Composability of (L^w, L) in the original abstract wrapped Floer setup

Ascertain whether, in Ganatra–Pardon–Shende’s Definition 2.11 of an abstract wrapped Floer setup, when the wrapping category \(\tilde{R}_L\) of a Lagrangian \(L\) consists only of the identity \(id_L\) and condition (iv) (factorization property) is applied with \(K\) a single Lagrangian, the pair \((L^w, L)\) is composable (i.e., belongs to \(L_1\)) for the resulting \(L^w\) appearing in the factorization \(L \leadsto L^w \leadsto L\), so that the corresponding elements in the decorated poset \(P\) can be endowed with the necessary order relation used in constructing sufficiently wrapped decorated posets.

Background

The paper revisits the original abstract wrapped Floer setup of Ganatra–Pardon–Shende (Definition 2.11) and proposes a modification that removes certain choices and the factorization axiom. In the original setup, each Lagrangian $L$ is equipped with a wrapping category $\tilde{R}_L$, and a factorization property is imposed to relate morphisms in $\tilde{R}_L$ to composable tuples of Lagrangians.

While geometrically wrapping categories typically contain many distinct objects, the original formal definition does not exclude the degenerate case $\tilde{R}_L = \{ id_L \}$. In the context of constructing sufficiently wrapped decorated posets (used to define the wrapped Fukaya category via localization), the authors point out a gap: even if the identity $id_L$ factorizes through some $L^w$ with $(L^w, K) \in L_1$, the definition does not guarantee $(L^w, L) \in L_1$. Without this composability, the desired order relation on the corresponding elements of the poset $P$ cannot be ensured, obstructing parts of the construction.