ScalingStacks

Let $Z$ be a curve.

Let us note some basics facts about oriented paths.

Let us note some basic properties of smooth and admissible paths and classes.

Note that $Z^{\operatorname{opp}\nolimits}$ and $Z$ have the same smooth paths. Note also that the notion of “opposite orientation” does not depend on the orientation of $Z$.

We now define a category associated with admissible paths.

We put ${\mathcal{S}}(Z,1)={\mathbf{F}}_{2}[{\mathcal{S}}^{\bullet}(Z,1)]$.

Let $L(Z)=(\bigoplus_{c\in T(Z)}{\mathbf{Z}}e_{c})/(\bigoplus_{c\in T(Z)}{\mathbf{Z}}(e_{c}+e_{\iota(c)}))$. We define a bilinear map $\langle-,-\rangle:R(Z)\times R(Z)\to L(Z)$ by

Note that $(m_{c}+m_{\iota(c)})(\beta)=0$ for all but finitely many $c$’s, hence the sum above is finite. More precisely, let $\zeta$ be a non-identity homotopy class of paths in $Z$. We have $\zeta(0+)\neq\zeta(1-)$ and

If $\zeta$ is admissible and non-identity, then $\zeta(1-)\neq\zeta(0+)$, hence

We define a group $\Gamma^{\prime}(Z)$, a central extension of $R(Z)$ by $L(Z)$. The set of elements of $\Gamma^{\prime}(Z)$ is $L(Z)\times R(Z)$ and the multiplication is given by

We put $\Gamma(Z)=(\bigoplus_{\Omega\in\pi_{0}(Z)}\frac{1}{2}{\mathbf{Z}}e_{\Omega})\times\Gamma^{\prime}(Z)$.

Note that $L(Z)$, $\Gamma^{\prime}(Z)$ and $\Gamma(Z)$ depend only on the $1$-dimensional space underlying $Z$ and on $\iota$.

Let $D$ be a subset of $T(Z)$ such that $D\cap\iota(D)=\emptyset$. We denote by $\Gamma(Z,D)$ the quotient of $\Gamma(Z)$ by the central subgroup generated by $\{e_{c}+\frac{1}{2}e_{\Omega}\}$, where $c\in D$ and $\Omega$ is the connected component of $Z$ containing $c$. The canonical map $\bigoplus_{\Omega\in\pi_{0}(Z)}\frac{1}{2}{\mathbf{Z}}e_{\Omega}\to\Gamma(Z,D)$ is injective and we identify $(\frac{1}{2}{\mathbf{Z}})^{\pi_{0}(Z)}$ with its image.

We put a partial order on $\Gamma(Z,D)$ by setting $g_{1}\geq g_{2}$ if $g_{1}g_{2}^{-1}\in(\frac{1}{2}{\mathbf{Z}}_{\geq 0})^{\pi_{0}(Z)}$.

We define $\bar{\Gamma}(Z,D)$ to be the quotient of $\Gamma(Z,D)$ by the central subgroup generated by $\frac{1}{2}e_{\Omega}-\frac{1}{2}e_{\Omega^{\prime}}$ for $\Omega,\Omega^{\prime}\in\pi_{0}(Z)$.

The image of $(\frac{1}{2}{\mathbf{Z}}_{\geq 0})^{\pi_{0}(Z)}$ in $\bar{\Gamma}(Z,D)$ is $\frac{1}{2}{\mathbf{Z}}$ (where $\frac{1}{2}e_{\Omega}\mapsto\frac{1}{2}$). Let $r\in\frac{1}{2}{\mathbf{Z}}$. We still denote by $r$ the image of $r$ in $\bar{\Gamma}(Z,D)$. Given $x\in\bar{\Gamma}(Z,D)$, we put $x+r=x\cdot r=r\cdot x$.

We define a partial order on $\bar{\Gamma}(Z,D)$ by setting $g_{1}\geq g_{2}$ if $g_{1}g_{2}^{-1}\in\frac{1}{2}{\mathbf{Z}}_{\geq 0}$.

Given $z\in Z_{o}$, we denote by $C(z)^{+}$ the set of $c\in C(z)$ such that there is an oriented path $\gamma$ in $Z$ with $m_{c}^{+}(\gamma)=1$. Note that $C(z)=C(z)^{+}\coprod\iota(C(z)^{+})$. Note also that given $\zeta$ an oriented homotopy class of paths in $Z$, we have

Given $E$ a subset of $Z_{o}$, we put $E^{+}=\coprod_{z\in E}C(z)^{+}$.

Let $M$ be a subset of $Z$. We denote by $L_{M}(Z)$ the subgroup of $L(Z)$ generated by elements $e_{c}$ with $\mathrm{pt}(c)\in M$. The restriction of the pairing $\langle-,-\rangle$ to $R_{M}(Z)\times R_{M}(Z)$ takes values in $L_{M}(Z)$ and we denote by $\Gamma^{\prime}_{M}(Z)$ the subgroup of $\Gamma^{\prime}_{M}(Z)$ with elements $(m,\alpha)$ where $m\in L_{M}(Z)$ and $\alpha\in R_{M}(Z)$. Finally, we define $\Gamma_{M}(Z)$ as the subgroup $(\bigoplus_{\Omega\in\pi_{0}(Z),\ M\cap\Omega\neq\emptyset}\frac{1}{2}{\mathbf{Z}}e_{\Omega})\times\Gamma^{\prime}_{M}(Z)$ of $\Gamma(Z)$.

We denote by $\Gamma_{M}(Z,D)$ (resp. $\bar{\Gamma}_{M}(Z,D)$) the image of $\Gamma_{M}(Z)$ in $\Gamma(Z,D)$ (resp. $\bar{\Gamma}(Z,D)$).

If $M\subset Z_{o}$, we put $\Gamma_{M^{+}}(Z)=\Gamma_{M}(Z,M^{+})$.

Let $f:Z\to Z^{\prime}$ be a morphism of curves.

It follows from the previous lemma that the morphism $f$ induces a functor $f:{\mathcal{S}}^{\bullet}(Z,1)\to{\mathcal{S}}^{\bullet}(Z^{\prime},1)$. We have constructed a functor ${\mathcal{S}}^{\bullet}(-,1)$ from the category of curves to the category of pointed categories.

Let us state a version of Lemma 7.1.20 for morphisms of curves.

Note that $f$ induces an injective morphism of groups $f:L(Z)\to L(Z^{\prime})$ and a map $f:\pi_{0}(Z)\to\pi_{0}(Z^{\prime})$.

The next lemma is an immediate consequence of Lemma 7.1.24.

It follows from Lemmas 7.3.15 and 7.1.25 that we have a morphism of groups $f:\Gamma(Z)\to\Gamma(Z^{\prime}),\ (r,(m,\alpha))\mapsto\bigl(f(r),(f(m),f(\alpha))\bigr)$ which restricts to an injective morphism of groups $\Gamma^{\prime}(Z)\to\Gamma^{\prime}(Z^{\prime})$.

Let $D$ be a subset of $T(Z)$ such that given $z\in\operatorname{pt}\nolimits(D)$, the composition $D\cap\operatorname{pt}\nolimits^{-1}(z)\to C(z)\to C(z)/\iota$ is bijective. The morphism $f:\Gamma(Z)\to\Gamma(Z^{\prime})$ induces a morphism $f:\Gamma(Z,D)\to\Gamma(Z^{\prime},f(D))$. Let $g,h\in\Gamma(Z,D)$. If $g<h$, then $f(g)<f(h)$. If $f:\pi_{0}(Z)\to\pi_{0}(Z^{\prime})$ is injective and $f(g)<f(h)$, then $g<h$.

Finally, the morphism $f:\Gamma(Z,D)\to\Gamma(Z^{\prime},f(D))$ induces a morphism $f:\bar{\Gamma}(Z,D)\to\bar{\Gamma}(Z^{\prime},f(D))$. Given $g,h\in\bar{\Gamma}(Z,D)$, we have $g<h$ if and only if $f(g)<f(h)$.

Let $Z_{1},\ldots,Z_{r}$ be the connected components of $Z$. There are isomorphisms of groups $R(Z_{1})\times\cdots\times R(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R(Z)$ and $L(Z_{1})\times\cdots\times L(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L(Z)$ given by the inclusions $Z_{i}\hookrightarrow Z$. They induce an isomorphism of groups

The inclusions $Z_{i}\hookrightarrow Z$ induce pointed functors ${\mathcal{S}}^{\bullet}(Z_{i},1)\to{\mathcal{S}}^{\bullet}(Z,1)$ and give rise to an isomorphism of pointed categories

Let $f:Z\to Z^{\prime}$ be a morphism of curves. We define a non-multiplicative “functor” $f^{\#}:\operatorname{add}\nolimits({\mathcal{S}}(Z^{\prime},1))\to\operatorname{add}\nolimits({\mathcal{S}}(Z,1))$. It commutes with coproducts but is not a functor, i.e., it is not compatible with composition for a general $f$. We put $f^{\#}(z^{\prime})=\coprod_{z\in f^{-1}(z^{\prime})}z$. Given $\zeta^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime},1)}(z^{\prime}_{1},z^{\prime}_{2})$ non-zero, we define $f^{\#}(\zeta^{\prime})$ to be

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  $\operatorname{id}\nolimits$ if $\zeta^{\prime}=\operatorname{id}\nolimits$
  

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  $0$ if $\zeta^{\prime}$ does not lift to an admissible class of paths in $Z$
  

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  the composition

  $$\coprod_{z\in f^{-1}(z^{\prime}_{1})}z\xrightarrow{\text{projection}}z_{1}\xrightarrow{\zeta}z_{2}\xrightarrow{\text{inclusion}}\coprod_{z\in f^{-1}(z^{\prime}_{2})}z$$

  where $\zeta:z_{1}\to z_{2}$ is the unique lift of $\zeta^{\prime}$, otherwise (cf Lemma 7.3.14).

We denote by $f^{-1}(\zeta^{\prime})$ the set of admissible lifts of $\zeta^{\prime}$. We have $f^{\#}(\zeta^{\prime})=\sum_{\zeta\in f^{-1}(\zeta^{\prime})}\zeta$.

Given $\zeta^{\prime}_{1}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime},1)}(z^{\prime}_{1},z^{\prime}_{2})$ and $\zeta^{\prime}_{2}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime},1)}(z^{\prime}_{2},z^{\prime}_{3})$ such that $f^{\#}(\zeta^{\prime}_{1})\neq 0$ and $f^{\#}(\zeta^{\prime}_{2})\neq 0$, we have $f^{\#}(\zeta^{\prime}_{2})f^{\#}(\zeta^{\prime}_{1})=f^{\#}(\zeta^{\prime}_{2}\zeta^{\prime}_{1})$ (cf Lemma 7.3.13).

Given $f^{\prime}:Z^{\prime}\to Z^{\prime\prime}$ a morphism of curves, we have $(f^{\prime}f)^{\#}=f^{\#}f^{\prime\#}$.

Since quotient maps are strict, we have the following consequence of Lemma 7.3.16.

The construction $Z\mapsto\operatorname{add}\nolimits({\mathcal{S}}(Z,1))$ and $f\mapsto f^{\#}$ defines a contravariant functor from the category of curves with strict morphisms to the category of ${\mathbf{F}}_{2}$-linear categories.

Lemma 7.3.17 and Proposition 7.3.18 have the following consequence.

Note that Proposition 7.3.19 provides an identification of ${\mathcal{S}}(Z/\!\!\sim,1)$ with a (non-full) subcategory of $\operatorname{add}\nolimits({\mathcal{S}}(Z,1))$.

Consider a chord diagram $({\mathcal{Z}},{\mathbf{a}})$ as in §7.2.4 with associated singular curve $Z$. Define

Proposition 7.3.19 shows that the algebra ${\mathcal{A}}(Z,1)$ is the opposite of Zarev’s algebra ${\mathcal{A}}_{Za}({\mathcal{Z}},1)$ [Za, Definition 2.6] (this will be explained for the more general algebras ${\mathcal{A}}(Z)$ in §7.4.11).

$\bullet\$Consider the chord diagram $({\mathbf{R}},\{\{1,3\},\{2,4\}\})$.

The associated singular curve $Z$ is the quotient of oriented ${\mathbf{R}}$ by the relation whose non-trivial equivalence classes are $1=\{1,3\}$ and $2=\{2,4\}$.

The full pointed subcategory of ${\mathcal{S}}(Z,1)$ with object set $\{1,2\}$ is generated by $\alpha,\alpha^{\prime}:1\to 2$ and $\beta:2\to 1$ with relations $\beta\alpha=\alpha^{\prime}\beta=0$. This corresponds to the well-known “torus algebra” in bordered Floer homology.

$\bullet\$Consider the chord diagram $(S^{1},\{\{\pm 1\},\{\pm i\}\})$.

The associated singular curve $Z$ is the quotient of oriented $S^{1}$ by the relation whose non trivial equivalence classes are $1=\{\pm 1\}$ and $2=\{\pm i\}$.

The full pointed subcategory of ${\mathcal{S}}(Z,1)$ with object set $\{1,2\}$ is generated by $\alpha,\alpha^{\prime}:1\to 2$ and $\beta,\beta^{\prime}:2\to 1$ with relations $\beta\alpha=\alpha^{\prime}\beta=\alpha\beta^{\prime}=\beta^{\prime}\alpha^{\prime}=0$. A curved $A_{\infty}$-deformation of this subcategory appears in [LiOzTh3].

We have