Non-unital monoidal category of contact manifolds and Legendrian correspondence

@misc{oh2026nonunital,
  title = {{Non-unital monoidal category of contact manifolds and Legendrian correspondence}},
  author = {Yong-Geun Oh and Junhyuk Park},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.15599},
  eprint = {2606.15599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15599},
}

TY  - UNPB
TI  - Non-unital monoidal category of contact manifolds and Legendrian correspondence
AU  - Yong-Geun Oh
AU  - Junhyuk Park
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2606.15599
UR  - https://arxiv.org/abs/2606.15599
AB  - There are two purposes of the present paper which are interrelated. The first goal is to construct the structure of a non-unital monoidal category Cont of contact manifolds, not necessarily coorientable, by developing the contact topology \emph{without contact forms}. The non-unital monoidal product is the functorial contact product ⋆, called star product, introduced in \cite{oh:shelukhin-conjecture}. We prove that the product ⋆ is associative and there exist a collection α= {α_X,Y,Z} of the \emph{associator} isomorphisms α_X,Y,Z: X ⋆ (Y⋆ Z) ≅ (X ⋆ Y) ⋆ Z for X, Y, Z ∈ Cont, that satisfy the pentagon axiom, i.e., that the triples (Cont, ⋆, α) form a nonunital monoidal category. The second goal is to develop the calculus of Legendrian correspondences, which are by definition embedded Legendrian submanifolds of the contact product Q ⋆ Q'. Legendrian correspondences will play the role of 1-morphisms in the $2$-categorical structure to be equipped with Cont whose two morphisms are contact instanton cohomologies HI(R_ab,R'_ab) associated to a pair of Legendrian correspondences R_ab, R'_ab ∈ Leg(Q_a,Q_b). With this future application in mind, we define the composition of Legendrian correspondences and prove that the composition of a generic pair is again embedded and hence canonically becomes a Legendrian correspondence.
ER  -