The following is a comment to David’s post “Nadler-Zaslow without Floer theory”. There was some error posting it as a comment probably because it is too long, so I turn it into a post.

Based on David’s comment about specializations, I think the following categorical calculations of Floer cohomology (and higher $A_\infty$-structures) should work, though there are some parts I’m not quite sure. Before describing this, some (easy) comments are: when we look at the projection of the Legendrian lifting of a Lagrangian graph $L_{df}$ in $T^*M$ to $M\times \mathbb{R}$, it is just the graph of $f$ in $M\times\mathbb{R}$; when we have two Lagrangian graphs $L_{df_1}$ and $L_{df_2}$, and consider the projection of $\tilde{L}_{df_1}^{s}\cup\tilde{L}_{df_2}$ for $s\in\mathbb{R}$, the moments when critical behavior happens are corresponding to $s\in CritVal(f_2-f_1)$, so in particular there are only differentials from large $s$ to some $s$ (note that the grading for Floer cohomology is opposite to Morse homology). Essentially we are doing Morse theory, and we wanted some algebraic framework (avoiding holomorphic discs) of this to extend to arbitrary Lagrangians, which can be thought as generalized Morse theory.

Given any two (exact) Lagrangians $L_1,L_2$, consider the infinity of the conical Lagrangians $\bigcup\limits_{s\in\mathbb{R}}Cone(ExCone(L_1)\cap \{\tau=1\}+s;s,-1)\subset T^*(M\times\mathbb{R}\times\mathbb{R})$ with coordinate $(x,\xi; t,\tau; s, \theta)$ ($ExCone(L_1)\cap\{\tau=1\}+s$ means shift the $t$-coordinate of $ExCone(L_1)\cap\{\tau=1\}$ by $s$) and $ExCone(L_2)\times\mathbb{R}\times\{0\}\subset T^*(M\times\mathbb{R}\times\mathbb{R})$, and denote these two Legendrians by $\Lambda_1$ and $\Lambda_2$ respectively. Consider $Sh_{\Lambda_1\cup\Lambda_2}(M\times\mathbb{R}\times\mathbb{R})$, then this category is clearly invariant under Lagrangian isotopies of $L_1$ and $L_2$ (relative to the infinity). Similarly $\lim\limits_{s\rightarrow\pm\infty}Sh_{\tilde{L}_1^{s}\cup \tilde{L}_2}(M\times\mathbb{R})$ is invariant under Lagrangian isotopies. There are restriction functors $\rho_{\pm\infty}$ from $Sh_{\Lambda_1\cup\Lambda_2}(M\times\mathbb{R}\times\mathbb{R})$ to $\lim\limits_{s\rightarrow\pm\infty} Sh_{\tilde{L}_1^{s}\cup \tilde{L}_2} (M\times\mathbb{R})$ respectively. The proposed “categorical” Floer complex is

$$\mathcal{CF}(L_1,L_2)=:Cone(\rho_{-\infty})$$

(and $\mathcal{CF}(L_2,L_1)$ should be $Cone(\rho_{+\infty})$). Here is a question that I wonder about.

Question: How to get a graded vector space from taking cone of categories?

Now let $s_1<s_2<…<s_n$ be the critical moments where $\tilde{L}_1^{s}\cap\tilde{L}_2\neq\emptyset$. Let $Sh_{s}$ denote for $Sh_{\tilde{L}_1^{s}\cup \tilde{L}_2} (M\times\mathbb{R})$ and $Sh_{(i)}$ denote for $Sh_{s}$ for any $s_{i}<s<s_{i+1}$. Then there are specialization functors

$$Sh_{s_i}\leftarrow Sh_{(i)}\rightarrow Sh_{s_{i+1}},$$

and the category $Sh_{\Lambda_1\cup\Lambda_2}(M\times\mathbb{R}\times (s_{i-1}, s_{i+1}))\simeq Sh_{(i-1)}\times_{Sh_{s_i}} Sh_{(i)}$. One can assign the intersection points of $L_1$ with $L_2$ at level $s_i$ the category

$$Cone(Sh_{(i-1)}\times_{Sh_{s_i}} Sh_{(i)}\rightarrow Sh_{(i-1)}).$$

(Maybe it’s safer to use $Cone(Sh_{(0)}\times_{Sh_{s_1}}\times\cdots Sh_{(i-1)}\times_{Sh_{s_i}} Sh_{(i)}\rightarrow Sh_{(0)}\times_{Sh_{s_1}}\times\cdots Sh_{(i-1)})$ and change the $\mathcal{C}_{[k,l]}$ below accordingly. Are there essential differences?)

In the following, let’s use the notation

$$\mathcal{C}_{[k,l]}=: Cone(Sh_{(k)}\times_{Sh_{s_{k+1}}}Sh_{(k+1)}\times\cdots\times_{Sh_{s_{l}}}Sh_{(l)}\rightarrow Sh_{(k)})$$

for any $l>k$.

There are natural exact triangles (is this notion correct for a category of categories? This is just like a filtration of complexes) for any $i<j<k$,

$$\mathcal{C}_{j,k}\rightarrow \mathcal{C}_{i,k}\rightarrow \mathcal{C}_{i,j}.$$

In particular, if $j=i+1, k=i+2$, then the above exact triangle determines a map

$$\mathcal{C}_{i,i+1}\rightarrow \mathcal{C}_{i+1,i+2}[1],$$

which should be viewed as counting flow lines from intersection points at level $s_{i+1}$ to intersection points at level $s_{i+2}$ (it seems that I get Morse homology, but one should be able to get Floer cohomology by assigning each point with $Cone(Sh_{(i-1)}\times_{Sh_{s_i}} Sh_{(i)}\rightarrow Sh_{(i)})$). Moreover, one could trace all the differentials

$$\partial: \mathcal{C}_{i,i+1}\rightarrow\mathcal{C}_{i+1, n}[1]$$

from the exact triangle $\mathcal{C}_{i+1,n}\rightarrow \mathcal{C}_{i,n}\rightarrow \mathcal{C}_{i,i+1}$

Here is an example illustrating this.

There are also similar constructions for (higher) compositions. For example, given $L_0, L_1, L_2$, to compute $Hom(L_1,L_2)\otimes Hom(L_0,L_1)\rightarrow Hom(L_0,L_2)$, one takes $\bigcup\limits_{s_1,s_2}Cone(ExCone(L_0)\cap\{\tau=1\}+s_1+s_2; s_1,-1;s_2,-1)$, $\bigcup\limits_{s_1,s_2}Cone(ExCone(L_1)\cap\{\tau=1\}+s_1;s_1,-1;s_2,0)$ and $ExCone(L_2)\times\mathbb{R}\times\{0\}\times\mathbb{R}\times\{0\}$ in $T^*(M\times\mathbb{R}^3)$ with coordinate $(x,\xi; t,\tau; s_1,\theta_1;s_2,\theta_2)$ and $ExCone(L_0)\cap\{\tau=1\}+s_1+s_2$ means shifting the $t$ coordinate by $s_1+s_2$. Let $\{s_1^{(i)}\}, \{s_2^{(i)}\}$ and $\{s_3^{(i)}\}$ be the set of critical values for the pairs $(\tilde{L}_1, \tilde{L}_2)$, $(\tilde{L}_0, \tilde{L}_1)$ and $(\tilde{L}_0, \tilde{L}_2)$ respectively. Then one draws the hyperplanes for these critical values as in this picture (these divide the $(s_1,s_2)$-plane into regions with different configurations of $(\tilde{L}_0^{s_1+s_2}, \tilde{L}_1^{s_1}, \tilde{L}_2)$. To calculate compositions, one finds all possible triangles like the ones in yellow, and finds all possible “Morse trees” as illustrated in blue. The formulation of the composition map is similar as before: along a Morse tree, one can form fiber product of sheaf categories, and take appropriate cones. For higher compositions $\mu_n$, one considers $n$-simplices and then the “dual” to them are Morse trees with $n-1$ inputs and 1 output.

If one can show that these give an $A_\infty$-structure, then these would give a purely algebraic description (avoiding counting discs) for $\mu_n$ in $Fuk(T^*M)$.

Note: Several changes have been made, especially the map (**) has been deleted since it doesn’t make sense.

Updated 4/17: From yesterday’s discussion with David (Berkeley), I learned that taking cone of categories is not well-defined, since the category of categories is not stable. He suggests that it might be more natural to construct a 2-category over the $E_1$-operad, rather than trying to construct an $A_\infty$-category, which is just a $\mathbb{C}$-linear category over the $E_1$-operad.