2.1 Categories

In this paper we will work in the setting of dg categories. The theory of ordinary triangulated categories suffers from limitations which make it unsuitable to many applications in modern geometry. These limitations are essentially of two kinds. First, there are issues arising when working within a given triangulated category, and that originate from the lack of functorial universal constructions, such as cones. Additionally, triangulated categories do not themselves form a well-behaved category. Among other things, this implies that we have no workable notion of limits and colimits of triangulated categories, which are essential to perform gluing of categories.

The theory of dg categories gives us means to overcome both kinds of limitations. As opposed to ordinary triangulated categories, dg categories fit inside homotopically enriched categories and this allows us to perform operations such as taking limits and colimits. There are two main ways to define dg categories and to understand the structure of the category that they form.

1. (i) Dg categories can be defined as ordinary categories strictly enriched over chain complexes. Dg categories then form an ordinary category equipped with a model structure. In fact, there are several meaningful options for the model structure: the one that is most relevant for us is the Morita model structure considered in [Reference TabuadaTab05] and [Reference ToënToe07].

2. (ii) Dg categories can also be defined as $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories: that is, as $\infty$ -categories that carry an action of the symmetric monoidal $\infty$ -category of chain complexes. Following this definition, dg categories form naturally an $\infty$ -category. This approach is taken up for instance in the important recent reference [Reference Gaitsgory and RozenblyumGR17].

Approaches (i) and (ii) each have several distinct advantages. It is often useful to have recourse to both viewpoints simultaneously, and this is what we shall do in the present paper. Categories strictly enriched over chain complexes are much more concrete objects, and their homotopy limits in the model category of dg categories can be calculated explicitly. On the other hand, perspective (ii) gives rise to more natural definitions and constructions and puts at our disposal the comprehensive foundations on $\infty$ -categories provided by the work of Lurie [Reference LurieLur09] and Gaitsgory–Rozenblyum [Reference Gaitsgory and RozenblyumGR17].

In fact, there is a well-understood dictionary between these two viewpoints. Inverting weak equivalences in the model structure of dg categories (i) yields an $\infty$ -category which is equivalent to the $\infty$ -category of dg categories (ii). As a consequence homotopy limits and colimits match, via this correspondence, their $\infty$ -categorical counterparts. A proof of this equivalence was given by Cohn [Reference CohnCoh13] and then in a more general context by Haugseng [Reference HaugsengHau15] (see § 2.1.1 below). Throughout the paper we adopt the flexible $\infty$ -categorical formalism coming from viewpoint (ii) while performing all actual calculations (e.g. of limits) in the model category of dg categories made available by viewpoint (i). We shall give more precise pointers to the literature below.

Before proceeding, however, we remark that we will be mostly interested in $\mathbb{Z}_{2}$ -graded, rather than $\mathbb{Z}$ -graded, dg categories. Some of the results which we will recall in the following have not been stated explicitly in the literature in the $\mathbb{Z}_{2}$ -graded setting. However, they can all be transported to the $\mathbb{Z}_{2}$ -graded setting without variations. We refer to Section 1 of [Reference Dyckerhoff and KapranovDK18] and Section 2 of [Reference DyckerhoffDyc17] for a detailed summary of the Morita homotopy theory of $\mathbb{Z}_{2}$ -graded dg categories, which is closely patterned on the $\mathbb{Z}$ -graded theory developed in [Reference ToënToe07].

The standard reference for $\infty$ -categories is [Reference LurieLur09]. We will follow closely the treatment of dg categories given in [Reference Drinfeld and GaitsgoryDG15] and [Reference Gaitsgory and RozenblyumGR17, I.1.10.3], to which we refer the reader for additional details. In particular, the notation and basic facts below are all taken from these two references. The only difference is that we will systematically place ourselves in the $\mathbb{Z}_{2}$ -graded setting.

We denote by $\text{Vect}^{(2)}$ the presentable, stable and symmetric monoidal $\infty$ -category of $\mathbb{Z}_{2}$ -periodic chain complexes of $\unicode[STIX]{x1D705}$ -vector spaces. We denote by $\text{Vect}^{(2),\text{fd}}$ its full subcategory of compact objects. For the rest of this section we will call:

1. – $\infty$ -categories tensored over $\text{Vect}^{(2),\text{fd}},$ $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories;

2. – cocomplete $\infty$ -categories tensored over $\text{Vect}^{(2)},$ cocomplete $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories.

These are dg categories in the sense of viewpoint (ii). We introduce the following notation.

1. – We denote by $\text{DGCat}^{(2),\text{non\text{-}cocmpl}}$ the $\infty$ -category of (not necessarily small) $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories.

2. – We denote by $\text{DGCat}_{\text{cont}}^{(2)}$ the $\infty$ -category of cocomplete $\unicode[STIX]{x1D705}$ -linear graded dg categories, and continuous functors between them; $\text{DGCat}_{\text{cont}}^{(2)}$ carries a symmetric monoidal structure, and we denote its internal Hom by $\text{Fun}_{\text{cont}}(-,-)$ .

3. – We denote by $\text{DGCat}_{\text{small}}^{(2)}$ the $\infty$ -category of small $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories; $\text{DGCat}_{\text{small}}^{(2)}$ also carries a natural symmetric monoidal structure and we denote its internal Hom by $\text{Fun}(-,-)$ .

4. – If $\text{C}$ is an object in $\text{DGCat}_{\text{cont}}^{(2)}$ , we denote by $\text{C}^{\unicode[STIX]{x1D714}}$ its subcategory of compact objects,

   $$\begin{eqnarray}\text{C}^{\unicode[STIX]{x1D714}}\in \text{DGCat}_{\text{small}}^{(2)}.\end{eqnarray}$$

We have the forgetful and the Ind-completion functors

$$\begin{eqnarray}\text{U}:\text{DGCat}^{(2),\text{non\text{-}cocmpl}}\longrightarrow \text{DGCat}_{\text{cont}}^{(2)}\quad \quad \text{Ind}:\text{DGCat}^{(2),\text{non\text{-}cocmpl}}\longrightarrow \text{DGCat}_{\text{cont}}^{(2)}.\end{eqnarray}$$

The Ind-completion functor can be defined, on objects, via the formula (see [Reference Gaitsgory and RozenblyumGR17, I.1], Lemma 10.5.6)

$$\begin{eqnarray}\text{C}\in \text{DGCat}^{(2),\text{non\text{-}cocmpl}}\mapsto \text{Ind}(\text{C})=\text{Fun}_{\unicode[STIX]{x1D705}}(\text{C}^{\text{op}},\text{Vect}^{(2)})\in \text{DGCat}_{\text{cont}}^{(2)},\end{eqnarray}$$

where $\text{Fun}_{k}(-,-)$ denotes the $\infty$ -category of $\text{Vect}^{(2),\text{fd}}$ -linear functors.

Remark 2.1. In the main text we will apply the Ind-completion functor only to small $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories: that is, to objects in the full subcategory

$$\begin{eqnarray}\text{DGCat}_{\text{small}}^{(2)}\subset \text{DGCat}^{(2),\text{non\text{-}cocmpl}}.\end{eqnarray}$$

2.1.1 Rigid models

As we have already mentioned, the two viewpoints (i) and (ii) on foundations of dg categories which we discussed in § 2.1 can be fully mapped onto each other. This will allow us to leverage the computational power of model structures, while retaining at the same time the flexibility of the $\infty$ -categorical formalism. A precise formulation of the equivalence between perspectives (i) and (ii) was given in work of Cohn [Reference CohnCoh13]. We recall this below with the usual proviso that we will state it in the $\mathbb{Z}_{2}$ -graded setting.

We will sometimes refer to categories strictly enriched over chain complexes as (ordinary) dg categories in order to differentiate them from $\unicode[STIX]{x1D705}$ -linear $\infty$ -categories. Let $\text{dgCat}_{\unicode[STIX]{x1D705}}^{(2)}$ be the model category of small categories strictly enriched over chain complexes equipped with the Morita model structure [Reference ToënToe07]. Let ${\mathcal{W}}$ be the set of weak equivalences for the Morita model structure: these are dg functors which induce quasi-equivalences at the level of ( $\mathbb{Z}_{2}$ -graded) module categories.

Proposition 2.3 [Reference CohnCoh13, Corollary 5.5].

There is an equivalence of $\infty$ -categories

$$\begin{eqnarray}N(\text{dgCat}_{\unicode[STIX]{x1D705}}^{(2)})[{\mathcal{W}}^{-1}]\simeq \text{DGCat}_{\text{small}}^{(2)},\end{eqnarray}$$

where $N(-)$ is the nerve.

We will also need an analogue of Cohn’s result in the setting of large categories, that is for $\text{DGCat}_{\text{cont}}^{(2)}$ . Let us denote by $\text{dgCat}_{\unicode[STIX]{x1D705},\text{cont}}^{(2)}$ the strictly enriched analogue of $\text{DGCat}_{\text{cont}}^{(2)}$ . This is the Morita model category of cocomplete categories strictly enriched over $\unicode[STIX]{x1D705}-\text{mod}^{(2)}$ . Then the analogue of Proposition 2.3 is discussed as Example 5.11 of [Reference HaugsengHau15]. In many situations having recourse to the model category of dg categories has great computational advantages. In particular, we will be able to compute $\infty$ -categorical (co)limits in $\text{DGCat}_{\text{cont}}^{(2)}$ and $\text{DGCat}_{\text{small}}^{(2)}$ by calculating the respective homotopy (co)limits of the strict models in $\text{dgCat}_{\unicode[STIX]{x1D705},\text{cont}}^{(2)}$ .

Working with rigid models is often also helpful when considering commutative diagrams, as it allows us to sidestep issues of homotopy coherence in $\infty$ -categories. Recall that a diagram of ordinary dg categories

commutes if there is a natural transformation $\unicode[STIX]{x1D6FC}:H\circ F\Rightarrow K\circ G$ , which becomes a natural equivalence when passing to homotopy categories. All commutative diagrams of dg categories in this paper will be understood as coming from commutative diagrams of ordinary dg categories as above.

2.1.2 Limits of dg categories

Homotopy limits of dg categories can be calculated explicitly. Colimits of dg categories are, on the other hand, quite subtle. It is therefore an important observation that, under suitable assumptions, colimits can actually be turned into limits. This key fact can be formulated in various settings. For instance, in the context of colimits of presentable $\infty$ -categories such a result follows from Propositions 5.5.3.13 and 5.5.3.18 of [Reference LurieLur09]. We will recall a formulation of this fact in the setting of dg categories due to Drinfeld and Gaitsgory. We follow closely § 1.7.2 of [Reference Drinfeld and GaitsgoryDG15], with the usual difference that we adapt every statement to the $\mathbb{Z}_{2}$ -graded setting.

Let $I$ be a small $\infty$ -category, and let $\unicode[STIX]{x1D6F9}:I\rightarrow \text{DGCat}_{\text{cont}}^{(2)}$ be a functor. For every pair of objects $i,j\in I$ and morphism $i\rightarrow j,$ we set

$$\begin{eqnarray}C_{i}:=\unicode[STIX]{x1D6F9}(i)\quad C_{j}:=\unicode[STIX]{x1D6F9}(j)\quad \unicode[STIX]{x1D713}_{i,j}:=\unicode[STIX]{x1D6F9}(i\rightarrow j):C_{i}\rightarrow C_{j}.\end{eqnarray}$$

Assume that each functor $\unicode[STIX]{x1D713}_{i,j}$ admits a continuous right adjoint $\unicode[STIX]{x1D719}_{j,i}$ . Then there is a functor $\unicode[STIX]{x1D6F7}:I^{\text{op}}\rightarrow \text{DGCat}_{\text{cont}}^{(2)}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}(i)=C_{i}\quad \text{and}\quad \unicode[STIX]{x1D6F7}(i\rightarrow j)=\unicode[STIX]{x1D719}_{j,i}.\end{eqnarray}$$

Proposition 2.4 [Reference Drinfeld and GaitsgoryDG15, Proposition 1.7.5].

The colimit

$$\begin{eqnarray}\underset{i\in I}{\varinjlim }C_{i}:=\underset{I}{\varinjlim }\unicode[STIX]{x1D6F9}\in \text{DGCat}_{\text{cont}}^{(2)}\end{eqnarray}$$

is canonically equivalent to the limit

$$\begin{eqnarray}\underset{i\in I^{\text{op}}}{\varprojlim }C_{i}:=\underset{I}{\varprojlim }\unicode[STIX]{x1D6F7}\in \text{DGCat}_{\text{cont}}^{(2)}.\end{eqnarray}$$

2.1.3 Sheaves of dg categories

We will make frequent recourse to the concept of sheaf of dg categories either on graphs, or on schemes equipped with the Zariski topology. For our purposes it will be enough to use a rather weak notion of sheaf, which does not take into consideration hypercovers. Let ${\mathcal{F}}$ be a presheaf of dg categories over a topological space $X$ . We say that ${\mathcal{F}}$ satisfies Čech descent if, for all open subsets $U\subset X$ and covers ${\mathcal{U}}=\{{U_{i}\}}_{i\in I}$ of $U$ , the restriction functor

(3)

realizes ${\mathcal{F}}(U)$ as the limit of the semi-cosimplicial object obtained by evaluating ${\mathcal{F}}$ on the Čech nerve of ${\mathcal{U}}$ . If ${\mathcal{F}}$ satisfies Čech descent, we say that ${\mathcal{F}}$ is a sheaf of dg categories. The dual notion of cosheaf of dg categories is defined by reversing all the arrows in (3).

There are two conditions under which (3) can be simplified, and that will occur in the examples we will be considering in the rest of the paper:

1. (a) when triple and higher intersections of the cover ${\mathcal{U}}$ are empty;

2. (b) when ${\mathcal{F}}$ evaluated on the triple and higher intersection of the cover ${\mathcal{U}}$ is equivalent to the zero category.

We remark that condition (a) is in fact a special case of condition (b), but it is useful to keep them distinct. In this paper we will consider (co)sheaves of categories on graphs, which have covering dimension one and where therefore condition (a) is automatically satisfied. Also, we will consider sheaves of matrix factorizations on three-dimensional toric varieties equipped with the Zariski topology: there, although triple overlaps will not be empty, condition (b) will apply.

Let ${\mathcal{U}}=\{U_{1},\ldots ,U_{n}\}$ be an open cover of $U$ and assume that either condition (a) or (b) are satisfied. Then the sheaf axiom (3) is equivalent to requiring that ${\mathcal{F}}(U)$ is the limit of the following much smaller diagram, which coincides with the truncation of the Čech nerve encoding the sheaf axiom in the classical setting of sheaves of sets

(4)

The same reasoning applies to cosheaves of categories, with the only difference that the arrows in (4) have to be reversed.

We will always work in settings where either (a) (in the case of the topological Fukaya category) or (b) (in the case of matrix factorizations) are satisfied, and therefore we always implicitly reduce to (4).

2.1.4 Schemes and stacks

If $X$ is a scheme or stack we denote by

$$\begin{eqnarray}{\mathcal{P}}\text{erf}^{(2)}(X)\in \text{DGCat}_{\text{small}}^{(2)},\quad {\mathcal{Q}}\text{Coh}^{(2)}(X)\in \text{DGCat}_{\text{cont}}^{(2)}\end{eqnarray}$$

the $\mathbb{Z}_{2}$ -periodization of the dg categories of perfect complexes and of quasi-coherent sheaves on $X$ . We refer the reader to § 1.2 of [Reference Dyckerhoff and KapranovDK18] for the definition of $\mathbb{Z}_{2}$ -periodization. All schemes appearing in this paper will be quasi-compact and with affine diagonal. All DM stacks will be global quotients of such schemes by affine groups. Using the terminology introduced in [Reference Ben-Zvi, Francis and NadlerBFN10], these are all examples perfect stacks. This implies in particular that, if $X$ is such a scheme or stack, the category of quasi-coherent sheaves over $X$ is equivalent to the Ind-completion of its category of perfect complexes

$$\begin{eqnarray}{\mathcal{Q}}\text{Coh}^{(2)}(X)\simeq \text{Ind}({\mathcal{P}}\text{erf}^{(2)}(X)).\end{eqnarray}$$

2.2 Ribbon graphs

For a survey of the theory of ribbon graphs see [Reference Mulase and PenkavaMP98], and § 3.3 of [Reference Dyckerhoff and KapranovDK18]. We will just review some standard terminology. A graph $X$ is a pair $(V,H)$ of finite sets equipped with the following extra data:

1. – an involution $\unicode[STIX]{x1D70E}:H\rightarrow H$ ;

2. – a map $I:H\rightarrow V$ .

We call $V$ the set of vertices, and $H$ the set of half-edges. Let $v$ be a vertex. We say that the half-edges in $I^{-1}(v)$ are incident to $v$ . The cardinality of $I^{-1}(v)$ is called the valency of the vertex  $v$ . The edges of $X$ are the equivalence classes of half-edges under the action of $\unicode[STIX]{x1D70E}$ . We denote by $E$ the set of edges of $\unicode[STIX]{x1D70E}$ . The set of external edges of $X$ is the subset $E^{o}\subset E$ of equivalence classes of cardinality one, which correspond to the fixed points of $\unicode[STIX]{x1D70E}$ . The internal edges of $X$ are the elements of $E-E^{o}$ . Subdividing an edge $e$ of $X$ means adding to $X$ a two-valent vertex lying on $e$ . More formally, let $e$ be equal to $\{h_{1},h_{2}\}\subset H$ . We add a new vertex $v_{e}$ to $V$ , and two new half-edges $h_{1}^{\prime }$ and $h_{2}^{\prime }$ to $H$ . We modify the maps $\unicode[STIX]{x1D70E}$ and $I$ by setting

$$\begin{eqnarray}\unicode[STIX]{x1D70E}(h_{1})=h_{1}^{\prime },\quad \unicode[STIX]{x1D70E}(h_{2})=h_{2}^{\prime },\quad I(h_{1}^{\prime })=I(h_{2}^{\prime })=v_{e}.\end{eqnarray}$$

It is often useful to view a graph as a topological space. This is done by modeling the external and the internal edges of $G$ , respectively, as semiclosed and closed intervals, and gluing them according to the incidence relations. We refer to this topological model as the underlying topological space of $X$ . When talking about the embedding of a graph $X$ into a topological space, we always mean the embedding of its underlying topological space.

If a graph $X$ is embedded in an oriented surface it acquires a canonical ribbon graph structure by ordering the edges at each vertex counter-clockwise with respect to the orientation. Conversely, it is possible to attach to any ribbon graph $X$ a noncompact oriented surface inside which $X$ is embedded as a strong deformation retract. See [Reference Mulase and PenkavaMP98] for additional details on these constructions. If $\unicode[STIX]{x1D6F4}$ is a Riemann surface, a skeleton or spine of $\unicode[STIX]{x1D6F4}$ is a ribbon graph $X$ together with an embedding $X\rightarrow \unicode[STIX]{x1D6F4}$ as a strong deformation retract.

3.1 B-branes

3.1.1 Toric Calabi–Yau threefolds

Let $\widetilde{N}$ be a $(n-1)$ -dimensional lattice, and let $P$ be a lattice polytope in $\widetilde{N}_{\mathbb{R}}=\widetilde{N}\otimes \mathbb{R}$ . Set $N:=\widetilde{N}\oplus \mathbb{Z}$ , and $N_{\mathbb{R}}:=N\otimes \mathbb{R}$ . Denote by $C(P)\subset N_{\mathbb{R}}$ the cone over the polytope $P$ placed at height one in $N_{\mathbb{R}}$ . More formally, consider

$$\begin{eqnarray}\{1\}\times P\subset N_{\mathbb{R}}\cong \mathbb{R}\oplus \widetilde{N}_{\mathbb{R}},\end{eqnarray}$$

and let $C(P)$ be the cone generated by $\{1\}\times P$ inside $N_{\mathbb{R}}$ . Let $F(P)$ be the fan consisting of $C(P)$  and all its faces. The affine toric variety $X_{P}$ corresponding to $F(P)$ has an isolated Gorenstein singularity. The toric resolutions of $X_{P}$ are in bijection with smooth subdivisions of the cone $C(P)$ . We will be interested in toric crepant resolutions, that is, resolutions with trivial canonical bundle.

Toric crepant resolutions of $X_{P}$ are given by unimodular triangulations of $P$ , i.e. triangulations of $P$ by elementary lattice simplices. Any such triangulation ${\mathcal{T}}$ gives rise to a smooth subdivision of the cone $C(P)$ . We denote by $C({\mathcal{T}})$ the set of cones on the simplices $T\in {\mathcal{T}}$ placed at height one in $N_{\mathbb{R}}$ . Let $F({\mathcal{T}})$ be the corresponding fan, and let $X_{{\mathcal{T}}}$ be the toric variety with fan $F({\mathcal{T}})$ . The variety $X_{{\mathcal{T}}}$ is smooth and Calabi–Yau. All toric crepant resolutions of $X_{P}$ are isomorphic to $X_{{\mathcal{T}}}$ for some unimodular triangulation ${\mathcal{T}}$ of $P$ .

The following definition will be useful later on, see for instance [Reference Brodsky, Joswig, Morrison and SturmfelsBJMS15] for additional details on this construction.

Definition 3.1. Assume that $n=3$ . We denote by $G_{{\mathcal{T}}}$ the tropical curve dual to the triangulation ${\mathcal{T}}$ of $P$ .

Let $\widetilde{M}=\text{Hom}(\widetilde{N},\mathbb{Z})$ and $M=\text{Hom}(N,\mathbb{Z})$ . The height function on $N$ is by definition the projection

$$\begin{eqnarray}N=\mathbb{Z}\times \widetilde{N}\rightarrow \mathbb{Z}.\end{eqnarray}$$

The height function corresponds to the lattice point $(1,0)\in M\cong \mathbb{Z}\times \widetilde{M},$ which determines a monomial function

$$\begin{eqnarray}W_{{\mathcal{T}}}:X_{{\mathcal{T}}}\rightarrow \mathbb{A}_{\unicode[STIX]{x1D705}}^{1}.\end{eqnarray}$$

The category of B-branes on the Landau–Ginzburg model $(X_{{\mathcal{T}}},W_{{\mathcal{T}}})$ is the category of matrix factorizations for $W_{{\mathcal{T}}}$ , $MF(W_{{\mathcal{T}}})$ . We review the theory of matrix factorizations next.

3.1.2 Matrix factorizations

Let $X$ be a scheme or a smooth DM stack and let $f:X\rightarrow \mathbb{A}_{\unicode[STIX]{x1D705}}^{1}$ be a regular function. The category of matrix factorizations for the pair $(X,f)$ was defined in [Reference Lin and PomerleanoLP13] and [Reference OrlovOrl12], extending the theory of matrix factorizations for algebras that goes back to classical work of Eisenbud [Reference EisenbudEis80]. These references make various assumptions on $f$ and $X$ , which are always satisfied in the cases we are interested in. In the following $X$ will always be smooth of finite type, and $f$ will be flat. We will work with a dg enhancement of the category of matrix factorizations, which has been studied for instance in [Reference Lunts and SchnürerLS16] and [Reference PreygelPre12]. We refer to these papers for additional details. We denote by $MF(X,f)$ the $\mathbb{Z}_{2}$ -periodic dg category of matrix factorizations of the pair $(X,f)$ . It will often be useful to work with Ind-completed categories of matrix factorization.

Definition 3.2. We denote by $MF^{\infty }(X,f)$ the Ind-completion of $MF(X,f)$ ,

$$\begin{eqnarray}MF^{\infty }(X,f)=\text{Ind}(MF(X,f))\in \text{DGCat}_{\text{cont}}^{(2)}.\end{eqnarray}$$

The category $MF^{\infty }(X,f)$ has the following important descent property.

Proposition 3.3 [Reference PreygelPre12, Proposition A.3.1].

Let $f:X\rightarrow \mathbb{A}^{1}$ be a morphism. Then the assignment

$$\begin{eqnarray}U\mapsto MF^{\infty }(U,f|_{U})\end{eqnarray}$$

determines a sheaf for the étale topology.

Using Proposition 3.3 we can give a very concrete description of $MF^{\infty }(X_{{\mathcal{T}}},W_{{\mathcal{T}}})$ , where $X_{{\mathcal{T}}}$ and $W_{{\mathcal{T}}}$ are as in § 3.1.1. In order to do so, we need to explain how to attach a matrix factorizations-type category to a certain class of planar graphs. This will require setting up some notation and preliminaries.

Let $I$ be a set of cardinality three, say $I=\{a,b,c\}$ . Denote

$$\begin{eqnarray}X_{I}=\text{Spec}(\unicode[STIX]{x1D705}[t_{i},i\in I])=\text{Spec}(\unicode[STIX]{x1D705}[t_{a},t_{b},t_{c}]),\end{eqnarray}$$

and let $f$ be the regular function

$$\begin{eqnarray}f=\times _{i\in I}t_{i}=t_{a}t_{b}t_{c}:X\longrightarrow \mathbb{A}_{\unicode[STIX]{x1D705}}^{1}.\end{eqnarray}$$

For all $j\in I$ , let $I_{j}$ be the subset $I-\{j\}\subset I$ . Let $U_{j}$ be the open subscheme $X-\{t_{j}=0\}$ , and let $\unicode[STIX]{x1D704}_{j}$ be the inclusion $U_{j}\subset X$ . Denote by

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{j}^{\ast }:MF^{\infty }(X_{I},f)\longrightarrow MF^{\infty }(U_{j},f|_{U_{j}})\end{eqnarray}$$

the restriction functor. Let $f_{j}$ be the regular function

$$\begin{eqnarray}f_{j}=\times _{i\in I_{j}}t_{i}:U_{j}\longrightarrow \mathbb{A}_{\unicode[STIX]{x1D705}}^{1}.\end{eqnarray}$$

Note that $f|_{U_{j}}$ is given by $t_{j}f_{j}$ .

Proposition 3.4. There are equivalences of dg categories

$$\begin{eqnarray}MF^{\infty }(U_{j},f|_{U_{j}})\simeq MF^{\infty }(U_{j},f_{j})\simeq {\mathcal{Q}}\text{Coh}^{(2)}(\mathbb{G}_{m}).\end{eqnarray}$$

Proof. Recall that objects of $MF(U_{j},f_{j})$ are pairs of free finite rank vector bundles on $U_{j}$ , and maps between them

having the property that

$$\begin{eqnarray}d_{1}\circ d_{2}=f_{j}\cdot Id_{V_{2}}\quad \text{and}\quad d_{2}\circ d_{1}=f_{j}\cdot Id_{V_{1}}.\end{eqnarray}$$

Thus the assignment

determines an equivalence

$$\begin{eqnarray}MF(U_{j},f|_{U_{j}})\simeq MF(U_{j},f_{j}).\end{eqnarray}$$

The first equivalence is obtained by Ind-completion.

The second equivalence follows from Knörrer periodicity. For a very general formulation of Knörrer periodicity see Theorem 9.1.7(ii) of [Reference PreygelPre12]. Let us assume for convenience that $I=\{1,2,3\}$ , and that $j=1$ . Then $U_{j}=\mathbb{G}_{m}\times \mathbb{A}_{\unicode[STIX]{x1D705}}^{2}$ , and $f_{j}=t_{2}\cdot t_{3}$ , where $t_{2}$ and $t_{3}$ are coordinate on the factor $\mathbb{A}_{\unicode[STIX]{x1D705}}^{2}$ . By Knörrer periodicity $MF(U_{j},f_{j})$ is equivalent to the $\mathbb{Z}_{2}$ -periodic category of perfect complexes on the first factor, $\mathbb{G}_{m}$ . This concludes the proof.◻

Remark 3.5. The equivalences constructed in Proposition 3.4 are given by explicit functors, and do not rely on further choices. For the second equivalence, this follows from the proof of Knörrer periodicity in [Reference PreygelPre12]. Abusing notation we sometimes denote by $\unicode[STIX]{x1D704}_{j}^{\ast }$ also the composition of the pullback with the equivalences from Proposition 3.4. Thus we may write

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{j}^{\ast }:MF^{\infty }(X_{I},f)\rightarrow MF^{\infty }(U_{j},f_{j}),\quad \unicode[STIX]{x1D704}_{j}^{\ast }:MF^{\infty }(X_{I},f)\rightarrow {\mathcal{Q}}\text{Coh}^{(2)}(\mathbb{G}_{m}).\end{eqnarray}$$

We can abstract from Remark 3.5 a formalism of restriction functors which will be useful in the next section. If $L$ is a set of cardinality two, denote

$$\begin{eqnarray}X_{L}=\text{Spec}(\unicode[STIX]{x1D705}[t_{l},l\in L][u,u^{-1}])\cong \mathbb{G}_{m}\times \mathbb{A}_{\unicode[STIX]{x1D705}}^{2},\end{eqnarray}$$

and let $f$ be the morphism

$$\begin{eqnarray}f=\times _{l\in L}t_{l}:X_{L}\longrightarrow \mathbb{A}_{\unicode[STIX]{x1D705}}^{1}.\end{eqnarray}$$

Definition 3.6. Let $I$ and $L$ be sets of cardinality three and two respectively, and assume that we are given an embedding $L\subset I$ . We denote by $R_{\infty }^{MF}$ the composite

where:

1. (i) $\{j\}=I-L$ , and $\unicode[STIX]{x1D704}_{j}^{\ast }$ is defined as in Remark 3.5;

2. (ii) the equivalence $MF^{\infty }(U_{j},f_{j})\simeq MF^{\infty }(X_{L},f)$ is determined by the isomorphism of $\unicode[STIX]{x1D705}$ -algebras

   $$\begin{eqnarray}\unicode[STIX]{x1D705}[t_{i},i\in I][t_{j}^{-1}]=\unicode[STIX]{x1D705}[t_{l},l\in L][t_{j},t_{j}^{-1}]\stackrel{\cong }{\longrightarrow }\unicode[STIX]{x1D705}[t_{l},l\in L][u,u^{-1}]\end{eqnarray}$$
   that sends $t_{l}$ to $t_{l}$ , and $t_{j}$ to $t_{u}$ .

3.1.3 Planar graphs and matrix factorizations

Let $G$ be a trivalent, planar graph. Assume for simplicity that $G$ does not contain any loop. We will explain how to attach to $G$ a matrix factorization-type category. We denote by $V_{G}$ the set of vertices of $G$ , and by $E_{G}$ the set of edges.

1. – Let $v\in V_{G}$ , and take a sufficiently small ball $B_{v}$ in $\mathbb{R}^{2}$ centered at $v$ . Then the set of connected components of $B_{v}-G$ has cardinality three, and we denote it $I_{v}$ .

2. – Let $e\in V_{G}$ , and take a sufficiently small ball $B_{e}$ centered at any point in the relative interior of $e$ . The set of connected components of $B_{v}-G$ has cardinality two, and we denote it $L_{e}$ .

Remark 3.7. Note that the sets $I_{v}$ and $L_{e}$ do not depend (up to canonical identifications) on $B_{e}$ and $B_{v}$ . Further, if a vertex $v$ is incident to an edge $e$ , there is a canonical embedding: $L_{e}\subset I_{v}$ .

We attach to each vertex and edge of $G$ a category of matrix factorizations in the following way.

1. – We assign to $v\in V_{G}$ the category

   $$\begin{eqnarray}{\mathcal{B}}(v):=MF^{\infty }(X_{I_{v}},f).\end{eqnarray}$$

2. – We assign to $e\in E_{G}$ the category

   $$\begin{eqnarray}{\mathcal{B}}(e):=MF^{\infty }(X_{L_{e}},f).\end{eqnarray}$$

By Remark 3.7, and Definition 3.6, if a vertex $v$ is incident to an edge $e$ we have a restriction functor

(5) $$\begin{eqnarray}\displaystyle R^{{\mathcal{B}}}:{\mathcal{B}}(v)\longrightarrow {\mathcal{B}}(e). & & \displaystyle\end{eqnarray}$$

If two vertices $v_{1}$ and $v_{2}$ are incident to an edge $e$ , we obtain a diagram of restriction functors

Running over the vertices and edges of $G$ , we obtain a Čech-type diagram in $\text{DGCat}_{\text{cont}}^{(2)}$

(6)

Definition 3.8. We denote by ${\mathcal{B}}(G)$ the equalizer of diagram (6) in $\text{DGCat}_{\text{cont}}^{(2)}$ .

We say that a subset $T\subset G$ is a subgraph of $G$ if:

1. – $T$ is a trivalent graph;

2. – if $e$ is an edge of $G$ such that $T\cap e$ is nonempty, then $e$ is contained in $T$ .

Note that if $T$ is a subgraph of $G$ we have inclusions $V_{T}\subset V_{G}$ and $E_{T}\subset E_{G}$ . We can define a restriction functor

$$\begin{eqnarray}{\mathcal{B}}(G)\longrightarrow {\mathcal{B}}(T).\end{eqnarray}$$

This is obtained by considering the natural map between Čech-type diagrams given by the obvious projections shown in the following diagram.

It is useful to extend to a general pair of graphs $T\subset G$ the notation for restriction functors that we introduced in (5) in the case of a vertex and a neighboring edge.

Definition 3.9. If $T$ is a subgraph of $G$ , we denote the restriction functor

$$\begin{eqnarray}R^{{\mathcal{B}}}:{\mathcal{B}}(G)\longrightarrow {\mathcal{B}}(T).\end{eqnarray}$$

The definition of ${\mathcal{B}}(G)$ allows us to encode the category of matrix factorizations of a toric Calabi–Yau LG model in a simple combinatorial package. We use the notation of § 3.1.1.

Theorem 3.10. Let $P$ be a planar lattice polytope, equipped with a unimodular triangulation ${\mathcal{T}}$ . Let $G_{{\mathcal{T}}}$ be the dual graph of ${\mathcal{T}}$ . Then there is an equivalence of dg categories

$$\begin{eqnarray}MF^{\infty }(X_{{\mathcal{T}}},W_{{\mathcal{T}}})\simeq {\mathcal{B}}(G_{{\mathcal{T}}}).\end{eqnarray}$$

Proof. Let $C$ be the set of maximal cones in the fan of $X_{{\mathcal{T}}}$ . Consider the standard open cover of $X_{{\mathcal{T}}}$ by toric affine patches: $\{{U_{\unicode[STIX]{x1D70E}}\}}_{\unicode[STIX]{x1D70E}\in C},$ $U_{\unicode[STIX]{x1D70E}}\cong \mathbb{A}_{\unicode[STIX]{x1D705}}^{3}$ . By Proposition 3.3 the category $MF^{\infty }(X_{{\mathcal{T}}},f_{{\mathcal{T}}})$ can be expressed as the limit of the Čech diagram for the open cover $\{{U_{\unicode[STIX]{x1D70E}}\}}_{\unicode[STIX]{x1D70E}\in C}$ : the vertices of this diagram are products of the categories

$$\begin{eqnarray}MF^{\infty }(U_{\unicode[STIX]{x1D70E}},f_{{\mathcal{T}}}|_{U_{\unicode[STIX]{x1D70E}}})\quad \text{and}\quad MF^{\infty }(U_{\unicode[STIX]{x1D70E}}\cap U_{\unicode[STIX]{x1D70E}^{\prime }},f_{{\mathcal{T}}}|_{U_{\unicode[STIX]{x1D70E}}\cap U_{\unicode[STIX]{x1D70E}^{\prime }}}),\quad \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\prime }\in C,\end{eqnarray}$$

and the arrows are products of pullback functors.

Note that there is a natural bijection $\unicode[STIX]{x1D719}$ between the set $V_{{\mathcal{T}}}$ of vertices of $G_{{\mathcal{T}}}$ and $C$ . Moreover the definition of $I_{v}$ gives an identification $X_{I_{v}}\cong U_{\unicode[STIX]{x1D719}(\unicode[STIX]{x1D70E})}$ , and thus a canonical equivalence

$$\begin{eqnarray}{\mathcal{B}}(v)\simeq MF^{\infty }(U_{\unicode[STIX]{x1D719}(v)},f_{{\mathcal{T}}}|_{\unicode[STIX]{x1D719}(v)}).\end{eqnarray}$$

Similarly, by Remark 3.5, if $v$ and $v^{\prime }$ are two vertices of $G_{{\mathcal{T}}}$ and $e$ is the edge connecting them, we obtain a commutative diagram

where the horizontal arrows are canonical equivalences, and $\unicode[STIX]{x1D704}^{\ast }$ is the restriction of matrix factorizations along the embedding

$$\begin{eqnarray}\unicode[STIX]{x1D704}:U_{\unicode[STIX]{x1D719}(v)}\cap U_{\unicode[STIX]{x1D719}(v^{\prime })}\rightarrow U_{\unicode[STIX]{x1D719}(v)}.\end{eqnarray}$$

Thus the diagrams computing ${\mathcal{B}}(G_{{\mathcal{T}}})$ and $MF^{\infty }(X_{{\mathcal{T}}},f_{{\mathcal{T}}})$ are equivalent, and this concludes the proof.◻

3.2 A-branes

As explained in [Reference Hori and VafaHV15], the mirror of a toric Calabi–Yau LG model $(X_{{\mathcal{T}}},W_{{\mathcal{T}}})$ is a punctured Riemann surface $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ embedded as an algebraic curve in $\mathbb{C}^{\ast }\times \mathbb{C}^{\ast }$ . The graph $G_{{\mathcal{T}}}$ is the tropicalization of $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ . Since we are interested in studying the A-model on $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ we can disregard its complex structure and focus on its topology, which is captured by the genus and the number of punctures (see for instance [Reference Bouchard and SułkowskiBS12] for an explicit algebraic equation of $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ ). These can be read off from $G_{{\mathcal{T}}}$ . The genus of $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ is equal to the number of relatively compact connected components in $\mathbb{R}^{2}-G_{{\mathcal{T}}}$ . The number of punctures of $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ is equal to the number of unbounded edges of $G_{{\mathcal{T}}}$ . See Figure 1.

Figure 1. The picture shows an example of the relationship between the triangulation ${\mathcal{T}}$ , the tropical curve $G_{{\mathcal{T}}}$ , and the mirror curve $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ .

When $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ has genus $0$ , the authors of [Reference Abouzaid, Auroux, Efimov, Katzarkov and OrlovAAEKO13] consider the wrapped Fukaya category of $\unicode[STIX]{x1D6F4}_{{\mathcal{T}}}$ , and prove that it is equivalent to the category of matrix factorizations of the mirror toric LG model. In this paper we will consider an alternative model for the category of A-branes on a punctured Riemann surface, called the topological Fukaya category. The construction of the topological Fukaya category was first suggested by Kontsevich [Reference KontsevichKon09] and was studied in [Reference Dyckerhoff and KapranovDK18, Reference NadlerNad15, Reference Sibilla, Treumann and ZaslowSTZ14] and elsewhere in the case of punctured Riemann surfaces. We summarize the theory of the topological Fukaya category in § 4 below.

4.1 The cyclic category and the topological Fukaya category

We briefly review the setting of [Reference Dyckerhoff and KapranovDK18]. We refer for more details to the original paper and to [Reference DyckerhoffDyc17].

Consider the map $\mathbb{A}_{\unicode[STIX]{x1D705}}^{1}\stackrel{z^{n}}{\longrightarrow }\mathbb{A}_{\unicode[STIX]{x1D705}}^{1}$ . We denote by ${\mathcal{E}}^{n}$ the ( $\mathbb{Z}_{n+1}$ -graded category) of matrix factorizations $MF(\mathbb{A}_{\unicode[STIX]{x1D705}}^{1},z^{n+1})\in \text{DGCat}_{\text{small}}^{(2)}$ , see § 2.3.5 of [Reference Dyckerhoff and KapranovDK18] for additional details. The categories ${\mathcal{E}}^{n}$ assemble to a cyclic object in $\text{DGCat}_{\text{small}}^{(2)}$ . We state the precise result below.

Proposition 4.4 [Reference Dyckerhoff and KapranovDK18, Proposition 2.4.1], [Reference DyckerhoffDyc17, Theorem 3.2].

There is a cocyclic object

$$\begin{eqnarray}{\mathcal{E}}^{\bullet }:N(\unicode[STIX]{x1D6EC})\rightarrow \text{DGCat}_{\text{small}}^{(2)}\end{eqnarray}$$

that is defined on objects by the assignment

$$\begin{eqnarray}\langle n\rangle \in \unicode[STIX]{x1D6EC}\mapsto {\mathcal{E}}^{n}\in \text{DGCat}_{\text{small}}^{(2)}.\end{eqnarray}$$

Let

$$\begin{eqnarray}(-)^{\text{op}}:\text{DGCat}_{\text{small}}^{(2)}\rightarrow \text{DGCat}_{\text{small}}^{(2)}\end{eqnarray}$$

be the auto-equivalence of $\text{DGCat}_{\text{small}}^{(2)}$ sending a category to its opposite category.

Definition 4.6. Denote by ${\mathcal{E}}_{\bullet }:\unicode[STIX]{x1D6EC}^{\text{op}}\rightarrow \text{DGCat}_{\text{small}}^{(2)}$ the cyclic object defined as the composite

$$\begin{eqnarray}N(\unicode[STIX]{x1D6EC}^{\text{op}})\stackrel{N(-)^{\ast }}{\longrightarrow }N(\unicode[STIX]{x1D6EC})\stackrel{{\mathcal{E}}^{\bullet }}{\longrightarrow }\text{DGCat}_{\text{small}}^{(2)}\stackrel{(-)^{\text{op}}}{\longrightarrow }\text{DGCat}_{\text{small}}^{(2)}.\end{eqnarray}$$

To any ribbon graph $X$ , we may associate a cyclic set ${\mathcal{L}}(X)\in {\mathcal{S}}et_{\unicode[STIX]{x1D6EC}}$ (called the cyclic membrane in [Reference Dyckerhoff and KapranovDK18]) as follows. Given a vertex $v$ of $X$ , denote by $E(v)$ the set of edges at $v$ , and set $B(v)=E(v)^{\ast }$ . Let $\unicode[STIX]{x1D6EC}^{B(v)}$ be the cyclic simplex corresponding to the cyclically ordered set $B(v)$ . For an edge $e$ of $X$ , we have a two element set $B(e)=V(e)^{\ast }$ , where $V(e)$ is the set of vertices that $e$ joins. Let $\unicode[STIX]{x1D6EC}^{B(e)}$ be the corresponding cyclic 1-simplex. Next, define the incidence category $X_{[0,1]}$ of $X$ : the set of objects of $X_{[0,1]}$ is the disjoint union of the set of vertices and the set of edges of $X$ , and there is a unique morphism $v\rightarrow e$ for each flag $(v,e)$ in $X$ (consisting of a pair of a vertex $v$ and an edge $e$ incident to $v$ ). The cyclic simplices $\unicode[STIX]{x1D6EC}^{B(v)}$ and $\unicode[STIX]{x1D6EC}^{B(e)}$ assemble into a functor $U_{X}:X_{[0,1]}^{\text{op}}\rightarrow {\mathcal{S}}et_{\unicode[STIX]{x1D6EC}}$ , and we define

(7) $$\begin{eqnarray}\displaystyle {\mathcal{L}}(X)=\varinjlim U_{X}\in {\mathcal{S}}et_{\unicode[STIX]{x1D6EC}}. & & \displaystyle\end{eqnarray}$$

Proposition 4.8 [Reference Dyckerhoff and KapranovDK18, Proposition 3.4.4].

The cyclic membrane construction

$$\begin{eqnarray}{\mathcal{L}}:{\mathcal{R}}ib\rightarrow {\mathcal{S}}et_{\unicode[STIX]{x1D6EC}},\quad X\in {\mathcal{R}}ib\mapsto {\mathcal{L}}(X)\end{eqnarray}$$

extends to a functor from the category ${\mathcal{R}}ib$ of ribbon graphs and contractions between them to the category of cyclic sets.

If $X$ and $X^{\prime }$ are ribbon graphs, a contraction $X\rightarrow X^{\prime }$ is a map between the underlying topological spaces having the property that the preimage of each point in $X^{\prime }$ is either a point, or a sub-tree of $X$ . We refer the reader to [Reference Dyckerhoff and KapranovDK18] for additional details on the definition of ${\mathcal{R}}ib$ and a proof of the proposition.

As explained in [Reference DyckerhoffDyc17] the functor constructed in Proposition 4.8 can be enhanced to a functor of $\infty$ -categories

$$\begin{eqnarray}{\mathcal{L}}:N({\mathcal{R}}ib)\rightarrow {\mathcal{S}}_{\unicode[STIX]{x1D6EC}},\end{eqnarray}$$

where ${\mathcal{S}}$ is the $\infty$ -category of spaces.

Let $\unicode[STIX]{x1D6F4}$ be a Riemann surface with boundary and let $X\subset \unicode[STIX]{x1D6F4}$ be a spanning ribbon graph. The implementation of Kontsevich’s ideas due to Dyckerhoff and Kapranov [Reference Dyckerhoff and KapranovDK18] (see also [Reference NadlerNad15] and [Reference Sibilla, Treumann and ZaslowSTZ14]) gives ways to compute a model for the Fukaya category of $\unicode[STIX]{x1D6F4}$ from the combinatorics of $X$ . We will refer to it as the topological (compact) Fukaya category of $X$ or sometimes as the topological (compact) Fukaya category of the pair $(\unicode[STIX]{x1D6F4},X)$ . The next definition gives the construction of the topological (compact) Fukaya category, see Definition 4.1.1 of [Reference Dyckerhoff and KapranovDK18].

Proposition 4.12. There is a natural equivalence

$$\begin{eqnarray}{\mathcal{F}}_{\text{top}}(X)\simeq \text{Fun}({\mathcal{F}}^{\text{top}}(X),\text{Vect}^{(2),\text{fd}}).\end{eqnarray}$$

4.2 The Ind-completion of the topological Fukaya category

In this section we introduce the Ind-completed version of the topological (compact) Fukaya category. This plays an important role in proving that ${\mathcal{F}}^{\text{top}}(-)$ exhibits an interesting sheaf-like behavior with respect to closed coverings of ribbon graphs that we study in § 5.2. We remark that Definitions 4.13 and 4.15 mirror exactly Proposition 4.4 and Definitions 4.6 and 4.9 from the previous section, the only difference being that we are now working with cocomplete dg categories.

Proposition 4.16. For all $X\in {\mathcal{S}}_{\unicode[STIX]{x1D6EC}}$ there is an equivalence

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{F}}^{{\mathcal{E}}}(X)\simeq {\mathcal{I}}{\mathcal{F}}_{{\mathcal{E}}}(X).\end{eqnarray}$$

Proof. This is a formal consequence of Proposition 2.4. By Remark 4.14, ${\mathcal{I}}{\mathcal{E}}_{\bullet }$ can be realized by taking right adjoints of the structure morphisms of ${\mathcal{I}}{\mathcal{E}}^{\bullet }$ . If ${\mathcal{L}}$ is in ${\mathcal{S}}_{\unicode[STIX]{x1D6EC}}$ , by the general formula for pointwise Kan extensions we have

$$\begin{eqnarray}{\mathcal{I}}{\mathcal{F}}^{{\mathcal{E}}}({\mathcal{L}})=\underset{\{L(\langle n\rangle )\rightarrow {\mathcal{L}}\}}{\varinjlim }{\mathcal{I}}{\mathcal{E}}^{n}\simeq \underset{\{L(\langle n\rangle )\rightarrow {\mathcal{L}}\}}{\varprojlim }{\mathcal{I}}{\mathcal{E}}_{n}={\mathcal{I}}{\mathcal{F}}_{{\mathcal{E}}}({\mathcal{L}})\quad \text{in}~\text{DGCat}_{\text{cont}}^{(2)}.\Box\end{eqnarray}$$

Definition 4.17. If $X$ is a ribbon graph, we set

$$\begin{eqnarray}{\mathcal{F}}_{\infty }^{\text{top}}(X):={\mathcal{I}}{\mathcal{F}}^{{\mathcal{E}}}({\mathcal{L}}(X))\simeq {\mathcal{I}}{\mathcal{F}}_{{\mathcal{E}}}({\mathcal{L}}(X)).\end{eqnarray}$$

We call ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ the Ind-completed topological Fukaya category of X.

By Lemma 2.2 Ind-completion preserves colimits, and this immediately implies the following statement.

Proposition 4.18. If $X$ is a ribbon graph there is an equivalence

$$\begin{eqnarray}{\mathcal{F}}_{\infty }^{\text{top}}(X)\simeq \text{Ind}({\mathcal{F}}^{\text{top}}(X)).\end{eqnarray}$$

Proposition 4.16 and Definition 4.17 indicate a key difference with the setting of small categories considered in § 4.1: if we work with the Ind-completed (co)cyclic objects ${\mathcal{I}}{\mathcal{E}}^{\bullet }$ and ${\mathcal{I}}{\mathcal{E}}_{\bullet },$ there is no difference between Fukaya and compact Fukaya category. However, it is important to remark that ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ is not equivalent in general to the Ind-completion of ${\mathcal{F}}_{\text{top}}(X)$ . Instead $\text{Ind}({\mathcal{F}}_{\text{top}}(X))$ is a full subcategory of ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ . We clarify the relationship between these two categories in Example 4.19 below.

Example 4.19. Consider the ribbon graph $X$ given by a loop with no vertices. We can tabulate the value of the topological (compact) Fukaya category of $X$ and of its Ind-completed version as follows.

1. – The category ${\mathcal{F}}^{\text{top}}(X)$ is equivalent to ${\mathcal{P}}\text{erf}^{(2)}(\mathbb{G}_{m})$ .

2. – The category ${\mathcal{F}}_{\text{top}}(X)$ is equivalent to the full subcategory of ${\mathcal{P}}\text{erf}^{(2)}(\mathbb{G}_{m})$ given by complexes with compact support, ${\mathcal{P}}\text{erf}_{cs}^{(2)}(\mathbb{G}_{m})$ .

3. – By Proposition 4.18 the category ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ is equivalent to ${\mathcal{Q}}\text{Coh}^{(2)}(\mathbb{G}_{m})$ .

Note that $\text{Ind}({\mathcal{F}}_{\text{top}}(X))\simeq \text{Ind}({\mathcal{P}}\text{erf}_{cs}^{(2)}(\mathbb{G}_{m}))$ is a strict subcategory of

$$\begin{eqnarray}{\mathcal{F}}_{\infty }^{\text{top}}(X)\simeq {\mathcal{Q}}\text{Coh}^{(2)}(\mathbb{G}_{m}).\end{eqnarray}$$

5.1 Restriction to open subgraphs

Let $X$ be a ribbon graph. With a small abuse of notation we denote by $X$ also its underlying topological space. We say that $Y\subset X$ is a subgraph if it is a subspace having the property that, if the intersection of $Y$ with an edge $e$ of $X$ is not empty or a vertex, then $e$ is contained in $Y$ . If $Y$ is a subgraph of $X$ , then it is canonically the underlying topological space of a ribbon graph, which we also denote $Y$ . Note that if $U$ and $V$ are open subgraphs of $X$ , then their intersection $U\cap V$ is also an open subgraph of $X$ .

Remark 5.2. By Proposition 4.18  ${\mathcal{F}}_{\infty }^{\text{top}}(U)$ and ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ are equivalent to the Ind-completions of ${\mathcal{F}}^{\text{top}}(U)$ and ${\mathcal{F}}^{\text{top}}(X)$ . There is a natural equivalence

$$\begin{eqnarray}C_{\infty ,U}\simeq \text{Ind}(C_{U}):{\mathcal{F}}_{\infty }^{\text{top}}(U)\rightarrow {\mathcal{F}}_{\infty }^{\text{top}}(X).\end{eqnarray}$$

In particular the corestriction $C_{\infty ,U}$ is a morphism in $\text{DGCat}_{\text{cont}}^{(2)}$ .

Remark 5.5. Let $X$ be a ribbon graph. Let $V\subset U\subset X$ be open subgraphs. Then the following diagram commutes.

Propositions 5.6 and 5.7 can be extended in the usual way to account for arbitrary open covers of $X$ : given any open cover of $X$ , the (Ind-completed) (compact) Fukaya category can be realized as the homotopy (co)limit of the appropriate Čech diagram of local sections. This clarifies that this formalism is indeed an implementation of the idea that the Fukaya category of a punctured surface should define either a sheaf or a cosheaf of categories on its spine.

5.2 Restriction to closed subgraphs

In this section we turn our attention to closed subgraphs and closed covers of ribbon graphs. In the context of the topological Fukaya category, restrictions to closed subgraphs have also been investigated by Dyckerhoff [Reference DyckerhoffDyc17]. To avoid producing parallel arguments here we will refer to the lucid treatment contained in § 4 of [Reference DyckerhoffDyc17].

We introduce next restriction functors of ${\mathcal{F}}^{\text{top}}$ and ${\mathcal{F}}_{\infty }^{\text{top}}$ to good closed subgraphs: we will sometimes refer to these as exceptional restrictions, in order to mark their difference from the (co)restrictions to open subgraphs that were discussed in the previous section.

Proof. Our definition of $S^{Z}$ follows § 4 of [Reference DyckerhoffDyc17]. Let $Z$ be a closed subgraph, $U=X-Z$ , and consider the open subgraph $V$ consisting of $Z$ together with all half-edges of $X$ that touch $Z$ . Then $X=U\cup V$ is an open covering, and by Proposition 5.6 we have the following push-out square.

Next, we may consider another graph $\overline{V}$ that is obtained from $V$ by adding a 1-valent vertex to each edge of $V$ that does not belong to $Z$ . There is a covering $\overline{V}=V\cup \unicode[STIX]{x1D6F9}$ , where $\unicode[STIX]{x1D6F9}$ is the disjoint union of several copies of the graph with one vertex and one half-open edge. Another application of Proposition 5.6 yields the following push-out square.

Next, we make two observations: one is that $\unicode[STIX]{x1D6F9}\cap V$ naturally contains $U\cap V$ , the second is that ${\mathcal{F}}^{\text{top}}(\unicode[STIX]{x1D6F9})\simeq 0$ . This implies that the first push-out diagram maps to the second, and so there is a map ${\mathcal{F}}^{\text{top}}(X)\rightarrow {\mathcal{F}}^{\text{top}}(\overline{V})$ . Lastly we use the fact that ${\mathcal{F}}^{\text{top}}(\overline{V})\simeq {\mathcal{F}}^{\text{top}}(Z)$ , since $Z$ is obtained from $\overline{V}$ by contracting edges. The resulting functor is $S^{Z}:{\mathcal{F}}^{\text{top}}(X)\rightarrow {\mathcal{F}}^{\text{top}}(Z)$ . The functor $S_{\infty }^{Z}$ is the Ind-completion of $S^{Z}$ .◻

Proposition 5.12. Let $X$ be a ribbon graph, let $U\subset X$ be a good open subset and let $Z=X-U$ . The following are cofiber sequences in $\text{DGCat}_{\text{small}}^{(2)}$ and in $\text{DGCat}_{\text{cont}}^{(2)},$ respectively

$$\begin{eqnarray}{\mathcal{F}}^{\text{top}}(U)\stackrel{C_{X}}{\rightarrow }{\mathcal{F}}^{\text{top}}(X)\stackrel{S^{Z}}{\rightarrow }{\mathcal{F}}^{\text{top}}(Z),\quad {\mathcal{F}}_{\infty }^{\text{top}}(U)\stackrel{C_{\infty ,X}}{\rightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(X)\stackrel{S_{\infty }^{Z}}{\rightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(Z).\end{eqnarray}$$

Definition 5.13. Let $X$ be a ribbon graph and let $Z\subset X$ be a good closed subgraph. Then we denote by

$$\begin{eqnarray}T_{\infty }^{X}:{\mathcal{F}}_{\infty }^{\text{top}}(Z)\rightarrow {\mathcal{F}}_{\infty }^{\text{top}}(X)\end{eqnarray}$$

the right adjoint of the exceptional restriction functor. We will call it the exceptional corestriction functor. When the source of the exceptional corestriction functor is not clear from the context we will use the notation $T_{\infty ,Z}^{X}$ .

Proposition 5.14 is a compatibility statement that relates the various restrictions that we have introduced so far, and it will be useful in the next section.

Proposition 5.14. Let $X$ be a ribbon graph. Let $U\subset X$ be an open subgraph and let $Z\subset X$ be a good closed subgraph. If $Z$ is contained in $U$ then the following diagram commutes.

Before proving Proposition 5.14 we introduce some preliminary notation and results.

Definition 5.16. Let $F:A\rightarrow {\mathcal{F}}_{\infty }^{\text{top}}(K_{x})$ be a functor. We say that $A$ is not supported on $x$ if the composite

$$\begin{eqnarray}A\stackrel{F}{\longrightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(K_{x})\stackrel{R_{\infty }^{U_{x}^{p}}}{\longrightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(U_{x}^{p})\end{eqnarray}$$

is equivalent to the zero functor.

Lemma 5.17. Let $x$ be a vertex of $X$ , and let $F:A\rightarrow {\mathcal{F}}_{\infty }^{\text{top}}(K_{x})$ be a functor. If $A$ is not supported on $x$ , then there is a natural equivalence

$$\begin{eqnarray}R_{\infty ,X}^{K_{x}}\circ C_{\infty ,K_{x}}^{X}\circ F\simeq F.\end{eqnarray}$$

Proof. We fix first some notation. If $\unicode[STIX]{x1D6E4}$ is a ribbon graph, and $W$ is a subset of the vertices of $\unicode[STIX]{x1D6E4}$ we set

(8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}(W):=\coprod _{w\in W}U_{w},\quad \unicode[STIX]{x1D6E4}^{2}(W):=\coprod _{w,w^{\prime }\in W}U_{w}\cap U_{w^{\prime }}. & & \displaystyle\end{eqnarray}$$

Also if $L$ and $L^{\prime }$ are objects in ${\mathcal{F}}_{\infty }^{\text{top}}(\unicode[STIX]{x1D6E4})$ , we denote their Hom-complex $\text{Hom}_{\unicode[STIX]{x1D6E4}}(L,L^{\prime })$ .

Let us now return to the statement of the lemma. Let $V$ be the set of vertices of $X$ . By Proposition 5.7 the Ind-complete topological Fukaya category ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ is naturally equivalent to the equalizerFootnote ²

(9)

in $\text{DGCat}_{\text{cont}}^{(2)}$ , where $r_{1}$ and $r_{2}$ are products of restriction functors. Using the notation introduced in (8), we can rewrite this equalizer as

Similarly if $W=V-\{x\}$ is the set of vertices of $K_{x}$ , we obtain an equalizer diagram.

(10)

The inclusion $W\subset V$ gives projections $P$ and $Q$ that fit in a morphism of diagrams.

(11)

Further, the restriction $R_{\infty ,X}^{K_{x}}$ coincides with the morphism between the equalizers ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ and ${\mathcal{F}}_{\infty }^{\text{top}}(K_{x})$ induced by (11). Denote by $(P)^{L}$ and $(Q)^{L}$ the left adjoints of $P$ and $Q$ . The functor $(P)^{L}$ is given by the obvious quasi-fully faithful embedding

$$\begin{eqnarray}{\mathcal{F}}_{\infty }^{\text{top}}(X(W))\stackrel{\subset }{\longrightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(X(V))\simeq {\mathcal{F}}_{\infty }^{\text{top}}(X(W))\times {\mathcal{F}}_{\infty }^{\text{top}}(U_{x}),\end{eqnarray}$$

and similarly for $(Q)^{L}$ .

We will prove the lemma in two steps. First, we show that the diagram

(12)

is commutative, that is, there are natural equivalences

$$\begin{eqnarray}r_{1}\circ (P)^{L}\circ F\simeq (Q)^{L}\circ r_{1}^{\prime }\circ F\quad \text{and}\quad r_{2}\circ (P)^{L}\circ F\simeq (Q)^{L}\circ r_{2}^{\prime }\circ F.\end{eqnarray}$$

We remark that the commutativity does not hold if we do not precompose with $F$ . It is sufficient to prove that, for all $v$ and $v^{\prime }$ in $V$ , the diagram commutes after composing on the left with the restriction

$$\begin{eqnarray}R_{\infty }^{U_{v}\cap U_{v^{\prime }}}:{\mathcal{F}}_{\infty }^{\text{top}}(X^{2}(V))\rightarrow {\mathcal{F}}_{\infty }^{\text{top}}(U_{v}\cap U_{v^{\prime }}).\end{eqnarray}$$

If both $v$ and $v^{\prime }$ are different from $x$ , then

$$\begin{eqnarray}R_{\infty }^{U_{v}\cap U_{v^{\prime }}}\circ r_{i}\circ (P)^{L}\simeq R_{\infty }^{U_{v}\cap U_{v^{\prime }}}\circ (Q)^{L}\circ r_{i}^{\prime },\end{eqnarray}$$

and so commutativity holds also when precomposing with $F$ . Assume on the other hand that $v=x$ . Then

$$\begin{eqnarray}R_{\infty }^{U_{x}\cap U_{v^{\prime }}}\circ r_{i}\circ (P)^{L}\circ F\simeq 0\simeq R_{\infty }^{U_{x}\cap U_{v^{\prime }}}\circ (Q)^{L}\circ r_{i}^{\prime }\circ F.\end{eqnarray}$$

The first equivalence follows from the support assumption on $A$ , and the second one is a consequence of the fact that $R_{\infty }^{U_{x}\cap U_{v^{\prime }}}\circ (Q)^{L}=0$ . Thus diagram (12) commutes as claimed.

The commutativity of (12) and the universal property of the equalizer give us a functor

$$\begin{eqnarray}\tilde{F}:A\longrightarrow {\mathcal{F}}_{\infty }^{\text{top}}(X).\end{eqnarray}$$

Note that both $P\circ (P)^{L}$ and $Q\circ (Q)^{L}$ are naturally equivalent to the identity functor, and thus

$$\begin{eqnarray}R_{\infty ,X}^{K_{x}}\circ \tilde{F}\simeq F.\end{eqnarray}$$

The second and final step in the proof consists in noticing that $\tilde{F}$ is equivalent to $C_{\infty ,K_{x}}^{X}\circ F$ . That is, for all $L_{A}$ in the image of $F$ , and $L_{X}$ in ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ , there is a natural equivalence

$$\begin{eqnarray}\text{Hom}_{X}(\tilde{F}(L_{A}),L_{X})\simeq \text{Hom}_{K_{x}}(L_{A},R_{\infty ,X}^{K_{x}}(L_{X})),\end{eqnarray}$$

where $\text{Hom}_{X}(-,-)$ and $\text{Hom}_{K_{x}}(-,-)$ denote respectively the hom spaces in ${\mathcal{F}}_{\infty }^{\text{top}}(X)$ and in  ${\mathcal{F}}_{\infty }^{\text{top}}(K_{x})$ . This can be checked by computing explicitly the Hom-complexes on both sides in terms of the equalizers (9) and (10), see Proposition 2.2 of [Reference Sibilla, Treumann and ZaslowSTZ14] for a similar calculation. As a consequence there is a chain of equivalences

$$\begin{eqnarray}R_{\infty ,X}^{K_{x}}\circ C_{\infty ,K_{x}}^{X}\circ F\simeq R_{\infty ,X}^{K_{x}}\circ \tilde{F}\simeq F\end{eqnarray}$$

and this concludes the proof. ◻

Let $Z$ be a good closed subgraph of $X$ . Recall that

$$\begin{eqnarray}T_{\infty ,Z}^{X}:{\mathcal{F}}_{\infty }^{\text{top}}(Z)\longrightarrow {\mathcal{F}}_{\infty }^{\text{top}}(X)\end{eqnarray}$$

is the right adjoint of $S_{\infty ,Z}^{X}$ .

Proof. It follows from Proposition 5.12 that

$$\begin{eqnarray}{\mathcal{F}}_{\infty }^{\text{top}}(Z)\stackrel{T_{\infty ,Z}^{K_{x}}}{\longrightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(K_{x})\stackrel{R_{\infty ,K_{x}}^{K_{x}-Z}}{\longrightarrow }{\mathcal{F}}_{\infty }^{\text{top}}(K_{x}-Z)\end{eqnarray}$$

is a fiber sequence, and thus the composite is the zero functor. Since $U_{x}^{p}$ is contained in $K_{x}-Z$ , the restriction $R_{\infty ,K_{x}}^{U_{x}^{p}}$ factors through $R_{\infty ,K_{x}}^{K_{x}-Z}$ . This implies the first claim. As for the second claim, let us show first that there is a natural equivalence

(13) $$\begin{eqnarray}\displaystyle T_{\infty ,Z}^{K_{x}}\simeq R_{\infty ,X}^{K_{x}}\circ T_{\infty ,Z}^{X}. & & \displaystyle\end{eqnarray}$$

Consider the commutative square on the right-hand side of the following diagram.

The functor induced between the fibers, which is denoted by the dashed arrow, is equivalent to the identity. This gives equivalence (13). As a consequence we obtain

$$\begin{eqnarray}C_{\infty ,K_{x}}^{X}\circ T_{\infty ,Z}^{K_{x}}\simeq C_{\infty ,K_{x}}^{X}\circ R_{\infty ,X}^{K_{x}}\circ T_{\infty ,Z}^{X}\simeq T_{\infty ,Z}^{X}.\end{eqnarray}$$

Indeed, the first equivalence follows from (13) and the second one from Lemma 5.17. This concludes the proof.◻

The proof of Proposition 5.14.

Let $W$ be the set of vertices of $X$ that do not belong to $U$ . Note that $U$ is a connected component of the open subgraph $(X-W)\subset X$ . By induction it is sufficient to prove the claim in the following two cases:

1. (i) when $U$ is equal to $K_{x}$ for some vertex $x$ of $X$ ;

2. (ii) when $U$ is a connected component of $X$ .

Proposition 5.18 gives a proof in the first case. Indeed, it is sufficient to take right adjoints in claim (ii) of Proposition 5.18 to recover the commutativity statement from Proposition 5.14 for this class of open subgraphs. The second case is easier. The complement $X-U$ is open and we have a splitting

$$\begin{eqnarray}{\mathcal{F}}_{\infty }^{\text{top}}(X)\simeq {\mathcal{F}}_{\infty }^{\text{top}}(U)\times {\mathcal{F}}_{\infty }^{\text{top}}(X-U),\end{eqnarray}$$

and the claim follows immediately from here.◻

7.1 Surface topology

This section contains some remarks on surface topology that will be useful in later constructions.

Denote by $\unicode[STIX]{x1D6F4}_{g,n}$ an oriented surface of genus $g$ with $n$ punctures. Since the topology of these surfaces enters our discussion in a relatively coarse way, we will often draw the punctures as if they were boundaries, but strictly speaking $\unicode[STIX]{x1D6F4}_{g,n}$ is a noncompact (if $n>0$ ) manifold without boundary. The surface $\unicode[STIX]{x1D6F4}_{g,n}$ has $n$ ends corresponding to the punctures.Footnote ³

If $\unicode[STIX]{x1D6F4}_{1}$ and $\unicode[STIX]{x1D6F4}_{2}$ are two oriented punctured surfaces, we may form a new surface by the well-known end connect sum operation.

The end connect sum can also be described as attaching a one-handle to $\unicode[STIX]{x1D6F4}_{1}\coprod \unicode[STIX]{x1D6F4}_{2}$ . If alternatively we think in terms of bordered surfaces, the operation consists of adding a strip connecting two boundary components. If $\unicode[STIX]{x1D6F4}_{i}$ has genus $g_{i}$ and $n_{i}$ punctures ( $i=1,2$ ), then $\unicode[STIX]{x1D6F4}\#_{p_{1},p_{2}}\unicode[STIX]{x1D6F4}_{2}$ has genus $g_{1}+g_{2}$ and $n_{1}+n_{2}-1$ punctures.

The effect of end connect sum on skeleta is straightforward.

Figure 3. Decomposition of $\unicode[STIX]{x1D6F4}_{g,n}$ into end connect sum of $g$ copies of $\unicode[STIX]{x1D6F4}_{1,1}$ and $n-1$ copies of  $\unicode[STIX]{x1D6F4}_{0,2}$ .

In this paper, we are interested in skeleta with a certain shape near the punctures.

Model the pair of pants as $\mathbb{C}-\{-2,2\}$ . If $x$ and $y$ are in $\mathbb{C}$ , and $\unicode[STIX]{x1D716}$ is a positive real number, we denote by $S(x,\unicode[STIX]{x1D716})\subset \mathbb{C}$ the circle of center $x$ and radius $\unicode[STIX]{x1D716}$ , and by $I(x,y)\subset \mathbb{C}$ the straight segment joining $x$ and $y$ . We call $\unicode[STIX]{x1D6E9}$ graph the skeleton of the pair of pants given by

$$\begin{eqnarray}S(0,3)\cup I(-3i,3i).\end{eqnarray}$$

We call dumbell graph the skeleton given by

$$\begin{eqnarray}S(-2,1)\cup I(-1,1)\cup S(2,1).\end{eqnarray}$$

The $\unicode[STIX]{x1D6E9}$ graph has a cycle at each of the three punctures, whereas the dumbbell graph has a cycle at only two: for the third puncture, the boundary of the corresponding component of $\unicode[STIX]{x1D6F4}\setminus X$ consists of the entire skeleton. On the other hand, in the $\unicode[STIX]{x1D6E9}$ graph the cycles for any two punctures are not disjoint, whereas in the dumbbell graph they are disjoint. These graphs are shown in Figure 4.

Figure 4. Two skeleta for the pair of pants. (a) $\unicode[STIX]{x1D6E9}$ graph. (b) Dumbbell graph.

In fact, whenever $X$ is a skeleton with a cycle at a particular puncture, $\unicode[STIX]{x1D6F4}$ and $X$ can be decomposed into an end connect sum in a manner similar to that of Example 7.3.

Figure 5. Decomposition of $\unicode[STIX]{x1D6F4}$ into an end connect sum, depending on a choice of path $\unicode[STIX]{x1D6FE}$ between two punctures. The dashed line is for visual reference.

We remark that the proof shows that if $r$ edges are incident to the cycle at $p$ , then there are essentially $r$ choices for how to decompose $\unicode[STIX]{x1D6F4}$ and $X$ as in the lemma.