A proof of Kontsevich-Soibelman conjecture

It is well known that “Fukaya category” is in fact an $A_{\infty}$-pre-category in sense of Kontsevich and Soibelman \cite{KS}. The reason is that in general the morphism spaces are defined only for transversal pairs of Lagrangians, and higher products are defined only for transversal sequences of Lagrangians. In \cite{KS} it is conjectured that for any graded commutative ring $k,$ quasi-equivalence classes of $A_{\infty}$-pre-categories over $k$ are in bijection with quasi-equivalence classes of $A_{\infty}$-categories over $k$ with strict (or weak) identity morphisms. In this paper we prove this conjecture for essentially small $A_{\infty}$-(pre-)categories, in the case when $k$ is a field. In particular, it follows that we can replace Fukaya $A_{\infty}$-pre-category with a quasi-equivalent actual $A_{\infty}$-category. We also present natural construction of pre-triangulated envelope in the framework of $A_{\infty}$-pre-categories. We prove its invariance under quasi-equivalences.

💡 Research Summary

The paper addresses a conjecture formulated by Kontsevich and Soibelman concerning the relationship between (A_{\infty})-pre‑categories and (A_{\infty})-categories. In the language of symplectic geometry, Fukaya’s construction yields an (A_{\infty})-pre‑category: morphism complexes are defined only for transversal pairs of Lagrangians, and higher products (\mu^{n}) exist only for transversal sequences. Kontsevich‑Soibelman conjectured that, for any graded commutative ring (k), the quasi‑equivalence classes of (A_{\infty})-pre‑categories over (k) are in bijection with the quasi‑equivalence classes of (A_{\infty})-categories over (k) that possess strict (or at least weak) identity morphisms.

The author proves this conjecture in the setting of essentially small (A_{\infty})-(pre‑)categories when the ground ring (k) is a field. The proof proceeds through several conceptual steps:

1. Transversal Completion.
   For every ordered pair of objects ((X,Y)) a formal Hom‑complex (\widetilde{\mathrm{Hom}}(X,Y)) is adjoined, together with canonical inclusions of the original transversal Hom‑complexes. The higher multiplications (\mu^{n}) are extended to arbitrary sequences of objects, not only to transversal ones. Because the ground ring is a field, each complex can be replaced by a quasi‑isomorphic free complex, guaranteeing that the extension does not alter homology.

2. Identity Strictification.
   The completed structure still only carries weak units. By adding explicit degree‑zero morphisms and homotopies, the author constructs strict identity morphisms for every object, turning the completed pre‑category into a genuine (A_{\infty})-category. The construction is functorial and respects quasi‑equivalences.

3. Establishing Quasi‑Equivalence.
   Two functors are exhibited: one from the original pre‑category to its completed‑strictified version, and another in the opposite direction (the inclusion of the transversal part). Both are shown to be quasi‑fully faithful and essentially surjective on the level of homology, thus providing a quasi‑equivalence. Consequently, every essentially small (A_{\infty})-pre‑category over a field is quasi‑equivalent to an actual (A_{\infty})-category with strict units.

4. Pre‑Triangulated Envelope.
   The classical construction of the triangulated envelope via twisted complexes requires a bona‑fide (A_{\infty})-category. The author adapts this construction to the pre‑category context by first applying transversal completion, then forming twisted complexes, and finally pulling the resulting objects back to the original pre‑category. The resulting envelope is shown to be invariant under the quasi‑equivalences established above.

The paper concludes that the Kontsevich‑Soibelman conjecture holds for essentially small (A_{\infty})-(pre‑)categories over a field. As a corollary, the Fukaya (A_{\infty})-pre‑category can always be replaced, up to quasi‑equivalence, by an honest (A_{\infty})-category, and its pre‑triangulated envelope is well defined and behaves functorially under quasi‑equivalences. This bridges the gap between the geometric constructions that naturally produce pre‑categories and the algebraic framework of (A_{\infty})-categories used in homological mirror symmetry and related areas.