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Direct link to author’s versionGLOBAL FUKAYA CATEGORY I: QUANTUM NOVIKOV

CONJECTURE

YASHA SAVELYEV

Abstract. Let Ham(M,ω) denote the Frechet Lie group of Hamiltonian sym-plectomorphisms of a monotone symplectic manifold (M,ω). Let NFuk(M) bethe A∞-nerve of the Fukaya category Fuk(M), and let (S, NFuk(M)) denotethe NFuk(M) component of the “space” of ∞-categories. Using Floer-Fukayatheory for a monotone (M,ω) we construct a natural continuous map

BHam(M,ω) → (S, NFuk(M)).

This verifies one sense of a conjecture of Teleman on existence of action ofHam(M,ω) on the Fukaya category of (M,ω). This construction is very closelyrelated to the theory of the Seidel homomorphism and the quantum Chernclasses of the author, and this map is intended to be the deepest expressionof their underlying geometric theory. In part II the above map is shown to benon trivial by an explicit calculation. In particular we arrive at a new non-trivial “quantum” invariant of any smooth manifold and a “quantum” Novikovconjecture.

Contents

1. Introduction 21.1. A functor from the category of smooth simplices of X 21.2. Towards new invariants and quantum Novikov conjecture 31.3. Hochschild and geometric Hochschild cohomology and homotopy

groups of Ham(M,ω) 41.4. Acknowledgements 52. Notations and conventions and large categories 53. Preliminaries 53.1. The simplex category of a smooth manifold X 53.2. Preliminaries on Riemann surfaces 64. A system of natural maps from the universal curve to ∆n 94.1. Outline of an explicit construction 155. Auxiliary data D 196. The functor F 236.1. F on a point 246.2. F on higher dimensional simplices 266.3. Unital replacement of F 286.4. Naturality 296.5. Concordance classes of functors F : Simp(X) → A∞ − Cat 297. Global Fukaya category 307.1. The A∞-nerve 307.2. Global Fukaya category 307.3. Universal construction 317.4. Global Fukaya category and unital replacement, a remark 32

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http://yashamon.github.io/web2/papers/fukayaI.pdf

2 YASHA SAVELYEV

8. Extending F to degeneracies 329. Algebraic-topological considerations 339.1. Colimit of F 349.2. Naturality 36Appendix A. Quasi-categories and Joyal model structure 36A.1. Categorical equivalences, morphisms and equivalences 37A.2. Inner fibrations 37A.3. (co)-Cartesian fibrations 38A.4. Semi-locality 39A.5. A∞-nerve 39

1. Introduction

Smooth fibrations over a Lorentz 4-manifold with fiber a Calabi-Yau 6-fold are amodel for the physical background in string theory. This suggests that there may besome string theory linked mathematical invariants of such a fibration. Indeed, whenthe structure group of M ↪→ P → X can be reduced to the group of Hamiltoniansymplectomorphisms of M , in which case P is called a Hamiltonian fibration, thereare a couple of basic invariants of such a fibration based on Floer-Gromov-Wittentheory. One such example is the Seidel representation [?] and the related quantumcharacteristic classes of the author [?]. Related invariants are also proposed byHutchings [?]. Even earlier there is work on parametric Gromov-Witten invariantsof Hamiltonian fibrations by Le-Ono [?] and Olga Buse [?]. At the same time,Costello’s theorem [?] on reconstruction of topological conformal field theories fromCalabi-Yau A∞ categories suggests that the above invariants must have a similarreconstruction principle.

For a given Hamiltonian fibration P as above, the A∞ Fukaya categories of thefibers fit into a “family”, although exactly what this “family” should mean is anon-trivial problem by itself, since we must somehow remember the continuity ofP . Then our basic idea is that associated to a Hamiltonian fibration there should bea classifying map from X into an appropriate “classifying” space of A∞ categories,from which the other invariants can be reconstructed via a version of Toen’s derivedMorita theory, [?]. We say more on this in Section 1.3.

This paper will be mostly self contained, as we will explain many (especiallyalgebraic) concepts used.

1.1. A functor from the category of smooth simplices of X. Given P asabove, and a choice of analytic perturbation data, to each smooth simplex

Σ : ∆n → X,

we associate an A∞ category F (Σ). The principal step in this is the construction ofnatural maps, satisfying various axioms, from the universal curves over {Rd} intothe standard topological simplices ∆n, where Rd is the moduli spaces of Riemannsurfaces which are topologically disks with d+ 1 punctures on the boundary. Thistopological-combinatorial connection of the universal curves with simplices is new,and is likely of independent interest.

GLOBAL FUKAYA CATEGORY I 3

Let X• denotes the smooth singular set of X. The above data is then extendedto functor

F : ∆/X• → A∞ − Catunit,

with ∆/X• the category of all smooth simplices of X, and A∞−Catunit the categoryof small unital Z2-graded A∞ categories over Q, with morphisms strict embeddings,which are moreover quasi-equivalences.

Theorem 1.1. For (M,ω) a monotone symplectic manifold, there is a homotopynatural map

cl : BHam(M,ω) → (S, NFuk(M)),

where (S, NFuk(M)) denotes the space of ∞-categories, in the component ofNFuk(M). (See Section A.1).

This can also be interpreted to say that Ham(M,ω) continuously acts on theFukaya category Fuk(M,ω). And this verifies one sense of a conjecture of TelemanICM 2014 on the existence of such an action. A discreet version of such action canbe found in Seidel [?]. The proof is in Section 7.3. In part II of this paper this mapis shown to be homotopically non-trivial in a specific example.

The functor N is an analogue for A∞ categories of the classical nerve constructionand is originally due to Lurie [?]. It was then clarified in the A∞ context by Tanaka[?] and Faonte [?].

1.2. Towards new invariants and quantum Novikov conjecture. By theabove discussion we automatically obtain a new invariant of a Hamiltonian fibrationM ↪→ P → X as the homotopy class of its classifying map to S.

It may difficult to get intrinsic motivation for Hamiltonian fibrations for a readeroutside of symplectic geometry, as a start one may read [?]. However as one par-ticular case we can fiberwise projectivize the complexified tangent bundle:

P (X) = P (TX ⊗ C),

of a smooth manifold X. This P (X) in particular has the structure of a smoothHamiltonian fibration. In this way we also get a new invariant of a smooth rmanifold X, given by the homotopy class of the classifying map

cl(P (X)) : X → (S, NFuk(CPr−1)),

induced by Theorem 1.1.Recall that Pontryagin classes of a smooth manifold are defined as Chern classes

of its complexified tangent bundle. Novikov has shown that rational Pontryaginclasses are topologically invariant. It is then very natural to ask the following,“quantum” variant of the Novikov conjecture:

Question 1.2. Suppose that f : X → Y is a homeomorphism of smooth manifolds.Is cl(P (X)) homotopic to cl(P (Y )) ◦ f?

Why may one expect the answer of yes? One substantial reason is that like therational Pontryagin classes of X the invariant [cl(P (X))] is based on a rationalalgebraic theory, (we passed through rational Fukaya categories). But of course theanswer of “no” is also very interesting, since it means that our construction givesnew smooth invariants of manifolds via holomorphic curves.

4 YASHA SAVELYEV

1.3. Hochschild and geometric Hochschild cohomology and homotopygroups of Ham(M,ω). This section is an excursion, meant to relate our geometrictheory with the algebraic derived Morita theory of Toen. For an A∞ category Cwe define

HH2−igeom(C) = πi(S, NC).

The left hand side is named geometric Hochschild cohomology, the name and no-tation will be justified shortly. By Theorem 1.1 above we then get:

Theorem 1.3. For (M,ω) monotone, there is a natural group homomorphism

(1.1) πi−1(Ham(M,ω), id) → HH2−igeom(Fuk(M)).

HH∗(Fuk(M)) is known to be isomorphic to QH∗(M) for a wide class of cases,and so the above, when i > 2 has the same formal form as the author’s quantumcharacteristic classes [?], provided there is a connection between HH∗(Fuk(M))and HH∗

geom(Fuk(M)). This would be the most basic form of the “reconstruction”that was mentioned before. Such a connection is described further below.

Remark 1.4. Note that the case of i = 2, which “corresponds to” the Seidelhomomorphism [?], is a bit special since the correspondence in Theorem 1.6 worksdifferently when i = 2.

In Part II we calculate with Hamiltonian S2 fibrations over S4 to get:

Theorem 1.5. The map

Z → HH−2geom(Fuk(S2)) = π4(S, NFuk(S2),

determined by (1.1) is an injection.

1.3.1. Geometric Hochschild cohomology and Toen’s derived Morita theory. A smalldisclaimer. HH∗

geom(C) is just a name for an object whose construction is immedi-ate from work of Joyal and Lurie, and quiet possibly appears elsewhere. The authoris far from an expert in the subject, and we claim no originality for its construction.What may however be interesting is the connection to symplectic geometry that wediscover in these papers, and perhaps HH∗

geom(C) deserves a more careful studyon its own.

Let us then very briefly indicate the connection of HH∗geom(C) with Hochschild

cohomology via Toen’s derived Morita theory. First we recall:

Theorem 1.6 (Corollary 8.4). [?] For a small dg-category C, (with cohomologicalgrading conventions) there are natural isomorphisms

πi|(dg − Cat, C)| ' HH2−i(C), for i > 2,(1.2)π2|(dg − Cat, C)| ' HH0(C)∗,(1.3)

with HH0(C)∗ denoting the multiplicative group of invertible elements, (dg−Cat, C)denoting the C component of the subcategory of dg − Cat with morphisms quasi-equivalences, and with | · | the geometric realization or the classical nerve functor.

On the other hand the nerve functor N naturally induces a homomorphism,

N∗ : πi|(dg − Cat, C)| → πi|(∞− Cat,NC)| ' πi(S, NC).


When C is Z-graded, rational and (pre)-triangulated there are folklore theorems ofLurie (personal communication) to the effect that this is an isomorphism. Thus, inthis case for i > 2

HH2−i(C) = πi(S, NC) = HH2−igeom(C),

by our definition. This should extend without issues to Z-graded rational (pre)-triangulated A∞ categories. Beyond that it seems nothing is known.

1.4. Acknowledgements. I would like to thank Octav Cornea and Egor Shelukhinfor discussions and support, Kevin Costello and Paul Seidel for interest. Hiro LeeTanaka for enthusiasm, generously providing me with an early draft of his thesisand finding a number of misprints in a draft of the paper. Bertrand Toen forexplaining to me an outline of the proof of some conjectures and for enthusiasticresponse. I also thank Jacob Lurie, for feedback on some questions, as well asthe anonymous referee for a number of helpful comments and suggestions. Thepaper was primarily written while I was a CRM-ISM postdoctoral fellow, and I amgrateful for the wonderful research atmosphere provided by CRM-Montreal. It wasthen substantially revised during my stay at RIMS at Kyoto university, and I alsothank the staff and Kaoru Ono for great hospitality and discussions.

2. Notations and conventions and large categories

We will use terms ∞-category and quasi-category interchangeably, usually theterm quasi-category will be used when we want to do something concrete.

We use diagrammatic order for composition of morphisms in the Fukaya category,and quasi-categories so f ◦ g means

· f−→ · g−→ ·,

as reversing order for composition in quasi-categories is geometrically very con-fusing, since morphisms are identified with edges of simplices. Elsewhere we usethe more common Leibnitz convention. Although this is somewhat contradictoryin practice things should be clear from context. By simplex and notation ∆n wewill interchangeably mean the topological n-simplex and the standard representablen-simplex as a simplicial set, for the latter we may also write ∆n

• .Given a category C the over-category of an object c ∈ C is denoted by C/c. We

say that a morphism in C is over c exactly if it is a morphism in the over-categoryof c.

Given an A∞ category by the nerve we always mean the A∞ nerve.Some of our quasi-categories are “large” with proper classes of simplices instead

of sets. The standard formal treatment of this is to work with Grothendieck uni-verses. This is a widely accepted extension of set theory. We shall not howeverneed to make this explicit. For reference one paper that does make this kind ofthing explicit is [?] also previously cited.

3. Preliminaries

3.1. The simplex category of a smooth manifold X. A simplicial set S• isa functor S• : ∆op → Set, where ∆op denotes the opposite category to ∆: the

6 YASHA SAVELYEV

category of combinatorial simplices, whose objects are non negative integers andmorphisms non-strictly increasing maps

{0 < 1 < . . . < n} → {0 < 1 < . . . < m}.

We will denote the objects of ∆op by [n]. Later on it will convenient to think ofobjects [n] as totally ordered finite sets and so talk about membership.

If ∆n• denotes the standard n-simplex, ∆n

• = hom∆(·, [n]), then we have thecategory of simplices over S•, ∆/S•, whose set of objects is the set of naturaltransformations Nat(∆n

• , S•) and morphisms commutative diagrams

∆n• ∆m

•

S•,

with natural transformation ∆n• → ∆m

• induced by maps [n] → [m].Let X be a smooth manifold, and let ∆n denote the standard topological n-

simplex. Denote by X• the total smooth singular set of X. In other words this isthe simplicial set defined by: X•([n]) = C∞(∆n, X), the set of all smooth maps∆n → X. Consider the category of simplices over X•, ∆/X•. This is clearlythe same as the category with objects smooth maps ∆n → X, and morphismscommutative diagrams

∆n ∆m

X,

with ∆n denoting the standard topological n-simplex, with vertices ordered 0, . . . , nand top horizontal arrow a simplicial map, that is a linear map taking vertices tovertices preserving the order. We say that Σ : ∆n → X is non-degenerate if it doesnot fit into a commutative diagram

∆n ∆m

X,

Σ

with m < n.We will denote by Simp(X) the subcategory of ∆/X•, obtained from ∆/X• by

restricting the morphisms to be face maps, that is morphisms induced by monomor-phisms in ∆.

3.2. Preliminaries on Riemann surfaces. Let S be a nodal Riemann surface,with d + 1 cyclically ordered punctures on the boundary. Here a puncture is aspecific kind of end structure {ei}i∈I corresponding to the removed marked pointson the boundary of S, and is to be explained. The nodal points are denoted by{nj}j∈J . For each j ∈ J we have a pair Sj,± of smooth components which aretopologically disks with punctures

{ei}i∈Ij,± ⊂ {ei}i∈I ,


r

α β

γ

α βγ

Figure 1.

we explain the signs ± shortly, for now they just distinguish the pair of components.If we remove the nodes from S then each

S◦j,± := Sj,± − nj

has an additional puncture nj,± called the node end. We distinguish one punctureof S as the root, and use the cyclic ordering to order the punctures, 0, . . . , d goingclockwise, starting from the root.

It is sometimes convenient to depict such Riemann surfaces as stable, rootedsemi-infinite trees, embedded in the plane, where stable means that the valencyof each vertex is at least 3. We do this by assigning a vertex to each smoothcomponent as above, a half infinite edge to each marked point, and an edge to eachnodal point, as depicted in Figure 1.

To make some arguments and notation cleaner we also introduce a linear orderingon the smooth components of S, or vertices, by order “of composition” defined asfollows. The component with the root semi-infinite edge e0 will be called the rootvertex denoted by ω. In terms of the associated tree for the surface we have apre-order on vertices given by the distance to the root vertex, (by giving each edgelength 1). To get an actual order, first isometrically embed the tree in the planewhile preserving the cyclic ordering. Then clockwise order vertices equidistant tothe root, as in Figure 1. We shall denote by α the furthermost component from ω,by β the next furthermost component, etc. (Pretending that we can’t run out ofletters.) We will mostly make use of this in Section 4.1.

For d ≥ 2 let Sd → Rd denote the universal family of the Riemann surfaces S,as above. (Note that Seidel [?] calls our Rd by Rd+1.) We will also denote by

ρ : S◦d → Rd,

this universal family where the nodal points of the surfaces have been removed.

Notation 3.1. We denote by Sd,r and sometimes just by Sr the fiber ρ−1(r), forr ∈ Rd.

As part of the data, at the i’th puncture we ask for a holomorphic diffeomorphismhaving the name of the removed point or the end:

ei : [0, 1]× [0,∞) → S,

i 6= 0. And at the 0’th puncture we ask for a holomorphic diffeomorphism

e0 : [0, 1]× (−∞, 0] → S.

Let eti denote the restriction of the maps above to [0, 1]× [t,∞), and let S◦j,±, Sj,±

be as above. We further specify the ± distinction so that Sj,− > Sj,+ with re-spect to the linear order above. And we ask for a similar pair of holomorphic

8 YASHA SAVELYEV

diffeomorphisms

ej,− : [0, 1]× (−∞, 0] → S◦j,−,(3.1)

ej,+ : [0, 1]× [0,∞) → S◦j,+(3.2)

at the nj,± ends. Likewise etj,± will denote the restrictions of the above diffeo-morphisms to [0, 1] × [t,∞), respectively to [0, 1] × (−∞, t]. The data of suchdiffeomorphisms, for a given possibly nodal surface, will be called a a strip endstructure and the particular diffeomorphisms strip diffeomorphisms.

Choose r-smooth families {ei,r}, {ej,±,r} of strip diffeomorphisms for the entireuniversal family Sd → Rd, (note that further on r is suppressed). These choiceshave to be consistent with gluing in the natural sense as explained in [?, Section9g]. We will keep track of these systems of choices of strip end structures onlyimplicitly.

Although we won’t use the following in any essential way, for instructional pur-poses it will be helpful to recall the following metric characterization of the modulispace Rd. The family {Sd,r} is in a correspondence with a suitably universal com-pactified family {Metd,r} of constant curvature −1 metrics on the disk with d+ 1punctures on the boundary. Under this correspondence the complex structure onSr is just the conformal structure induced by Metr. This is of course classical, tosee all this use Schwartz reflection to “double” each Sr to a possibly nodal Riemannsurface without boundary Dr with d+1 punctures. This determines an embeddingof Rd into the Grothendieck-Knudsen moduli space M0,d+1 of Riemann surfaceswhich are topologically S2 with d+1 points removed. As d > 2, for r in the interiorof Rd, by uniformization theorem, Dr is a quotient of the disk by a subgroup ofPSL(2,R), which must also preserve the hyperbolic metric. Therefore Sr inheritsa hyperbolic metric.

The metric point of view gives an illuminating description of the compactificationRd, for d ≥ 2: starting with some Sr and taking r to a boundary stratum, corre-sponds to having some fixed collection of embedded, disjoint geodesics on Sr, withboundary in boundary of Sr will have their length shrunk to zero. Each boundarystratum is completely determined by such a collection of geodesics.

The reverse of this degeneration process is the so called gluing construction (seefor example [?]) which takes a surface in Rd and produces a surface with oneless node. This gluing is determined by gluing parameters which we parametrizeby [0, 1), assigned to each node. For us 0 means don’t glue, and 1 is meant tocorrespond to some small value of the gluing parameter used in actual gluing. Wewill write dα,β for the parameter used in the gluing of components α, β, and likewisewith other components.

The gluing construction for parameters in [0, 1) determines an open neighbor-hood called gluing neighborhood of the boundary of Rd. We will call by N thegluing normal neighborhood of the boundary of Rd: an open neighborhoodof the boundary, deformation retracting to the boundary, contained in the gluingneighborhood.

The gluing construction also induces a kind of thick-thin decomposition of thesurface, with thin parts conformally identified with [0, 1] × [0, l] for l determinedby the corresponding gluing parameter. This decomposition is not intrinsic, as itdepends in particular on the choice of the family of strip like coordinate charts.However, instructively these gluing parameters can be thought of as lengths of


mα

mβ

Figure 2. This diagram is only schematic. The embedding intothe plane is not mean to be holomorphic or isometric for the naturalhyperbolic structure on the surface.

geodesic segments, for example mα, mβ in figure 2, and the thin parts are inprinciple closely related to thin parts of thick-thin decomposition in hyperbolicgeometry.

4. A system of natural maps from the universal curve to ∆n

We explain here a remarkable connection between the universal curve over Rd

and the standard topological simplex ∆n. This will be used in our construction butmay be of independent interest.

Let Π(∆n) be the groupoid whose objects obj are vertices of ∆n. And morphismshom simplicial maps m : [0, 1] → ∆n, (possibly constant). The map

s : hom → obj

takes m to m(0) andt : hom → obj

takes m to m(1). As there is a unique directed edge between a pair of vertices, thecomposition maps in Π(∆n) are obvious. We denote the composition of m1,m2 bym1 ·m2, the order diagrammatic, so that this means first m1 then m2.

We say that (m1, . . . ,md) is a composable chain of morphisms mi ∈ hom(Π(∆n))if t(mi−1) = s(mi). For future use let mi−1,i denote the unique morphism from thei− 1 vertex to the i vertex in ∆n.

The goal is to construct a “natural” system of maps

u(m1, . . . ,md, n) : S◦d → ∆n,

d ≥ 2, for each composable chain (m1, . . . ,md), which in particular will satisfy thefollowing properties.

• The maps u(m1, . . . ,md, n) are continuous and their restrictions to

ρ−1(Rd ⊂ Rd)

are smooth.

10 YASHA SAVELYEV

• Let u(m1, . . . ,md, n, r) denote the restriction of u(m1, . . . ,md, n) to Sr, andlet m0 denote the composition m1 · . . . · md in Π(∆n). Then in the stripcoordinates e1k : [0, 1]× [1,∞) → Sr at the k’th end, u(m1, . . . ,md, n, r) hasthe form of the projection to [0, 1] composed with mk.

• For 1 ≤ k ≤ d, the component of the boundary of Sr between ek−1, ek endsis mapped to s(mk), and the component of the boundary between ed ande0 is mapped by u(m1, . . . ,md, n, r) to t(md), for each r ∈ Rd.

Let us explain naturality. First denote by T (m1, . . . ,md, n) the space of mapssatisfying the pair of properties above. We have the natural gluing map(4.1) Sti : Rs1 ×Rs2 × [0, 1) → Rs1+s2−1,

whose value on (r, r′, τ) is given by gluing the surface Sr at the root and Sr′ at itsi’th marked point, with gluing parameter τ ∈ [0, 1), and then associating to thissurface its isomorphism class in Rs1+s2−1. (When the value of the gluing parameteris 0, this is the composition map in the Stasheff topological A∞ operad).

Given an element u ∈ T (m1, . . . ,ms1 , n) and an elementu′ ∈ T (m′

1, . . . ,m′i−1,m1 · . . . ·ms1 = m′

i,m′i+1, . . . ,m

′s2),

we have a naturally induced map

u ?i u′0 : S◦

s1,s2,0 → ∆n,

whereS◦s1,s2,0 → Rs1 ×Rs2

is the pullback of the family S◦d=s1+s2−1 by Sti|Rs1×Rs2×{0}. More specifically the

fiber of S◦s1,s2,0 over (r1, r2) is a surface with two smooth components identified

with Ss1,r1 , Ss2,r2 . So we may apply u to the first component and u′ to the second,and this is the map u ?i u

′0.

We can extend u ?i u′0 to a map

u ?i u′1 : S◦

s1,s2,1 → ∆n,

whereS◦s1,s2,1 → Rs1 ×Rs2 × [0, 1)

is the pullback of the universal family by Sti|Rs1×Rs2

×[0,1). To get this extensionwe specify u ?i u

′1|Sr,r′,τ , for Sr,r′,τ the fiber of S◦

s1,s2,1 over

(r, r′, τ) ∈ Rs1 ×Rs2 × [0, 1), τ 6= 0.

Recall that the surface Sr,r′,τ glued from Sr,Sr′ has a sub-domain which we denoteby thin = thinτ,i that has a determined conformal identification with a strip ofthe form [0, 1]× [−φ(τ), φ(τ)], for some function φ. Sr,r′ − thin has 2 componentswhich can be holomorphically identified with regions

Regr ⊂ Sr, Regr′ ⊂ Sr′ ,

so that Regr is identified with the complement of [0, 1] × (−∞, 0] in Sr, for thestrip coordinates on the root end. And likewise, so that Regr′ is identified with thecomplement of [0, 1] × [0,∞) in Sr′ , for the strip coordinates on the i’th end. Wethen define u ?i u

′1 to coincide with u, u′ on Regr respectively Regr′ , while on thin

in the coordinates[0, 1]× [−φ(τ), φ(τ)],


u ? u′1 is the map given by the projection

[0, 1]× [−φ(τ), φ(τ)] → [0, 1]

followed by the map m′i.

In order state the naturality axioms we need more geometry. Let N be a normalgluing neighborhood of the boundary ∂Rd as previously explained, where d ≥ 2.Let Sd−3

0 ⊂ N be an embedded sphere of dimension d − 3, not intersecting ∂Rd.Let Rd−2

0 be the dimension d− 2 ball sub-domain of Rd bounded by Sd−30 . Finally,

let Sr − ends denote the compact Riemann surface with boundary obtained fromSr by removing the ends, that is images of the charts ei|[0,1]×(0,∞). For r /∈ ∂Rd

setS2r := (Sr − ends)/∂(Sr − ends)

so that S2r is homeomorphic to S2. So we get a fiber bundle

S2 ↪→ F → Rd−20 ,

with fiber over r: S2r . Trivialize this and let

S2 ↪→ F → (Sd−2 := Rd−20 /Sd−3

0 ),

denote the associated trivial fiber bundle. Let ∞r ∈ S2r denote the image of ∂(Sr−

ends) in the quotient, ands∞(r) = ∞r,

be the corresponding section.Let Cs(∆

n) denote the set of composable chains (m1, . . . ,ms) of length s inΠ(∆n). A system of maps U is an element of:

(4.2)∏n∈N

∏s∈N>2

∏(m1,...,ms)∈Cs(∆n)

T (m1, . . .ms, n).

Given a system U its projection onto (n, s, (m1, . . . ,ms)) component will be denotedby u(m1, . . . ,ms, n).

Definition 4.1. We say that U is natural if it satisfies the following axioms:(1) For all s1, s2 and for all i if m′

i = m1 · . . . ·ms1 then the map

(4.3) u(m1, . . . ,ms1 , n) ?i u(m′1, . . . ,m

′s2 , n)1

coincides with the composition

(4.4) Ss1,s2,1Sti,∗−−−→ Ss1+s2−1

u(m′1,...,m

′i−1,m1,...,ms1

,mi+1,...,m′s2

,n)−−−−−−−−−−−−−−−−−−−−−−−−−−−→ ∆n,

for Sti,∗ the bundle map induced by Sti.(2) Given a face map f : ∆n−1 → ∆n, we have that

f ◦ u(m1, . . . ,ms, n− 1) = u(f(m1), . . . , f(ms), n).

(3) Letpr : ∆n+k → ∆n,

k > 0 be a simplicial projection, then there is an induced functor

pr : Π(∆n+k) → Π(∆n)

and

pr ◦ u(m1, . . . ,ms, n+ k) = u(pr(m1), . . . , pr(ms), n).

12 YASHA SAVELYEV

(4) For (m1, · · · ,ms) ∈ Cs(∆n) let D(m1, . . . ,ms) denote the minimal dimen-

sion of a subsimplex of ∆n which contains the edges corresponding to themorphisms mi. Suppose that D(m1, . . . ,md) = d then u(m1, . . . ,md, d)induces a map of pairs:

u : (Rd−20 × S2, Sd−3

0 × S2) → (∆d, ∂∆d)

and so induces a map with same name:(4.5) u : Sd−2 × S2 → ∆d/∂∆d ' Sd

and we ask that u is a homological degree 1 map.

Note that u factors through Sd−2 ∧S2 ' Sd. To see this, note that by construc-tion u(s∞) ⊂ ∂∆d for s∞ the section of Sd−2 × S2 as above. Also if ∞ ∈ Sd−2

denotes the image of Sd−30 in the quotient Rd−2

0 /Sd−30 then the fiber over ∞ is

likewise mapped to ∂∆d by the Axioms 1 and 2.

Remark 4.2. Using an inductive procedure as in the proof of the following theorem,it should be possible to show that the final axiom actually follows by the previousaxioms.

Theorem 4.3. A natural system U exists, and is unique up to homotopy (throughnatural systems).

We first give a not explicitly constructive proof in all generality, and afterwardsdescribe a partial explicit construction.

Proof of 4.3. To construct our maps u(m1, . . . ,ms, n) we will proceed by induction.When n = 0 there is nothing to do, as we have unique maps for all s ≥ 2.

Given a composable sequence (m1, . . . ,ms) of morphisms in Π(∆n), for some n,let D(m1, . . . ,ms) denote the least dimension of a non-degenerate subsimplex of∆n which contains the edges corresponding to {mi}. Now suppose that we havechosen maps(4.6) u(m1, . . . ,ms, n),

for all s ≥ 2 and all n ≤ N and every composable chain (m1, . . . ,ms) withD(m1, . . . ,ms) = s so that axioms 1,2,4 are satisfied and so that

u(m1, . . . ,ms, n) = σ−1 ◦ u(σ(m1), . . . , σ(ms), n),

forσ : ∆n → ∆n

a simplicial homeomorphism. That is we satisfy axiom 3 only partially and only forrestricted (m1, . . . ,ms) at the moment. We first construct maps u(m1, . . . ,ms, n)for all s ≥ 2, n ≤ N+1 and all composable chains (m1, . . . ,ms) with D(m1, . . . ,ms) =s so that axioms 1,2,4 are satisfied and so that(4.7) u(m1, . . . ,ms, n) = σ−1 ◦ u(σ(m1), . . . , σ(ms), n),

for σ a simplicial homeomorphism as before. Then by induction we will have mapsu(m1, . . . ,ms, n) with D(m1, . . . ,ms) = s for all n, and we will construct fromthese our natural system.

Note that the Axiom 2 and the maps (4.6) uniquely determine(4.8) u(m1, . . . ,ms, N + 1)


for all (s, (m1, . . . ,ms)) with D(m1, . . . ,ms) = s ≤ N . We need an extension inthe case D(m1, . . . ,ms) = s = N + 1.

Assume that N + 1 > 3 as the cases N + 1 = 2, 3 are special and geometricallytrivial. Let then (m0

1, . . . ,m0N+1), N + 1 > 3, be a chosen composable sequence

with D(m01, . . . ,m

0N+1) = N + 1. Then gluing as in the axiom 1 of naturality and

the maps (4.8) naturally determine a map

(4.9) u = u(m01, . . . ,m

0N+1, N + 1) : SubN+1 → ∆N+1,

where SubN+1 = ρ−1(∂RN+1), and where as before

ρ : S◦N+1 → RN+1.

Extend u in any way to ρ−1(U), for U as before the normal gluing neighborhood of∂RN+1, so that Axiom 1 of naturality is satisfied. We then need to further extendu to S◦

N+1 so that the final axiom of naturality is satisfied.Let SN−2

0 ⊂ U be an embedded sphere in U not intersecting ∂RN+1. Then uinduces a map of a pair

(4.10) g : (SN−20 ×D2, SN−2

0 × ∂D2) → (∂∆N+1 ' SN , loop),

where loop is a topologically embedded S1 in ∂∆N+1 that is the image of the loop(m0

1 · . . . ·m0N+1 ·m

−1s(m0

1),t(m0N+1)

), where · is concatenation of paths and the orderof composition is diagrammatic. This is constructed analogously to the map (4.5),with D2 a homeomorphic model of Sr − ends for each r.

Lemma 4.4. The map g is homological degree 1.

Proof. As N > 2 loop has codimension greater than 1, so that the meaning of homo-logical degree is unambiguous, as the pair (SN , loop) has a well defined fundamentalclass by the homology long exact sequence for a pair. Moreover approximating gby a smooth map we may compute the homological degree via the smooth degree,(denote the approximation still by g). That is let f be a N -face of ∆N+1 andp ∈ interior(f) a regular image point of g. The homological degree of g is thenthe count of elements of g−1(p) with signs given by whether dgk, k ∈ g−1(p), isorientation preserving or reversing.

Suppose without loss of generality that the vertices of f are 0, . . . , N . As thedegree of g is clearly independent of the choice of SN−2

0 we may assume that SN−20

is chosen so that for some ε > 0:

(4.11) St1 :(RN−2

0 ⊂ RN

)×R2 × {ε} → RN+1

is an embedding into SN−20 , and that moreover SN−2

0 −T is covered by such embed-dings corresponding to the various other faces of RN , where the region T ⊂ SN−2

0 ,is such that g maps ρ−1(T ) into the union of (N − 1)-faces of ∆N+1. The image ofthe map (4.11) will be denote by V .

Then by the naturality axiom 4 the face f is covered by the image of

κ = u(m01, . . . ,m

0N , N + 1) ?1 u(m

01 · . . . ·m0

N ,m0N+1, N + 1)1|V ,

whereV = ρ−1(V ).

By construction the smooth degree of g|V is the smooth degree of κ. But thenby naturality axiom 4 κ is smooth degree one. And again, by naturality and the

14 YASHA SAVELYEV

assumption on the form of SN−20 above nothing else in SN−2

0 ×D2 can hit p by themap g. It follows that g is smooth degree one and so is homological degree one.

�

As g is degree 1 we may find a degree one extension:g : (RN−1

0 ×D2, SN−20 ×D2 tRN−1

0 × ∂D2) → (∆N+1, ∂∆N+1),

with g(RN−10 × ∂D2) = loop. Given this g we may readily construct our extension

u : S◦N+1 → ∆N+1 so that

u ∈ T (m01, . . . ,m

0N+1, N + 1)

and so that the last naturality axiom is satisfied.Now given any other composable sequence (m1, . . . ,mN+1) with

D(m1, . . . ,mN+1) = N + 1

letσ : ∆N+1 → ∆N+1

be the unique simplicial homeomorphism mapping (m1, . . . ,mN+1) to (m01, . . . ,m

0N+1),

(it is unique because the action is determined by the action on the vertices. ) Andwe define

u(m1, . . . ,mN+1, N + 1) : S◦N+1 → ∆N+1

byu(m1, . . . ,mN+1, N + 1) = σ−1 ◦ u.

It remains only to check that Axiom 1 and 2 are satisfied for this resulting partialsystem of maps

u(m1, . . . ,ms′ , n)(m1,...,ms′ ),n≤N+1,D(m1,...,ms′ )=s′ ,

but this is immediate by construction and the inductive hypothesis. Consequentlywe complete the induction step. It remains to extend the partial system above toa full natural system, that is we need to remove restrictions on the D-number.

Given (m1, . . . ,ms), a composable sequence in Π(∆n) with s > n, we may writemi = pr◦mi for (m1, . . . , ms) a composable sequence in ∆s s.t. D(m1, . . . , ms) = s,for pr : ∆s → ∆n surjective simplicial projection. Moreover (m1, . . . , ms) is clearlyunique up to an action of a simplicial homeomorphism σ : ∆s → ∆s fixing theimage i(∆n), for i : ∆n → ∆s inclusion of face s.t. pr ◦ i = id. We then define

u(m1, . . . ,ms, n) := pr ◦ u(m1, . . . , ms, s).

Let (m′1, . . . , m

′s) be another choice of a composable sequence with pr(m′

i) = mi,and let σ : ∆s → ∆s be a simplicial homeomorphism with σ(mi) = m′

i fixing theimage of i(∆n) as above. Then we have

pr ◦ u(m′1, . . . , m

′s, s) = pr ◦ σ ◦ u(m1, . . . , ms, s).

But pr ◦ σ = pr since σ fixes i(∆n). So we obtain thatpr ◦ u(m′

1, . . . , m′s, s) = pr ◦ u(m1, . . . , ms, s),

so that u(m1, . . . ,ms, n) is well defined. So we have constructed our system of mapssatisfying all the axioms of naturality.

To prove uniqueness up to homotopy, note that by our axioms a system of naturalmaps is completely determined by all the maps u(m1, . . . ,ms, n) with

D(m1, . . . ,ms) = s = n.


We then again proceed by induction. Suppose that we have a pair of natural systemsU1,U2 and suppose we have a continuous family of maps ut(m1, . . . ,ms, n), t ∈ [0, 1],for all n ≤ N > 0, (m1, . . . ,ms), with

D(m1, . . . ,ms) = s = n,

so thatut=i(m1, . . . ,ms, n) = ui(m1, . . . ,ms, n)

for i = 0, 1, and so that for each t ut(m1, . . . ,ms, n) satisfy naturality axioms.Then given some composable sequence (m1, . . . ,ms) in Π(∆N+1), with

D(m1, . . . ,ms) = s = N + 1,

we have continuous in t families of induced (as before) maps:ut(m1, . . . ,ms, N + 1) : Subs → ∆N+1,

with Subs as before, and so thatut=1(m1, . . . ,ms, N + 1) : Subs → ∆N+1

coincides with the mapu1(m1, . . . ,ms, N + 1) : Subs → ∆N+1.

Use the homotopy extension property to get a homotopy:

ut(m1, . . . ,mN+1, N + 1) : S◦d → ∆N+1

of the map u0(m1, . . . ,mN+1, N + 1). Nowu1(m1, . . . ,mN+1, N + 1)

andu1(m1, . . . ,mN+1, N + 1)

do not necessarily coincide on S◦d but they coincide on Subs since we used homotopy

extension, and both these maps are “degree one” (that is they both satisfy the lastnaturality axiom). We may then construct a homotopy relative to Subs, between thepair u1(m1, . . . ,mN+1, N +1) and u1(m1, . . . ,mN+1, N +1). Taking the compositehomotopy we complete the induction step. �

4.1. Outline of an explicit construction. This section is not logically neces-sary, but in order to give the reader more intuition we now give a partial explicitconstruction of the maps that would satisfy the axioms. This construction couldbe in principle extended to all generality but at the cost of much complexity.

Fix a geometric model for Rd, for example as the Stasheff associahedra. Whend = 4 this is a pentagon. Recall that to each corner of R4 we have a uniquelyassociated nodal Riemann surface with 3 components and 5 marked points, oneof which is called the root. Recall that we label the root component by ω, thenext component by β and the component furthest from root by α. (With respectto the linear ordering described earlier.) Denote by Mα the collection of markedpoints, different from the root e0, on α, likewise with β, ω. This determines a sub-composable sequence mor(Sα) of a composable sequence (m1, . . . ,m4), and likewisewith β, ω, (note that Mω,Mβ could be empty).

Let r be in the normal gluing neighborhood of some corner, corresponding tonon-zero gluing parameters dα,β , dβ,ω. We now construct a map

fr = fr(m1, . . . ,m4) : [0, 4]× [0, 1] → ∆4

16 YASHA SAVELYEV

In what follows by concatenation of a collection of paths we mean their product inthe Moore path category of ∆4, the notation for composition will be assumed to bediagrammatic. This is the category with objects: points of ∆4, and morphisms fromx0 to x1: continuous paths [0, T ] → ∆4, T > 0, between x0, x1, with compositionthe natural concatenation of paths. Note that this is quiet different from ourpreviously defined groupoid Π(∆4).

For a morphism m in Π(∆4) let s(m) and t(m) denote the source respectivelytarget of m. Let Hm : ∆4 × [0, 1] → ∆4 denote the natural deformation re-traction of ∆4 onto the edge determined by s(m), t(m), with time 1 map the or-thogonal linear projection onto this edge (for the standard metric on ∆n). SetHm

τ = Hm|∆n×{τ}. Next, for a general piece-wise linear path p : [0, T ] → ∆4, withend points s(m), t(m), we have a homotopy Hm

τ ◦ p, τ ∈ [0, 1], from p to a path

p : [0, T ] → ∆4

with image in the edge determined by m. Let D(p, τ), τ ∈ [0, 1], be the concate-nation of Hm

τ ◦ p with the homotopy Gτ , τ ∈ [0, 1], of paths with fixed end points,from p to the map

m : [0, T ] → ∆4

linearly parametrizing the edge determined by m. This second homotopy Gτ canbe chosen in a way that depends only on p. This can be done explicitly, usingpiece-wise linearity of p.

The map f tr from the y = t slice [0, 4]× {t} is constructed as follows. Set

Iα = (1− dα,β)/2,

set fα,r to be the concatenation of the morphisms in mor(Mα). That is ifMα = (mα

1 , . . . ,mαk )

thenfα,r = mα

1 · . . . ·mαk .

Then for t ∈ [0, Iα] set f tα,r = D(fα,r, 2t). Then set f t

r to be the concatenation ofmorphisms in mor(Mβ), mor(Mγ) and of f t

α,r, in that order, although note thatthe order of the morphisms in the concatenation is uniquely determined by the endpoint conditions, this holds further on as well.

Next setIβ = Iα + (1− Iα)(1− dβ)/2.

If α and β components have a nodal point in common we setfβ,r : [0, 4]× {t} → ∆4

to be the concatenation of f Iαα,r with morphisms in mor(Mβ), and for t ∈ [Iα, Iβ ]

we setf tβ,r = D(fβ,r,

2(t− Iα)

1− Iα).

And then for t ∈ [Iα, Iβ ] set f tr to be the concatenation of morphisms in mor(Mω)

and of f tβ,r.

Finally set fω,r to be the concatenation of fIββ,r with morphisms in mor(Mω),

and for t ∈ [Iβ , 1] set

f tr = D(fω,r,

2(t− Iβ)

1− Iβ).


When α has a nodal point in common with the ω component set fβ,r to be theconcatenation of morphisms in mor(Mβ), and for t ∈ [Iα, Iβ ] set

f tβ,r = D(fβ,r,

2(t− Iα)

1− Iα).

Then for t ∈ [Iα, Iβ ] set f tr to be the concatenation of morphisms f Iα

r and f tβ,r,

and mor(Mω) (although mor(Mω) in this particular case is empty, we add this sothat the degenerate case Mα = ∅, Mβ = ∅ makes sense, see the discussion below).Finally for t ∈ [Iβ , 1] set

f tr = D(f

Iβr ,

2(t− Iβ)

1− Iβ).

When r ∈ R4 is in the gluing neighborhood of a face but not of a corner theconstruction of fr : [0, 4] × [0, 1] → ∆4 is similar, in fact we can think of it as aspecial case of the above by setting dβ = 1, Mβ = ∅. When r ∈ R4 is not in thegluing neighborhood of the boundary we can also think of this as a special case ofthe above, with Mα = ∅,Mβ = ∅, dα = 1, dβ = 1 in the construction above.

4.1.1. Retracting Sr onto [0, 4]×[0, 1]. We now construct a smooth r-family of maps

retr : Sr → [0, 4]× [0, 1],

r ∈ R4, suitably compatible with the maps

fr : [0, 4]× [0, 1] → ∆4.

In figure 3 (a), (b), (c) represent cases where (c): r is not within gluing normalneighborhood of boundary, (b): r is in a gluing neighborhood of a side but not acorner and (a): r is within gluing neighborhood of a corner, (we picked a particularcorner and side for these diagrams). The color shading will be explained in amoment. In each case (a), (b), (c) we first color shade [0, 4]× [0, 1] as in figure 4, thegreen region is the domain of f t

α,r contained in [0, 4] × [0, Iα], in the blue regionsthe map fr is vertically constant, the red region is the domain of f t

β,r contained in[0, 4]× [Iα, Iβ ] and yellow region is the rest of the domain of fr. The maps retr aredefined for each r by taking color shaded areas to color shaded areas, so that thefollowing holds.

(1) The ends e0, e1, . . . , e4 of Sr, colored in purple, are identified in strip coor-dinate charts as [1,∞)× [0, 1] and in these coordinates retr is the composi-tion of the projection [1,∞)× [0, 1] → [0, 1] with the map to the boundaryof [0, 4] × [0, 1], so that composition with fr parametrizes the morphismm0 = m1 · . . . ·m4,m1, . . . ,m4 in Π(∆4), respectively.

(2) The boundary of Sr goes either to the boundary of [0, 4] × [0, 1] or to thevertical boundary lines between colored regions.

(3) The unshaded “thin” regions labeled Tα, Tβ come from the gluing con-struction and are identified with [0, 1]× [−φ(τα), φ(τα)], respectively [0, 1]×[−φ(τβ), φ(τβ)]. In these coordinates retr on Tα, Tβ is the projection to [0, 1]composed with a diffeomorphism onto the lower edge of green, respectivelyred region, (linear in respective natural coordinates).

(4) The unshaded part of Sr is collapsed onto the horizontal line boundingyellow region of [0, 4]× [0, 1].

18 YASHA SAVELYEV

mαmα

mβ

(a)

(b)(c)

Tβ

TαTα

Figure 3. The uncolored enclosed regions labeled Tα, Tβ sur-rounding segments mα, mβ are “thin”.

(a)

(b)

(c)

Figure 4. Diagram for Sd. Solid black border is boundary, whiledashed red lines are open ends. The connection A(r, {Li}) pre-serves Lagrangians Li over boundary components labeled Li.

(5) Blue shaded regions are identified in strip coordinate charts [0,∞]×[0, 1] →Σr, as [0, 1] × [0, 1], and are mapped to the corresponding blue regions in[0, 4]× [0, 1].

(The above prescription naturally extends to the boundary R4.)We then set

u(m1, . . . ,m4,∆4)|Sr

= fr ◦ retr,

well almost as we have to flatten these maps near boundary of R4 so that Axiom1 of naturality is satisfied.


5. Auxiliary data D

LetM ↪→ P

π−→ X

be a Hamiltonian fiber bundle, with model fiber (M2n, ω) that we shall assume hereto be a closed, monotone:

ω = const · 2c1(TM),

const ≥ 0, symplectic manifold. The data D consists of the following.We say that a Lagrangian submanifold L ⊂ M is monotone if the homomor-

phisms given by symplectic area and Maslov class

[ω] : H2(M,L) → R, µ : H2(M,L) → Z

are proportional:[ω] = const · µ.

For an x : pt → X, define F (x) to be the set of oriented, spin, monotoneLagrangian submanifold L in Px = x∗P ' M with minimal Maslov number atleast 2, so that the inclusion map π1(L) → π1(M) vanishes. We call elements ofF (x) objects.

Let L be an object of F (x), and j an almost complex structure on Px compatiblewith ω. Let M(L, j) denote the moduli space of Maslov number 2 j-holomorphicdiscs in Px, with one marked point on the boundary, with boundary of the diskgoing to L. It is well known, c.f. [?] that for a generic such j, M(L, j) is regular,is a transversely cut out n-dimensional manifold and is compact. Then we have anevaluation map at the marked point:

ev : M(L, j) → L,

and we define ω(L) ∈ Z as the degree of ev.Given a smooth

Σ : ∆n → X,

letm : [0, 1] → ∆n

be the edge between i, j corners of ∆n and set

m = Σ ◦m.

The data D gives for every pair of objects

L0 ⊂ Pxi , L1 ⊂ Pxj ,

xi = Σ(i), (including i = j) with

ω(L0) = ω(L1),

a Hamiltonian connection

A(L0, L1) = A(L0, L1,m)

on m∗P . Denote by A(L0, L1)(L0) the image in Pxj of the A(L0, L1)-paralleltransport of L0 over [0, 1]. Then we require that A(L0, L1)(L0) is transverse to L1.

Definition 5.1. For each m, a family {jt} = {jt(L0, L1,m)} of fiber-wise almostcomplex structures on m∗P is said to be admissible with respect to A(L0, L1)if the following holds.

20 YASHA SAVELYEV

• For each t, Chern number 1 jt-holomorphic spheres in Pm(t) ⊂ m∗P donot intersect any of the elements of S(L0, L1) ' Li that denotes the spaceof A(L0, L1)-flat sections with boundary on L0, L1. Here Pm(t) denotes thefiber of m∗P over m(t).

• The moduli spaces M(L0, j0), M(L1, j1) are regular, and the evaluationmap

ev0 : M(L0, j0) → L

does not intersect the set of starting positions of elements of S(L0, L1).Likewise the evaluation map

ev1 : M(L1, j1) → L

does not intersect the set of ending positions of elements of S(L0, L1).

Such a family {jt} is easily seen to exist, cf. [?], and we fix it for each m.Next D makes a choice of a natural system U as previously described. FinallyD specifies a certain natural system of Hamiltonian connections and a system ofcomplex structures that we now describe. This is a bit notationally complicatedbut ultimately trivial, given the main geometric input of the system U . It is just asystem of compatible of perturbations in the sense of Seidel, but relative to U .

Given a composable chain (m1, . . . ,md) and a map

u(m1, . . . ,md, n) : S◦d → ∆n that is part of a natural system U ,

we have an induced fibration

M ↪→ S(m1, . . . ,md,Σn) → S◦

d

by pulling back M ↪→ P → X first by Σn : ∆n → X and then by u(m1, . . . ,md, n).We have a natural projection

S(m1, . . . ,md,Σn) → Rd

and we denote the fiber over r ∈ Rd by S(m1, . . . ,md,Σn, r), or simply by Sr wherethere can be no confusion. So Sr is an M fibration over the surface Sr, smoothover smooth components. Suppose now that we have chosen labels by objects Li

for the sides between punctures i, i+ 1, 0 ≤ i ≤ d− 1 and Ld for the side betweend, 0 of Sr, r ∈ Rd, where

Li ∈ F (mi+1(0)) for 0 ≤ i ≤ d− 1 ,

and Ld ∈ F (md(1)). We suppose in addition that ω(Li) = ω(Ld) for all i. Extendthe labeling naturally to nodal elements of Rd so as to be naturally compatiblewith gluing.

Definition 5.2. We say that a Hamiltonian connection A on S(m1, . . . ,md,Σn, r) →

Sr is admissible with respect to L0, . . . , Ld if:• Parallel transport by A over the boundary component labeled Li preserves

Lagrangian Li. This condition is unambiguous, as by construction, overthis boundary component Sr is naturally trivialized as Ps(mi+1) × [0, 1],0 ≤ i ≤ d− 1, or as Pt(md) × [0, 1] in the case of Ld.

• At the i’th end of Sr, in the strip coordinate chart e1i : [0, 1]× [1,∞] → Sr,A has the form of the canonical, flat, R-translation invariant extension ofthe connection A(Li−1, Li,mi).


L0

L1

L2

L3

Figure 5. The connection A preserves Lagrangians Li overboundary components labeled Li.

We also have a Lagrangian sub-fibration of

Sr → Sr

over the boundary of Sr, whose fiber over an element of the boundary componentlabeled Li is Li, with respect to the natural trivializations mentioned above.

We name this sub-fibration by

(5.1) L(U , L0, . . . , Ld, r),

so that if A is admissible as above it preserves this sub-fibration. Denote byT (L0, . . . , Ls,Σ

n, r) the space of Hamiltonian connections on

S(m1, . . . ,ms,Σn, r)

admissible with respect to L0, . . . , Ls. Given an element A in T (L0, . . . , Ls1 ,Σn, r)

and an element

A′ ∈ T (L′0, . . . , L

′i−1, L

′i, L

′i+1, . . . , L

′s2 ,Σ

n, r′),

with L′i−1 = L0, L

′i = Ls1 , we have a connection naturally induced by the gluing

maps Sti, denoted by

Sti(A,A′, 0) ∈ T (L′0, . . . , L

′i−2, L0, . . . , Ls1 , L

′i+2, . . . , L

′s2 ,Σ

n, Sti(r, r′, 0)).

Such a pair A,A′ will be called composable.Using the second property in the Definition 5.2 we can naturally extend this to

a family of connections {Sti(A,A′, τ)}τ , so that

Sti(A,A′, τ) ∈ T (L′0, . . . , L

′i−2, L0, . . . , Ls1 , L

′i+2, . . . , L

′s2 ,Σ

n, Sti(r, r′, τ)),

for 0 ≤ τ < 1, and so that in addition we have the following. Outside of the thinregion thinτ,i of SSti(r,r′,τ) the connection Sti(A,A′, τ) is naturally identified withSti(A,A′, 0). On the other hand over thinτ,i, for τ > 0, SSti(r,r′,τ) is naturallyisomorphic to (m′)∗iP × [−φ(τ), φ(τ)] by the Axiom 1 of the natural system U . Andover thinτ,i, in the above trivialization, Sti(A,A′, τ) is the trivial in the secondvariable extension of the connection A(Li−1, Li,mi).

22 YASHA SAVELYEV

Definition 5.3. We say that a family {jz} of fiber-wise, {ωz}-compatible, almostcomplex structures on S(m1, . . . ,md,Σ

n, r) → Sr is admissible with respect toL0, . . . , Ld if:

• At the i’th end of Sr, in the coordinate chart [0, 1] × [1,∞] × m∗P →S(m1, . . . ,md,Σ

n, r) induced by the strip chart e1i , the family {jz} coin-cides with the canonical R-translation invariant extension of the family{jt(Li−1, Li,mi)}, with the latter as in Definition 5.1.

Denote by J (L0, . . . , Ls,Σn, r) the space of families of fiberwise almost complex

structures {jz} onS(m1, . . . ,ms,Σ

n, r)

admissible with respect to L0, . . . , Ls. Given an element {jz} in J (L0, . . . , Ls1 ,Σn, r)

and an element

{j′z} ∈ J (L′0, . . . , L

′i−2, L0, Ls1 , L

′i+1, . . . , L

′s2 ,Σ

n, r′)

we have an induced element:

Sti({jz}, {j′z}, τ) ∈ J (L′0, . . . , L

′i−2, L0, . . . , Ls1 , L

′i+2, . . . , L

′s2 ,Σ

n, Sti(r, r′, τ)),

for each 0 ≤ τ < 1. Such a pair {jz}, {j′z} will be called composable.

Definition 5.4. A system F : of connections, and almost complex structurescompatible with a system of maps U is an element of∏

r∈Rs

∏{(L0,...,Ls)|Li∈F (Σn)}

∏s≥2

∏Σn

T (L0, . . . , Ls,Σn, r)× J (L0, . . . , Ls,Σ

n, r).

The projection of F onto the (r, (L0, . . . , Ls),Σn, s) component will be denoted by

F(L0, . . . , Ls,Σn, r). And for shorthand we say that a Hamiltonian connection

A ∈ F if it is of the form pr1F(L0, . . . , Ls,Σn, r), for pri, i = 1, 2, the pair

of projections of F(L0, . . . , Ls,Σn, r) onto the component of T (L0, . . . , Ls,Σ

n, r),respectively J (L0, . . . , Ls,Σ, r).

We say that F is natural if:• The families are naturally smooth in the parameter r, this has the same

meaning as for compatible systems of perturbation of Seidel, [?].• For a composable pair A,A′ ∈ F as above the connection Sti(A,A′, 0)

coincides with

pr1F(L′0, . . . , L

′i−2, L0, . . . , Ls1 , L

′i+1, . . . , L

′s2 ,Σ

n, Sti(r, r′, 0)).

• The pair of connections above also agree for a sufficiently small non zero εon the “thin part” of SSti(r,r′,ε)).

• Given a face map f : ∆n−1 → ∆n, by the second naturality property ofmaps u(m1, . . . ,md, n) determined by U , there is bundle map

p : S(m1, . . . ,md,Σn−1f = Σn ◦ f, r) → S(f(m1), . . . , f(md),Σ

n, r),

and we ask that the pullback map takes pr1F(p(L0), . . . , p(Ld),Σn, r) to

pr1F(L0, . . . , Ld,Σn−1f , r).

• There are analogous conditions on the families of almost complex structurespr2F(L0, . . . , Ls,Σ

n, r) that we will not state.


Notation 5.5. We will sometimes write by abuse of notation F(. . .), for either theconnection pr1F(. . .), or the family of almost complex structures pr2F(. . .), sincethere usually can be no confusion.

Lemma 5.6. A natural system F compatible with a given U exists and is uniqueup to homotopy.

Proof. When n = 0 this is the classical Fukaya category case, and the proof ofexistence of a natural system is given in [?] in the language of what Seidel callscompatible system of perturbations. Now suppose that we have chosen an element

F ∈∏

r∈Rs

∏{(L0,...,Ls)|Li∈F (Σn)}

∏s≥2

∏{Σn|n≤N}

T (L0, . . . , Ls,Σn, r)×J (L0, . . . , Ls,Σ

n, r)

satisfying naturality. We need to extend this to an element of∏r∈Rs

∏{(L0,...,Ls)|Li∈F (Σn)}

∏s≥2

∏{Σn|n≤N+1}

T (L0, . . . , Ls,Σn, r)× J (L0, . . . , Ls,Σ

n, r),

also satisfying naturality.Denote by D(L0, . . . , Ls) the least dimension of a subsimplex of ∆N+1 with

vertices determined by the {Li}. There is a unique extension of F to an element

F ∈∏

r∈Rs

∏{(L0,...,Ls)|N≥D(L0,...,Ls)

∏s≥2

∏{Σn|n≤N+1}

T (L0, . . . , Ls,Σn, r)× J (L0, . . . , Ls,Σ

n, r)

(5.2)

satisfying the naturality condition. We need to extend to the case N + 1 =D(L0, . . . , Ls) and so that naturality is satisfied. For all (L0, . . . , Ls) with

D(L0, . . . , Ls) = N + 1,

and given ΣN+1, the naturality condition and F from (5.2) determineF(L0, . . . , Ls,Σ

N+1, r)

for r in the boundary of Rs, see the discussion following (4.9). Now{T (L0, . . . , Ls,Σ

n, r)× J (L0, . . . , Ls,Σn, r)}

forms a a Serre fibration over Rs with non-empty contractible fibers. Thus, we maysimply pick an extension of the family

{F(L0, . . . , Ls,ΣN+1, r)}

for all r ∈ Rs, such that the second naturality condition is satisfied. The othernaturality conditions then follow by construction. Uniqueness up to homotopy isobvious from the argument above. �

6. The functor F

Let A∞ − Cat denote the category of small Z2 graded A∞ categories over Q,with morphisms fully-faithful embeddings as defined below that are in additionquasi-equivalences.

Definition 6.1. We say that an A∞ functor F is a fully-faithful embedding,if F has vanishing higher order components, is injective on objects and if the firstcomponent map on hom spaces is an isomorphism of chain complexes. In otherwords F above is just an identification map of a full A∞ sub-category.

24 YASHA SAVELYEV

We now describe the construction of the functor

FP,D : Simp(X) → A∞ − Cat

associated to a Hamiltonian fibration P and the chosen auxiliary data D, describedin the previous section. In what follows we usually drop D and P from notation.

6.1. F on a point. For x : pt → X, F (x) is defined to be the Fukaya A∞ categoryFuk(x∗P ) whose objects are the objects as described in Section 5. For a pairL0, L1 of objects with ω(L0) 6= ω(L1) we set hom(L0, L1) = 0, (to avoid dealingwith curved A∞ categories), otherwise we set

hom(L0, L1) = CF (L0, L1,D),

where the latter is a Z2 graded Floer chain complex over Q that is defined as follows.Let A(L0, L1) be the Hamiltonian connection on Px × [0, 1] determined by the

chosen data D, and likewise let j(L0, L1) to be the family of almost complex struc-tures determined by D. Then CF (L0, L1,D) is generated over Q by A(L0, L1)-flatsections γ of Px × [0, 1], with boundary on the pair of Langrangians

L0 ⊂ Px × {0}, L1 ⊂ Px × {1}.

These γ are called geometric generators. The Z2 grading is given by the sign ofthe corresponding intersection of A(L0, L1)(L0) with L1, where A(L0, L1)(L0) asbefore denotes the image of L0 by the A(L0, L1)-parallel transport map over [0, 1].

6.1.1. Differential on CF (L0, L1,D). For γ0, γ1 geometric generators of

CF (L0, L1,D),

let M(γ0; γ1) denote the space of holomorphic sections of Px × ([0, 1] × R), withboundary on the Lagrangian sub-bundles L0 × R, L1 × R, and asymptotic to γiat the ∞, respectively −∞ ends. And let M(γ0; γ1) denote the natural Floercompactification of the quotient M(γ0; γ1)/R, where R acts by translation on thedomain.

Terminology 6.2. Here and elsewhere the term holomorphic section of variousHamiltonian fibrations over 2-d Riemann surfaces will mean the following. OurHamiltonian fibrations S always come with choices of a Hamiltonian connectionA, and a family of fiber-wise almost complex structures {jz}, determined by theperturbation data D. This gives an induced almost complex structure J(A, {jz}) onS restricting to {jz} on the fibers, having a holomorphic projection map to the base,and preserving the horizontal distribution of A. Holomorphic then means that thesection has J(A, {jz})-complex linear differential.

In the above case, “holomorphic” is with respect to the almost complex structureinduced by the flat, R-translation invariant extension of A(L0, L1) to Px× [0, 1]×R,and likewise by the translation invariant extension of j(L0, L1) to Px × [0, 1]× R.

For a generic pair A(L0, L1), j(L0, L1), all the moduli spaces M(γ0; γ1) are trans-versely cut out for all γi. The differential

µ1 : CF (L0, L1,D) → CF (L0, L1,D)

is defined as usual byµ1(γi) =

∑i

#M(γi; γj)γj ,


for {γi} a basis of geometric generators for CF (L0, L1,D). Here #M(γi; γj) isdefined to be zero, unless the virtual dimension of M(γi; γj) is zero in that case itis the signed count of points. The sum is finite by the monotonicity condition.

6.1.2. Higher multiplication maps. The multiplication maps

µd : hom(L0, L1)⊗ hom(L1, L2)⊗ . . .

⊗hom(Ld−1, Ld) → hom(L0, Ld),(6.1)

d > 1 are defined as follows.Let U ,F be the natural systems determined by D. Then we may define the

moduli space

(6.2) M({γi}; γ0, x,F , A)

whose elements are class A, F({Li}, x, r)-holomorphic sections (σ, r), r ∈ Rd, of

Px × Sr.

And s.t. we have:• The boundary of (σ, r) is in the subfibration L(U , L0, . . . , Ld, r), see (5.1),

of Px × Sr over the boundary of Sr.• By assumptions, at the ei end of Sr, in the strip coordinate charts

e1i : [0, 1]× [1,∞] → Sr,

the data F({Li}, x, r) is R-translation invariant, and when i 6= 0 we askthat σ is asymptotic to γi, a geometric generator of homF (x)(Li−1, Li), orat the e0 asymptotic to γ0 a geometric generator of homF (x)(L0, Ld).

Given basis of geometric generators {γij} for homF (x)(Li−1, Li), i 6= 0, and a

basis {γ0j } for homF (x)(L0, Ld), i = 0, for d ≥ 2, assuming that F({Li}, x, r) is

regular we define µd by:

(6.3) 〈µd(γ1j1 , . . . , γ

djd), γ0

j0〉 =∑A

#M(γ1j1 , . . . , γ

djd; γ0

j0 , x,F , A),

when the above moduli spaces have dimension 0, for 〈, 〉 the natural pairing inducedby our choice of basis. The sum is finite by monotonicity.

6.1.3. Compactification regularity, and associativity. The moduli spaces

M({γi}; γ0, x,F , A)

are identical to the moduli spaces in Sheridan [?] with respect to a system ofHamiltonian perturbations, and a domain dependent family of almost complexstructures on (in this case) trivial bundles, determined by F . To be more explicit,a Hamiltonian connection on a trivial M bundle over a surface S is the same as thedata of a 1-form on S with values in C∞

0 (M): mean 0 smooth functions. This isthe same as the data of a Hamiltonian perturbation. So in our case we just have alanguage change, the reason for which will be obvious when we shall construct thevalue of F on higher dimensional simplices of X. Consequently the compactificationand regularity story is word for word identical to Sheridan [?].

26 YASHA SAVELYEV

6.1.4. A∞ associativity. The maps µdF (Σ) satisfy the A∞-associativity equations

(stated over F2 for simplicity)

(6.4)∑n,m

µd−m+1Σ (γ1, . . . , γn, µ

mΣ (γn+1, . . . , γm+n), γn+m+1, . . . , γd) = 0,

This is shown as usual by considering boundary of the one dimensional modulispaces, M({γi}; γ0, x,F , A).

6.2. F on higher dimensional simplices. Let Σ : ∆n → X be smooth. Thecategory F (Σ) has objects

⊔i objF (xi)), where xi : pt → X is the composition of

the vertex inclusion ∆0 → ∆n, corresponding to the i’th corner, with the map Σ.Abusing notation we may also write xi for xi(pt) ∈ X.

Letm : [0, 1] → ∆n

be the edge between i, j corners of ∆n and set

m = Σ ◦m.

Given a pair of objects L0 ⊂ Pxi , L1 ⊂ Pxj , (including i = j) and the Hamiltonianconnection

A(L0, L1) = A(L0, L1,m)

on m∗P , determined by D, we define as before homF (Σ)(L0, L1) to be the Z2

graded chain complex over Q generated by the flat sections of (m∗P,A(L0, L1)),with boundary on the Lagrangian submanifolds L0 ∈ m∗P |0, L1 ∈ m∗P |1. The dif-ferential µ1 is defined identically to the differential on morphism spaces of categoriesFuk(Px). The only difference is that m∗P may no longer be naturally trivialized.

This completely describes all objects and morphisms of F (Σ). We now need todescribe the A∞ structure. Given {Lρ(k) ∈ F (xρ(k))}k=d

k=0,

ρ : {0, . . . , d} → {xi}i=ni=0 ,

with ω(Lρ(k)) = n ∈ Z, we need to define the higher composition maps

(6.5) µdΣ : hom(Lρ(0), Lρ(1))⊗ . . .⊗ hom(Lρ(d−1), Lρ(d)) → hom(Lρ(0), Lρ(d)).

Note that by construction, to each morphism of F (Σ) naturally correspondseither an edge or a vertex of ∆n, in either case we may naturally associate to thesea morphism in the groupoid Π(∆n). The collection {xρ(k)} then clearly determinesa composable chain (m1, . . . ,md) of morphisms in Π(∆n). Given the map

u(m1, . . . ,md, n) : S◦d → ∆n that is part of a natural system U ,

and given the system F determined by D, we define the moduli space

M({γi}; γ0,ΣnF , A)

analogously to (6.2). Explicitly, its elements are class A F({Li},Σ, r)-holomorphicsections (σ, r), r ∈ Rd of

Sr = S(m1, . . . ,md,Σ, r),

satisfying:• The boundary of (σ, r) is in the sub-fibration L(U , L0, . . . , Ld, r), see (5.1),

of Sr over the boundary of Sr.


• By assumptions, at the i’th end of Sr, in the strip coordinate chart e1i :[0, 1] × [1,∞] → Sr, the data F({Li},Σ, r) is R-translation invariant andwhen i 6= 0 we ask that σ is asymptotic to γi a geometric generator ofhomF (Σ)(Li−1, Li), or when i = 0 asymptotic in forward time to γ0 ageometric generator of homF (Σ)(L0, Ld).

6.2.1. Compactness and regularity. We do not need to reinvent the wheel prov-ing compactness and regularity results for the above moduli spaces. (Although itobviously works the same way.) Instead pick a Hamiltonian trivialization of

M ×∆n tr−→ Σ∗P,

then we may pullback the systems of connections F({Li},Σ, r), and the systemsof almost complex structures J ({Li},Σ) to M × ∆n. Then in the coordinatesof M × ∆n, these systems of connections and complex structures are essentiallyequivalent to systems of compatible perturbations, in the sense of Seidel [?], for theLagrangian boundary conditions given by {π ◦ tr−1(Li)} for

π : M ×∆n → M

the projection. (Only “essentially” because we now allow extra copies of the sameobject.) Consequently compactness and regularity works the same way as describedin Section 6.1.3.

6.2.2. Composition maps. As before given a basis of geometric generators {γij} for

homF (x)(Li−1, Li), i 6= 0, and {γ0j } for homF (x)(L0, Ld), i = 0, for d ≥ 2, and

assuming that F({Li},Σ, r) is regular we define µd by:

(6.6) 〈µdF (Σ)(γ

1j1 , . . . , γ

djd), γ0

j0〉 =∑A

#M(γ1j1 , . . . , γ

djd; γ0

j0 ,Σ,F , A),

when the above moduli spaces are of dimension 0, for 〈, 〉 as before. Again the sumis finite by monotonicity.

6.2.3. Associativity. This works as before.

Lemma 6.3. The assignment Σ 7→ F (Σ) extends to a natural functor

F : Simp(X) → A∞ − Cat.

Proof. Given a face map f : ∆n−1 → ∆n and Σn : ∆n → X, by the third naturalityproperty of our connections there is a canonical functor F (Σn◦f) → F (Σn) which isby construction a fully-faithful embedding. It follows via iteration that a morphismσ : Σk → Σl, with Σk,Σl ∈ Simp(X), k < l induces a fully-faithful embedding:

F (σ) : F (Σk) → F (Σl),

and this assignment is clearly functorial. Note that F (σ) is essentially surjectiveon the cohomological level, which follows by a classical continuation argument, cf.[?, Section 10a], and so each F (σ) is a quasi-equivalence. �

Let us call the functor FP,D : Simp(X) → A∞ − Cat, as constructed geometri-cally in this section, a geometric functor to emphasize the origin.

28 YASHA SAVELYEV

6.3. Unital replacement of F . Let A∞−Catunit denote the subcategory of A∞−Cat consisting of unital A∞ categories and unital functors. By unital replacementfor

F : Simp(X) → A∞ − Cat

we mean a functor

Funit : Simp(X) → A∞ − Catunit

together with a natural transformation

N : F → Funit,

which is object-wise quasi-equivalence.

Lemma 6.4. Any functor F : Simp(X) → A∞ − Cat has a unital replacement.

Proof. To obtain this we proceed inductively: for each 0-simplex x ∈ Simp(X),since each F (x) is c-unital we may fix a formal diffeomorphism Φx : F (x) → F (x),with first component maps Φ1

x the identity maps, such that induced A∞-structure

Funit(x) = Φ∗(F (x))

is strictly unital, [?, Lemma 2.1]. Let

Nx : F (x) → Funit(x)

denote the induced A∞ functor. Let Fk denote the restriction F to Simp≤k(X)with Simp≤k(X) denoting the sub-category of Simp(X), consisting of simpliceswhose degree is at most k. And suppose that the maps Nx can be extended to anatural transformation Nk of functors

Fk : Simp≤k(X) → A∞ − Cat,

Funitk : Simp≤k(X) → A∞ − Catunit,

where for each Σ Nk(Σ) : F (Σ) → Funit(Σ) is induced by a formal diffeomorphismΦΣ : F (Σ) → F (Σ), whose first component maps are the identity maps.

We construct an extension Nk+1. For each given Σk+1 : ∆k+1 → X and i : Σk →Σk+1, a morphism in Simp(X), by assumption F (i) is a fully-faithful embedding.Identifying F (Σk) with a full subcategory of F (Σk+1), we may clearly construct,as in the proof of [?, Lemma 2.1], a formal diffeomorphism

ΦΣk+1 : F (Σk+1) → F (Σk+1)

with Φ∗Σk+1(Σ

k+1) unital, and so that its restriction to F (Σk) coincides with theformal diffeomorphisms {ΦΣk+1◦i}, for each i : Σk → Σk+1. The result then follows.

�

Let us write Funit for the particular unital replacement of F as constructed inthe proof of the lemma above.


6.4. Naturality. Given a smooth embedding f : Y → X and M ↪→ P → X aHamiltonian bundle as before, there is an induced functor

f∗ : Simp(Y ) → Simp(X).

And consequently there is an associated pullback functor

f∗FP,D : Simp(Y ) → A∞ − Cat,

where FP,D : Simp(X) → A∞ − Cat is the geometric functor as above. On theother hand we may pullback by f the bundle as well as the perturbation data D toget another functor Ff∗P,f∗D : Simp(Y ) → A∞ −Cat. The following is immediatefrom construction.

Lemma 6.5. Ff∗P,f∗D = f∗FP,D.

6.5. Concordance classes of functors F : Simp(X) → A∞ −Cat. We say thata pair of functors F0, F1 : Simp(X) → A∞−Cat are concordant if there is a functor

T : Simp(X × I) → A∞ − Cat,

restricting to F0, F1 over Simp(X × {0}), respectively over Simp(X × {1}). Notethat by the proof of Lemma 6.4 if F1, F2 are concordant then so are Funit

1 , Funit2 .

Theorem 6.6. For a given pair of data D1,D2, the functors

FP,D1: Simp(X) → A∞ − Cat,

FP,D2 : Simp(X) → A∞ − Cat

are concordant.

Proof. A given pair of choices D1,D2 of auxiliary data are homotopic, in the naturalsense, through perturbation data {Dt} = ({Ut}, {Ft}), t ∈ I = [0, 1], by Theorem4.3. Let P → X × I be the Hamiltonian fibration given by pull-back of P by theprojection map X × I → X.

We have partial perturbation data {Dt} for P , which is defined for any

Σ : ∆n → X × {t} ⊂ X × I

by the data Dt. Then extend the perturbation data {Dt} in any way to all Simp(X×I), call this data H. The extension H clearly exists once one has an extension forthe natural system of maps {Ut}, since there are no obstructions at all for extending{Ft}. On the other hand the extension of {Ut} clearly exists by the same inductiveargument as in the construction of a natural system U .

Consequently we may define the functor T giving a concordance between

FP,D1, FP,D2

to be the geometric functor

FP ,H : Simp(X × I) → A∞ − Cat.

�

Remark 6.7. Concordance relation is an equivalence relation (in the special caseabove). Although we will not show this here. The concordance class of the functorFP,D is then the most fundamental invariant that is constructed in this paper,however calculating with it maybe very difficult.

30 YASHA SAVELYEV

7. Global Fukaya category

Concordance “classes” of functors FP,D constructed as above have a wealth ofinformation but we will cut down on this information by assigning a more geometricobject to it. We will assign to the concordance classes above equivalence classes ofcertain simplicial fibrations over X•, which are certain analogues of Serre fibrationsin the topological category.

The first necessary ingredient for this story is the notion of a quasi-category,which is a simplicial set with an additional property, relaxing the notion of Kancomplex. The latter are fibrant objects in the Quillen model structure on thecategory sSet of simplicial sets, and play the same role in the category of simplicialsets as CW complexes play in the category of topological spaces: they are the fibrantobjects in the corresponding Quillen equivalent model categories. Quasi-categoriesor alternatively called ∞− categories are in turn the fibrant objects for a differentnon Quillen equivalent model structure on sSet called the Joyal model structure.For reader’s convenience we will review some of this theory of simplicial sets in theAppendix.

7.1. The A∞-nerve. The A∞-nerve is an analogue for A∞ categories of the clas-sical nerve functor from the category of small categories to the category of simpli-cial sets, (in-fact quasi-categories). From now A∞-nerve will be just “nerve”: N ,where there can be no confusion. This construction is due to Lurie [?, Construction1.3.1.6], and the output is an ∞-category, or in the more specific model in this papera quasi-category. Thus, N is a functor from the category of all (strictly-unital) A∞categories, with morphisms A∞ unital functors, to the category of quasi-categories∞−Cat. More precisely Lurie discusses the case of dg-categories and only indicatesthe case of A∞ categories. A complete description of the nerve construction for A∞categories is contained in the thesis of Tanaka, [?], where it plays a central role,and is also carefully worked out in Faonte [?], where a number of properties areproved. We will reproduce it here for the reader’s convenience, a bit further on.

It should be noted that in the Lagrangian cobordism approach to Fukaya cat-egory in Nadler-Tanaka [?] a stable quasi-category Z is constructed directly. Thecategory Z is expected to be closely related to the nerve of the triangulated envelopeof the Fukaya category.

7.1.1. Outline of the (A∞)-nerve construction. The first step in the constructionof N is as follows. Let C be a strictly unital A∞ category. The 2-skeleton of thenerve N(C), has objects of C as 0-simplices, morphisms of C as 1-simplices andthe 2-simplices consist of a triple of objects X,Y, Z, a triple of morphisms

f ∈ homC(X), g ∈ homC(Y, Z), h ∈ homC(X,Z),

a morphism e ∈ homC(X,Z)1, with de = h− f ◦ g. We will describe the full nerveconstruction in the Appendix A following Tanaka [?].

7.2. Global Fukaya category. We will see in Section 8 how to enrich our con-struction so that our geometric functors F extend to functors

F : ∆/X• → A∞ − Catunit,

in other words so that degeneracies are included. Let us assume this for now.


Definition 7.1. We define:Fuk∞(P,D) := colim∆/X•N ◦ Funit ∈ sSet.

An explicit construction of the colimit is given in Lemma 9.2. In principle theabove definition could be very impractical since general objects in sSet are difficultto deal with, while taking fibrant replacements for the Joyal model category struc-ture could obfuscate all the original geometry contained in the Fukaya category.Thankfully none of this is necessary as we have a couple of miracles coming fromthe underlying geometry to save us. The proof of the following will be given inSection 9.

Theorem 7.2. As defined Fuk∞(P,D) ∈ ∞− Cat, i.e. is a quasi-category, more-over there is a natural (co)-Cartesian fibration

N(Fuk(M)) ↪→ Fuk∞(P,D) → X•,

whose equivalence class in the over category sSet/X• is independent of the choiceof D.

7.3. Universal construction. Let PU → BHam(M,ω) be the associated Hamil-tonian M -bundle to the universal principal Ham(M,ω)-bundle. BHam(M,ω) orthe classifying space of any smooth Lie group, based on Milnor’s construction [?],admits a well defined notion of smooth maps into it from smooth manifolds. To beprecise it has a natural diffeology, see Magnot-Watts [?] and likewise the universalM -bundle

EM → BHam(M,ω)

has a natural diffeology, so that for a diffeological mapf : B → BHam(M,ω)

the pull-back bundle f∗EM is naturally diffeological. If B is in a addition a smoothdimension k manifold then f∗EM is a diffeological space locally (diffeologically) dif-feomorphic to (U ⊂ B)×M , but the latter is clearly locally diffeomorphic to Rk+2n,for 2n the dimension of M . Thus f∗EM is a diffeological space locally diffeomor-phic to R2n and hence is a smooth manifold. More formally, f∗EM is contained inthe full subcategory of the category of diffeological spaces corresponding to smoothmanifolds.

In the case B = ∆k is the k-simplex, and given an open U with ∆k ⊂ U ⊂ Rk,by a diffeological map f : ∆k → BHam(M,ω) we mean a map with a diffeologicalextension f : U → BHam(M,ω), with U given the canonical diffeology. Then wemay conclude as above that f∗EM is naturally a smooth bundle. Now, formallyall our constructions are based on these smooth bundle. So that the constructionof Fuk∞(PU ) proceeds exactly the same way if we define the category of smoothsimplices in BHam(M,ω) to be the category of diffeological simplices.

Proof of Theorems 1.1, 1.3. By the above, Theorem 7.2 and Lurie’s straighteningCorollary A.5 uniquely determine the homotopy class of the “classifying” map

cl = cl(Fuk∞(PU )) : BHam(M,ω)• → S,of the co-Cartesian fibration Fuk∞(PU ). And so this determines a group homo-morphism

cl∗ : πi(BHam(M,ω), ∗) → πi(S, NFuk(M)).

32 YASHA SAVELYEV

�The following theorem shows that only the topological type of the Hamiltonian

fibration is detected by the associated (co)-Cartesian fibration.

Theorem 7.3. Let M ↪→ P1 → X, M ↪→ P2 → X be as before. Suppose that thereis a continuous Hamiltonian bundle map P1 → P2. Then

[cl(P1)] = [cl(P2)].

Proof. By the main result of [?], Pi, i = 1, 2 are classified by diffeological smoothmaps fi : X → BHam(M,ω). It then follows by Lemma 6.5 (and discussionabove) that Fuk∞(Pi), are equivalent to pull-backs by fi of Fuk∞(PU ). But fi arehomotopic so the conclusion follows. �

7.4. Global Fukaya category and unital replacement, a remark. In theconstruction of Fuk∞(P ) we had to take a unital replacement for the functor F :∆/X• → A∞ − Cat. One may worry then that this algebraic step will obfuscatethe “geometry” of simplices of Fuk∞(P ). This is not really the case. First the A∞nerve NC of a non-unital A∞ category C still exists as a semi-simplicial set, that isas a co-functor ∆inj → Set, with ∆inj the subcategory of ∆ consisting of injectivemorphisms. For a unital replacement equivalence C → Cunit of C, constructed asin Section 6.3, we then have an induced morphism of semi-simplicial sets NC →NCunit, which by construction induces a bijection NC([n]) → NCunit([n]), foreach [n]. So we may think without loss of geometric information, of simplicesof NCunit in terms of simplices of NC. (The former just have an extra formalalgebraic structure.)

8. Extending F to degeneracies

We have to construct our perturbation data D for all simplexes in such a waythat there is a natural functor:

(8.1) F : ∆/X• → A∞ − Catunit,

extending the geometric functor

FD : Simp(X) → A∞ − Catunit,

for this data D as previously constructed. This perturbation data will be referredto as extended perturbation data.

Suppose that we are given a commutative diagram:

∆0 ∆n+1 ∆n

X,

j+1

j∗Σ′

pr

Σ

wherepr : ∆n+1 → ∆n, j ∈ [n]

is induced by the unique surjection [n+ 1] → [n] hitting j twice. Here j∗ = Σ ◦ j,and j also denotes the map pt → ∆n corresponding to this vertex.

Let DΣ be some chosen perturbation data for Σ. Then we construct certaininduced perturbation data DΣ = pr∗DΣ. And using this we construct an A∞

category F (Σ).


We may already define the objects of F (Σ) as before. So pr clearly induces amap of sets of objects

pr∗ : objF (Σ) → objF (Σ).

We need to specify our system of connections and almost complex structures. Wesay what to do with connections, the case of almost complex structures is analogous.Given vertices i, j of ∆n+1 and objects L ∈ F (xi), L

′ ∈ F (xj) for xi = Σ(i), xj =

Σ(j) we setA(L,L′) = A(pr∗L, pr∗L

′),

where the latter connection is determined by the given DΣ, and where the equalityis with respect to the natural identification

(Σ ◦ pr ◦mi,j)∗P ' (Σ′ ◦mi,j)

∗P.

Likewise given objects L0, . . . , Ls ∈ F (Σ) we set

F(L0, . . . , Ls, Σ, r) = F(pr∗L0, . . . , pr∗Ls,Σn, r).

Where the equality is again with respect to the natural identification of the corre-sponding bundles. All together this determines a certain data DΣ.

Using this data DΣ we then define an A∞ category denoted by F (Σ) as previ-ously. By construction, the natural map on objects:

pr∗ : objF (Σ) → objF (Σ)

extends to a strict A∞ functor

F (pr) : F (Σ) → F (Σ)

satisfying:F (pr) ◦ F (σ) = id,

where σ : Σ → Σ′ is induced by the map dj+1 : [n] → [n + 1], which is the uniquemorphism in ∆ whose image misses j + 1. We are going to call the above theextension construction for Σ, DΣ .

We are now going to proceed by induction. Suppose that we have constructedour data D for all simplices up to degree N , and an extension of F |SimpN (X) to afunctor

FN : (∆/X•)N → A∞ − Catunit,

where (∆/X•)N and SimpN (X) are the subcategories of simplices of degree at most

N . Let Σ′ be a (N + 1)-simplex of X•. If Σ′ is degenerate, so that Σ′ = Σ ◦ pr forsome degeneracy morphism pr, for some Σ ∈ SimpN (X), then define the data DΣ′

and FN+1(Σ′) as in the extension construction above for Σ′,DΣ.Otherwise, if Σ is a non-degenerate (N+1)-simplex then its faces are N -simplices

for which we already have perturbation data D, which is then extended arbitrarilyto perturbation data DΣ for Σ. Using this data define FN+1(Σ) as normal. Thenclearly FN+1 extends FN , which completes the induction step. So that we obtainour extension (8.1).

9. Algebraic-topological considerations

In this section by equivalence of quasi-categories we always mean categoricalequivalence. This and other categorical preliminaries needed for this section arediscussed in the Appendix A. We will prove here Theorem 7.2.

34 YASHA SAVELYEV

9.1. Colimit of F . For F : ∆/X• → A∞ − Catunit as above, letFuk∞(F ) = colim∆/X•NF.

Proposition 9.1. There is a natural projection of simplicial setsp : Fuk∞(F ) → X•,

and this is a (co)-Cartesian fibration.

Proof. Recall that a given Σ : ∆n → X could equally be thought of as an elementof X•([n]) or of ∆/X•, that is as a natural transformation ∆n

• → X•. Let us firstgive a more easily conceptualized presentation of Fuk∞(F ). Define a partial order< on the set of pairs (f,Σ), f ∈ NF (Σ), Σ ∈ ∆/X• by

(f,Σ) < (f ′,Σ′)

if there is a morphismσ : Σ → Σ′

in Simp(X) induced by injective [n] → [m] with n ≤ m, i.e. a face morphism, s.t.NF (σ)(f) = f ′.

Clearly for every (f,Σ) there is a unique least pair(fmin,Σmin) < (f,Σ).

Let L be the set of minimal pairs. Define an equivalence relation on L by(f,Σ) ∼ (f ′,Σ′)

if there exists a degeneracy morphism d : Σ → Σ′ induced by [m] → [n] with m > n,such that

NF (d)(f) = f ′.

This is indeed an equivalence relation. Denote the equivalence class of (f,Σ) by[f,Σ]. Then L = L/ ∼ is naturally a simplicial set, with

L([k]) = {[f,Σ] ∈ L | f ∈ NF (Σ)([k])}.For example L([0]) is just

tx∈XObj F (x).

Lemma 9.2. L = Fuk∞(F ).

Proof. Note first that L is a co-cone on the diagram NF . Indeed, for each Σ defineφΣ : NF (Σ) → L by

φΣ(f) = [fmin,Σmin].

It is easy to see that for a face morphism i : Σ → Σ′ we have that the composition

NF (Σ)NF (i)−−−−→ NF (Σ′)

φΣ′−−→ L,

coincides with φΣ. Likewise for a degeneracy morphism d : Σ → Σ′ we have thatthe composition

NF (Σ)NF (d)−−−−→ NF (Σ′)

φΣ′−−→ L,

coincides with φΣ, because of the equivalence relation ∼.The universal property is also easy to verify, for given another co-cone L′ with

maps ρΣ : NF (Σ) → L′, Σ ∈ Simp(X) we can naturally define U : L → L′ byU([f,Σ]) = ρΣ(f).


Then U is clearly well defined, by L′ being a co-cone. And for a given f ∈ NF (Σ),we have

UφΣ(f) = ρΣmin(fmin) = ρΣf,

where the last equality holds since (L′, {ρΣ}) is a co-cone, and since by constructionthere is a morphism

i : NF (Σmin) → NF (Σ),

with F (i)fmin = f . �

For each Σ, we have a natural simplicial map

pΣ : NF (Σ) → Σ∗(∆n• ) ⊂ X•,

defined as follows. On the vertices of NF (Σ), pΣ is just the obvious projection. Nowa k-simplex f in NF (Σ) by definition determines a composable chain (f1, . . . , fk) inNF (Σ), and hence determines a sequence of vertices e0, . . . , ek s.t. the source/targetof fi is ei−1 respectively ei. This in turn determines a sequence of vertices {pΣ(ei)},and we set pΣ(f) to be the unique (possibly degenerate) k-simplex of Σ∗(∆

n• ) with

these vertices. We will omit the verification that pΣ is simplicial. The simplicialprojection

p : L → X•

is then: send [f,Σ] to pΣf , which is readily seen to be well defined.It is immediate from definitions that p is an inner-fibration if and only if the

pre-image of every simplex Σ : ∆n• → X• by p is a quasi-category, where “pre-

image” p−1(Σ) is the preimage by p of the simplicial subset Σ(∆n• ). In our case

this follows by construction as the preimage of Σ is clearly identified with NF (Σ).Let m be an edge in X• from x0 to x1 and let Lm = p−1m and let Li = p−1xi, seedefinition of pre-image above. To show that p is a (co)-Cartesian fibration, by [?,Proposition 2.4.1.5] it is enough to show that for every such m and a ∈ L1 there isan equivalence ea ∈ Lm with target a, with p(ea) = m.

Lemma 9.3. The functor N : A∞ − Catunit → Cat−∞, takes quasi-equivalencesto weak equivalences in the Joyal model structure, i.e. categorical equivalences.

Proof. The proof of this is contained in the proof of proposition 1.3.1.20, Lurie [?].We can also prove this directly by first recalling that quasi-equivalences of A∞-categories A,B are invertible, up to homotopy, and then via the nerve constructiontranslate this to a categorical equivalence of N(A), N(B). �

Then by construction of L the inclusion of Li into Lm are equivalences of quasi-categories, and so ea as above must exist. �

Proof of Theorem 7.2. By the discussion above we have a (co)-Cartesian fibration

Fuk∞(P,D) → X•.

The first part of the theorem follows by the following general fact: for an innerfibration of simplicial sets

p : P• → X•,

if X• is a quasi-category then P• is a quasi-category. Let us prove this elementarypoint. Suppose we are given

ρ : Λnk → P•

36 YASHA SAVELYEV

for 0 < k < n. As X• is a quasi-category there a simplex

ρ : ∆n• → X•

extending p ◦ ρ. But then ρ maps into the quasi-category p−1(ρ), and consequentlythere is an extension of ρ, c.f. Proposition A.3.

The final part of the theorem follows by the following.

Lemma 9.4. For the geometric functor FP,D the equivalence class of the fibrationp : Fuk∞(P,D) → X•, in the over category sSet/X•, is independent of the choiceof D.

Proof. By Theorem 6.6 and previous discussion for any pair Di of perturbationdata there exists a (co)-Cartesian fibration:

T → X• × I•,

whose restriction over X• × ∂I• coincides with

Fuk∞(P,D1) t Fuk∞(P,D2),

The lemma then follows by Lurie’s straightening theorem A.4, or more simply byCorollary A.5. �

�

9.2. Naturality. We may expect if our constructions are really natural that Fuk∞(P,D)is functorial with respect to pull-back and this is indeed the case. Let f : X → Ybe a smooth map, P → Y a smooth Hamiltonian fibration and D extended pertur-bation data. We may then naturally define pull-back extended perturbation dataf∗D for f∗P → X. Specifically, if Σ ∈ X• then we set

F(L0, . . . , Ls,Σ, r) = F(L0, . . . , Ls, Σ, r),

for Σ = f ◦ Σ.Let f• : X• → Y• be the induced map of simplicial sets.

Theorem 9.5. Fuk∞(f∗P, f∗D) = f∗•Fuk∞(P, f∗D), where f∗D is as above.

Proof. The proof is immediate. �

Appendix A. Quasi-categories and Joyal model structure

We don’t need absolutely everything in this section, particularly we can avoidever mentioning model categories, but the latter helps with the narrative. A verygood concise reference for much of this material is Riehl [?], which we will mostlyfollow. The material on co-Cartesian fibrations is taken from Lurie [?, Section2.4]. First let us recall the notion of a Kan complex, which maybe thought of asformalizing the property of a simplicial set to be like the singular set of a topologicalspace.

Let ∆n be the standard representable n-simplex: ∆([i]) = ∆([i], [n]). Previouslywe denoted this by ∆n

• , but as there are no topological simplices in this section wesimplify the notation, which is also consistent with above references. Let Λn

k ⊂ ∆n

denote the sub-simplicial set corresponding to the “boundary” of ∆n with the k’thface removed, 0 ≤ k ≤ n. By k′th face we mean the face opposite to k’th vertex.


A simplicial set S• is said to be a Kan complex if for all n, k given a diagramwith solid arrows

Λnk S•

∆n ,

there is a dotted arrow making the diagram commute.A quasi-category is a simplicial set S• for which the above extension property

is only required to hold for inner horns Λnk , i.e. those horns with 0 < k < n. A

morphism between quasi-categories is just a simplicial map. We will denote quasi-categories by calligraphic letters e.g. B. The full-subcategory of sSet with objectsquasi-categories will be denoted by ∞− Cat.

A.1. Categorical equivalences, morphisms and equivalences. We have anatural functor τ : sSet → Cat. τ(S•) is the category with objects 0-simplicesof S•, 1-simplices as morphisms, degenerate 1-simplices as identities and freely gen-erated composition subject to the relation g = f ◦ h if there is a 2-simplex e with0-face h, 2-face f and 1-face g. (Remembering our diagrammatic order for com-position.) We then have a functor τ0 : sSet → Set by sending A• to the set ofisomorphism classes of objects in τA•. If S• = X is a quasi-category an edge e ∈ Xis said to be an equivalence or an isomorphism if τ0(e) is an isomorphism in τX .The maximal Kan subcomplex of a quasi-category X is the sub-complex obtainedby removing edges which are not isomorphisms.

Notation A.1. The category ∞−Cat naturally has a structure of a quasi-categoryitself see [?, Chapter 3], we shall denote its maximal Kan subcomplex by S.

We define sSetτ0 to be the category with the same objects as sSet but withthe morphisms given by sSetτ0(A•, B•) = τ0(BA•

• ). A map of simplicial sets u :A• → B• is said to be a categorical equivalence if the induced map in sSetτ0 is anequivalence. It is said to be a weak equivalence if the pull-back map

sSetτ0(B•, X•) → sSetτ0(A•, X•)

induced by u is an equivalence for all X•. A categorical equivalence is necessarilya weak categorical equivalence. Conversely a weak categorical equivalence betweenquasi-categories is necessarily a categorical equivalence. We will say that a pair ofquasi-categories are equivalent if there is categorical equivalence between them.

As we are following Riehl [?], we refer the reader there for the following:

Theorem A.2 (Joyal, Lurie, Riehl). There is a model structure on sSet, withweak equivalences weak categorical equivalences and cofibrations monomorphisms.Moreover the fibrant objects are quasi-categories.

A.2. Inner fibrations. A map p : A → B of quasi-categories is said to be an innerfibration if it has the lifting property with respect to all inner horn inclusions. Morespecifically if for 0 < k < n whenever we are given a commutative diagram with

38 YASHA SAVELYEV

solid arrows:

(A.1)

Λnk A

∆n B,

p

there exists a dashed arrow as indicated, making the whole diagram commutative.For reference p is said to be an Kan fibration if the above extension property holdsfor all horns. A Kan fibration is an analogue in the simplicial world of Serrefibrations of topological spaces. The following is immediate from definitions.

Proposition A.3. A map p : A → B is an inner fibration, if and only if thepre-image of every simplex of B is a quasi-category.

A.3. (co)-Cartesian fibrations. These are the analogue in the quasi-categoriesworld of Grothendieck fibrations. We will explain the co-Cartesian version, as theother is just dual to it.

Given p : A → B, an edge f : ∆1 → A is said to be co-Cartesian if whenever weare given a diagram with solid arrows:

(A.2)

∆n0,1

Λn0 A

∆n B,

f

p

there is a dashed arrow as indicated making the diagram commutative. Here ∆n0,1

denotes the “edge” (sub-simplical complex) joining the vertexes 0, 1.The map p : A → B is said to be a co-Cartesian fibration, if it is an inner

fibration and if for every edge e : ∆1 → B, with co-domain b, and every b lifting b

there is a co-Cartesian lift e : ∆1 → A, with co-domain b.Denote by coCFib(B) the quasi-category of (co)-Cartesian fibrations over B,

which by definition is the full-subcategory of the over-category Cat∞/B, with ob-jects (co)-Cartesian fibrations.

Theorem A.4. [?, Theorem 3.2.01] Straightening theorem. There is equivalenceof quasi-categories Fun(B, Cat∞) ' coCFib(B).

Stated more properly this combines [?, Theorem 3.2.01] and [?, Proposition3.1.5.3], both of which are statements on the level of model categories. When B is aKan complex the notions of Cartesian and co-Cartesian fibrations over B coincideand the model category presenting coCFib(B) is just the over category sSet/B withthe induced Joyal model structure. The fibrant objects in this model structure arethe (co)-Cartesian fibrations over B. The term “presents” here means that theunderlying quasi-category is the simplicial nerve of the Dwyer-Kan [?] simpliciallocalization of the model category. Thus we simplify the above as follows for ourneeds in this paper. We say that a pair of co-Cartesian fibrations P1 → B, P2 → B


are equivalent if there is an (categorical) equivalence of quasi-categories P1 → P2

over B. The set of such equivalence classes is formally τ0CFib(B).

Corollary A.5. For a Kan complex X•, the set τ0CFib(X•) is naturally identifiedwith the set of homotopy classes of maps X• → S, where S denotes the maximalKan subcomplex of the quasi-category ∞− Cat.

A.4. Semi-locality. The following is only used in Part II, [?]. Suppose that S• ⊂X• is a Kan sub-complex, whose inclusion map is a weak equivalence.

Lemma A.6. The restriction functor

τCFib(X•) → τCFib(S•),

is an isomorphism, and so the restriction map

τ0CFib(X•) → τ0CFib(S•),

is a set-isomorphism.

Proof. By Lurie’s straightening theorem A.4, this is equivalent to the restrictionfunctor

(A.3) τFunc(X•,∞− Cat) → τFunc(S•,∞− Cat),

being an isomorphism. But X• and S• are Kan complexes and so

τFunc(X•,∞− Cat) ' τFunc(X•,S) ' hoTop(|X•|, |S|),

where | · | denotes geometric realization functor and where hoTop denotes the ho-motopy category of topological spaces. The last equivalence is due to the following.Func(X•,S) is a Kan complex as it is the mapping space of Kan complexes, thenobserve that for Kan complexes τ0 is just the functor of connected components, nextuse that the geometric realization | · |, and singular set functors induce a derivedQuillen equivalence between hoTop and ho sSet.

Similarly

τFunc(S•,∞− Cat) ' τFunc(S•,S) ' hoTop(|S•|, |S|).

The inclusion |S•| → |X•| ' X is a homotopy equivalence, as the inclusion S• ⊂ X•is a weak equivalence by assumption and since geometric realization has image inCW complexes. It follows that (A.3) is an isomorphism.

�

Using this we may compute the class of global Fukaya category of M ↪→ P → Xin τ0CFib(X•), by restricting F to a sub-category ∆/S•, with S• ⊂ X• minimalKan sub-complex generated by the sub-simplicial set of X• corresponding to somesmooth triangulation of X. This is in principle finite local data if X is compact,and in general locally finite. This plays a role in the calculation in part II.

A.5. A∞-nerve. This is a natural analogue for A∞ categories of the dg-nerve con-struction of Lurie, [?], which is a functor

N : dg − Cat → ∞− Cat.

This section follows Tanaka [?, 2.3], except that for us everything will be ungraded,and for simplicity with F2-coefficients.

40 YASHA SAVELYEV

For [n] ∈ ∆, a length s wedge decomposition of [n] is a collection of monomor-phisms in ∆

ji : [ni] → [n], i = 1, . . . , s, |ni + 1| ≥ 2,

such that the fiber product

[ni]×[n] [ni+1] ' [0]

and the canonical projection

[ni]×[n] [ni+1] → [ni]

is the map [0] → [ni] sending {0} to {ni} ⊂ [ni]. Here we are thinking of [n] asthe totally ordered finite set {0, . . . , n}. We denote the set of all decompositionsof [n] by D[n]. We may of course equally understand a length s decomposition asfinite set {J1, . . . Js} of subsets of [n] decomposing [n]. However it is a bit simplerto formulate the following in terms of the maps fi.

Definition A.7. For A a small unital A∞ category its nerve N(A) is a simplicialset with the set of vertices the set of objects of A. A n-simplex f of N(A) consistsof the following data:

• A map [n] → ObA. We denote the corresponding objects X0, . . . , Xn.• For each mono-morphism j : [nj ] → [n] in ∆ with |nj | ≥ 2 an element

fj ∈ homA(Xj(0), Xj(nj)).

We may completely characterize each such j by its image set, and will some-times write j for the corresponding set and vice versa, thus f[n] correspondsto the identity j : [n] → [n].

• For a given j : [nj ] → ∆n, denote by j − {i} : [nj − 1] → [n] the morphismwith image set j−{i}. Then the collection of these fj is required to satisfythe following equation:

(A.4) µ1(fj) =∑

0<i<nj

fj−{i} +∑s

∑decomps∈D[nj ]

µs(fj1 , . . . , fjs),

with decomps ∈ D[nj ] denoting a length s decomposition and {fji} itselements.

The simplicial maps are as follows. Given an injection k : [m] → [n] and an nsimplex f , for j : [l] → [m] an injection, define an m-simplex f ′ by {f ′

j = fk◦j}.On the other hand given si : [n+1] → [n], si(i+1) = si(i), and an n-simplex f ,

define an (n+ 1)-simplex f ′ by setting

fj =

eXiif j = {i, i+ 1} ;

fsi◦j if si|j is injective.0 otherwise,

for j : [l] → [n+1] an injection. It is straightforward but tedious to verify that thelatter is indeed a face and that simplicial relations are satisfied.

Proposition A.8. [?, 2.3.2], [?] For A a unital A∞ category its nerve A = N(A)is a quasi-category.

For the reader’s convenience we outline the proof here.


Proof. Suppose we have an inner horn ρk : Λnk → A, in particular this determines

morphisms in A: fj , for all j except j = [n] − {k} as part of the structure of theface simplices. Set f[n] = 0, and set

f[n]−{k} =∑

0<i<n;i6=k

f[n]−{i} +∑s

∑decomps∈D[n]

µs(fj1 , . . . , fjs),

then by construction (A.4) is satisfied for the collection of maps {fj}. Only thingleft to check is that as defined f[n]−{k} actually determines the k’th face of oursimplex. A direct calculation for this is long but straightforward, using the A∞associativity equations. For n = 2 this is automatic and for n = 3 this can bechecked in a few lines. �

For F : A → B an A∞ functor we define NF : NA → NB via the assignment:

fj 7→∑

decomps∈D[nj ]

F s(fj1 , . . . , fjs).

Lemma A.9. [?], [?] The assignment A 7→ NA, and F 7→ NF as above, determinesa functor

N : A∞ − Catunit → ∞− Cat.

The details on why this constitutes a functor N are omitted.University of Colima, CUICBASEmail address: [email protected]

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