Blocked construction interface for multi-object Fukaya categories and polygon morphisms

This item records a blocked construction interface, not an assertion that a general Fukaya category has been constructed. Choose a smooth symplectic manifold $(M,\omega )$, not assumed compact or boundaryless here, a smooth $\omega$-tame almost-complex structure $J$, a coefficient field $\mathbb{k}$, and a set of compact branes. The analytic regime and its compactness-at-infinity or boundary hypotheses must be stated for the selected target; this interface selects no universal regime. A brane here is an oriented compact embedded Lagrangian with a chosen relative-spin structure; it may also carry a declared finite-rank $\mathbb{k}$-local system and a grading datum. A local system means a locally constant sheaf of finite-dimensional $\mathbb{k}$-vector spaces with isomorphisms as parallel transports. For an integer grading, let $\mathrm{Lag}(TM,\omega )$ be the bundle whose fiber at $p$ is the set of Lagrangian subspaces of $(T_{p}M,{\omega }_p)$, choose a covering $\stackrel{}{\mathrm{Lag}}\to \mathrm{Lag}(TM,\omega )$, and require a grading to be a lift of the Gauss map $p\mapsto T_{p}L$; a different convention, including a $\mathbb{Z}\mathrm{/}2$ grading, must be stated explicitly.

For a transverse pair $L_0,L_1$, use the intersection-point generator model from Lagrangian intersection and Hamiltonian chord data. Fix the morphism convention that an intersection generator $x\in L_0\cap L_1$ carries the fiber factor ${\mathrm{Hom}}_{\mathbb{k}}(E_{0,x},E_{1,x})$, where $E_{j,x}$ is the fiber of the chosen local system on $L_j$ at $x$; a chord from $L_0$ to $L_1$ carries ${\mathrm{Hom}}_{\mathbb{k}}(E_{0,\gamma (0)},E_{1,\gamma (1)})$. Over a field of characteristic different from two, choose for each generator a rank-one graded $\mathbb{k}$-module $o_x$ as its orientation line and coherent identifications of these lines under polygon gluing; in characteristic two take $o_x=\mathbb{k}$ in degree zero and suppress signs. Thus each generator summand includes its Hom factor tensor $o_x$, and its degree is fixed by the declared brane grading and orientation convention. Polygon coefficients use the boundary parallel transports of the local systems. A Hamiltonian-chord model requires its own perturbation convention. Complete the candidate morphism modules in the energy filtration of ${\mathrm{\Lambda }}_{\nu ,\mathbb{k}}^0$ from Universal Novikov completion and energy filtration, whose convolution ring and filtration are checked in Novikov convolution is a filtered ring operation; the selected regime must make this completion and its energy weights well-defined. For an ordered sequence $L_0,\dots ,L_k$ with $k\ge 1$, the intended operations have type $m_k:CF(L_0,L_1){\otimes }_{{\mathrm{\Lambda }}_{\nu ,\mathbb{k}}^0}\cdots {\otimes }_{{\mathrm{\Lambda }}_{\nu ,\mathbb{k}}^0}CF(L_{k-1},L_k)⟶CF(L_0,L_k),$ and each object has curvature element $m_0^L\in CF(L,L)$. For each fixed cyclic sequence $L_0,\dots ,L_k$, the candidate polygon moduli use the stable disk domains and evaluation maps from Stable J-holomorphic disks with boundary markings: the boundary arcs have their specified Lagrangian labels; in the local convention, inputs at $z_i$ for $1\le i\le k$ lie in $L_{i-1}\cap L_i$, and the output at $z_0$ lies in $L_k\cap L_0$. For each relative class $\beta$ on a fixed combinatorial domain, as defined in the stable-disk item, a proposed count is weighted by its symplectic area in the selected Novikov convention; sums over classes must converge in the energy filtration. The local relative-area lemma Relative disk area is invariant under relative homotopy proves homotopy invariance only on a fixed combinatorial domain, so compatibility across changing domain types remains an obligation. A chosen regularization would define the operation maps $m_k$. The category bar module is the algebraic direct sum of composable finite words in the shifted morphism modules, with length-zero summands indexed by objects; each $m_k$ extends by insertion to a degree-one coderivation ${\widehat{m}}_k$. The category-level A-infinity identities are the components of $D^2=0$ for $D={\sum }_{k\ge 0}{\widehat{m}}_k$, with $m_0$ retained for a curved model. In the $k=2$ diagram, the arrow indicates the output direction from $m_2(x_1,x_2)$ to $x_0$; the diagram supplies no regularity or gluing argument.

Here the positive boundary order is $z_0,z_1,z_2$: $x_1\in L_0\cap L_1$, $x_2\in L_1\cap L_2$, and $x_0\in L_2\cap L_0$ is the output generator in $CF(L_0,L_2)$. This is a labeling convention only.

Write $C(L)=CF(L,L)$ for a chosen self-Floer chain model, including the perturbation needed to define it. A cohomological unit is a class $[e_L]\in \ker (m_1)\mathrm{/}\mathrm{im}(m_1)$ in this self-Floer model, defined only when $m_1^2=0$, that acts as an identity for the induced $m_2$. In the shifted source convention, a strict unit $e_L\in C(L)[1{]}^{-1}$ satisfies $m_{k+1}(x_1,\dots ,e_L,\dots ,x_k)=0$ for $k\ge 2$ or $k=0$, and $m_2(e_L,x)=(-1{)}^{d(x)}{m}_2(x,e_L)=x$; here $d(x)$ is the unshifted cohomological degree, so its shifted degree is $d(x)-1$ and the equivalent shifted exponent is $\mathrm{\mid }x\mathrm{\mid }+1$. A multi-object strict-unit system is additional data.

A claimed category must state the selected analytic regime, all brane/orientation/grading/local-system conventions, coefficient field and energy filtration, polygon domains and compactification, regularization or perturbations, convergence of operation sums, units, and coherent gluing. A one-dimensional compactified polygon moduli space must have its codimension-one boundary identified with the signed broken-polygon configurations that give the A-infinity identities; over ${\mathbb{F}}_2$ signs may be suppressed, but regularity, compactness, and gluing are still required. The FOOO source also states pairwise filtered A-infinity bimodule structures for transverse pairs (Theorem 3.9 and Proposition 3.10), but these do not provide coherent composition operations for every sequence of objects or the category-level polygon identities required here. Kontsevich's cited 1994 discussion is preliminary historical motivation and is not used as a construction proof. No such general multi-object construction is proved here; compactness, transversality or virtual regularization, orientations, convergence, units, and composition remain explicit unresolved obligations.

For the generator factors above, take the orientation line $o_x$ to be a chosen rank-one graded $\mathbb{k}$-module with coherent gluing identifications; over a field of characteristic two take $o_x=\mathbb{k}$ in degree zero. Thus in the intersection model $CF(L_0,L_1)={⨁}_{x\in L_0\cap L_1}{\mathrm{\Lambda }}_{\nu ,\mathbb{k}}^0{\otimes }_{\mathbb{k}}{\mathrm{Hom}}_{\mathbb{k}}(E_{0,x},E_{1,x}){\otimes }_{\mathbb{k}}{o}_x\mathrm{.}$ This direct sum is finite: transversality makes $L_0\cap L_1$ discrete, while compactness of the embedded Lagrangians makes it compact. In the chord model use the analogous direct sum over the selected nondegenerate chords, with the endpoint fibers stated above; finiteness and energy control for that generator set are part of the chosen analytic regime. The displayed algebraic tensor products in the operation type must extend continuously to the energy-filtered completed tensor products selected by that regime.

For a stable map $u$ with normalized components $u_C$, define its total symplectic area by $E_{\omega }(u)={\sum }_C{\int }_C{u}_C^{\ast }\omega$. For a fixed combinatorial relative class $\beta$, this value is homotopy-invariant by Relative disk area is invariant under relative homotopy. A proposed count in class $\beta$ is weighted by a declared Novikov monomial $q^{\gamma (\beta )}$ whose valuation obeys $\nu (\gamma (\beta ))=E_{\omega }(\beta )$; sums over classes must converge in the energy filtration. The class-to-exponent map and any grading convention must be specified, and compatibility of relative classes across changing domain types remains an obligation.

Symplectic and almost-complex curve data

Relatively spin Lagrangian data

Stable J-holomorphic disks with boundary markings

Lagrangian intersection and Hamiltonian chord data

Universal Novikov completion and energy filtration

Relative disk area is invariant under relative homotopy

Novikov convolution is a filtered ring operation

Tame J-holomorphic area equals metric Dirichlet energy

FOOO single-object filtered curved A-infinity algebra