V-functors f : A Bbetween the underlying V-categories. Using [A, B], we now define a V-categoryA, B whose objects are morphisms f : A B of pseudo T-algebras. If f0


23/25


and f1 are morphisms of pseudo T-algebras, A, B(f0, f1) is defined as theequalizer

A, B

(f0, f1)

f0f1// [A, B](f0, f1)

1(bT)//

(a)0

// [TA, B](b Tf0, f1 a)

Of course, this is the internal version of (2): the 2-morphisms : f0 f1 ofCT are the elements of the set |A, B(f0, f1)|.

The composition morphism

A, B(f0, f1) A, B(f1, f2)m

A,Bf0f1f2

// A, B(f0, f2)is the universal arrow for the coequalizer A, B(f0, f2), whose existence isguaranteed by the commutativity of the diagram

A, B

(f0, f1)

A, B

(f1, f2)

f0f1f1f2// [A, B](f0, f1) [A, B](f1, f2)

m[A,B]f0f1f1 // [A, B](f0, f2)

2(bT)//

(a)0

// [TA, B](b Tf0, f2 c)

Indeed, we have

2(b T) m[A,B]f0f1f1

(f0f1 f1f2)

= m[TA,B]bTf0,bTf1,f2a

((b T) f0f1 2(b T) f1f2)

= m[TA,B]bTf0,bTf1,f2a

((b T) f0f1 ( a)1 f1f2)

= m[TA,B]bTf0,f1a,f2a

(1(b T) f0f1 ( a) f1f2)

= m[TA,B]

bTf0,f1a,f2a (( a)

0 f

0f1

( a) f1f2

)

= ( a)0 m[A,B]f0f1f1

(f0f1 f1f2).

Associativity of this composition is proved by a calculation along the samelines for the iterated composition map

A, B(f0, f1) A, B(f1, f2) A, B(f2, f3) A, B(f0, f3).The unit vf : A, B(f, f) of the V-category A, B is the universalarrow for the coequalizer A, B(f, f), whose existence is guaranteed by thecommutativity of the diagram

uf// [A, B](f, f)

(bT)//

(a)

// [TA, B](b Tf, f a)

Indeed, both (b T) uf and ( a) uf are equal to


24/25

24 Ezra Getzler

V( , [TA, B](b Tf, f a)).

In a remark at the end of Section 3 of [13], Im and Kelly dismiss theexistence of the V-categories of pseudo algebras. The construction presented

here appears to get around their objections.

References

[1] Clemens Berger and Ieke Moerdijk. Resolution of coloured operads andrectification of homotopy algebras. arXiv:math.AT/0512576.

[2] J. M. Boardman and R. M. Vogt. Homotopy invariant algebraic structureson topological spaces. Springer-Verlag, Berlin, 1973.

[3] Marta C. Bunge. Coherent extensions and relational algebras. Trans.Amer. Math. Soc., 197:355390, 1974.

[4] Kevin J. Costello. Topological conformal field theories and Calabi-Yaucategories. arXiv:math.QA/0412149.

[5] Brian Day. On closed categories of functors. In Reports of the MidwestCategory Seminar, IV, Lecture Notes in Mathematics, Vol. 137, pages138. Springer, Berlin, 1970.

[6] Eduardo J. Dubuc. Kan extensions in enriched category theory. LectureNotes in Mathematics, Vol. 145. Springer-Verlag, Berlin, 1970.

[7] Benjamin Enriquez and Pavel Etingof. On the invertibility of quantiza-tion functors. J. Algebra, 289(2):321345, 2005.

[8] Zbigniew Fiedorowicz. The symmetric bar construction.http://www.math.ohio-state.edu/fiedorow/symbar.ps.gz.

[9] Wee Liang Gan. Koszul duality for dioperads. Math. Res. Lett., 10(1):109124, 2003.

[10] E. Getzler. Batalin-Vilkovisky algebras and two-dimensional topologicalfield theories. Comm. Math. Phys., 159(2):265285, 1994.

[11] E. Getzler and M. M. Kapranov. Modular operads. Compositio Math.,110(1):65126, 1998.

[12] Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra. J.Amer. Math. Soc., 13(1):149208, 2000.

[13] Geun Bin Im and G. M. Kelly. A universal property of the convolutionmonoidal structure. J. Pure Appl. Algebra, 43(1):7588, 1986.

[14] Peter T. Johnstone. Sketches of an elephant: a topos theory compendium.Vol. 1, volume 43 of Oxford Logic Guides. The Clarendon Press OxfordUniversity Press, New York, 2002.

[15] G. M. Kelly. Structures defined by finite limits in the enriched context.I. Cahiers Topologie Geom. Differentielle, 23(1):342, 1982. Third Col-loquium on Categories, Part VI (Amiens, 1980).

[16] F. E. J. Linton. Coequalizers in categories of algebras. In Sem. on Triplesand Categorical Homology Theory (ETH, Zurich, 1966/67), pages 7590.Lecture Notes in Math., Vol. 80. Springer, Berlin, 1969.
http://www.math.ohio-state.edu/~fiedorow/symbar.ps.gzhttp://www.math.ohio-state.edu/~fiedorow/symbar.ps.gzhttp://www.math.ohio-state.edu/~fiedorow/symbar.ps.gzhttp://www.math.ohio-state.edu/~fiedorow/symbar.ps.gz


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[17] Saunders Mac Lane. Categorical algebra. Bull. Amer. Math. Soc., 71:40106, 1965.

[18] Martin Markl, Steve Shnider, and Jim Stasheff. Operads in algebra, topol-ogy and physics, volume 96 of Mathematical Surveys and Monographs.

American Mathematical Society, Providence, RI, 2002.[19] F. Marmolejo. Distributive laws for pseudomonads. Theory Appl. Categ.,

5:No. 5, 91147 (electronic), 1999.[20] Sergei Merkulov. PROP profile of deformation quantization and graph

complexes with loops and wheels. arXiv:math.QA/0412257.[21] Gregory Moore and Nathan Seiberg. Classical and quantum conformal

field theory. Comm. Math. Phys., 123(2):177254, 1989.[22] Kiiti Morita. Duality for modules and its applications to the theory of

rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A,6:83142, 1958.

[23] Ross Street. Fibrations and Yonedas lemma in a 2-category. In CategorySeminar (Proc. Sem., Sydney, 1972/1973), pages 104133. Lecture Notesin Math., Vol. 420. Springer, Berlin, 1974.

ezra getzler- operads revisited

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