One natural idea is that we only develop Floer theory for sufficiently smooth Lagrangians (eg. immersed Lagrangians, isolated $T^{2}$-cones, etc), and formally treat Lagrangian currents using approximation by smooth objects. Suppose $L_{i}$ are sufficiently smooth Lagrangian branes in the same $D^{b}Fuk(X)$ class, and $L_{i}\to L$ in the varifold/current topology, and assume the brane structures provide a Cauchy sequence in some appropriate sense, then one formally declare the Lagrangian current $L$ as carrying an object in the same $D^{b}Fuk(X)$ class. A weak Lagrangian brane would then tautologically be an equivalence class of Cauchy sequences. The same Lagrangian current may in principle support many different formal brane structures, not necessarily all in the same derived category class.

In this perspective, weak Lagrangian branes are indirect constructions, whose properties amount to quantitative properties of sufficiently smooth Lagrangians that can be bounded in terms of a priori quantities such as the distance on the branes, the flat norm on the currents, the Hausdorff distance between the Lagrangians, etc.

One concrete notion of distance is as follows (cf. [35, Definition 2.2], see also [10, section 5]). We can look for the $\alpha,\beta$ representing generators in $HF^{0}(L,L^{\prime})$ and $HF^{0}(L^{\prime},L)$ with cohomological compositions equal to the identity; in the almost calibrated case $CF^{-1}(L,L^{\prime})=0$, so $\alpha,\beta$ are unique up to scaling. Since all bounding cochains and $A_{\infty}$ products have non-negative Novikov exponents, and the sum of Novikov exponents add up to zero, we must have some negative Novikov exponent for $\alpha$ or $\beta$. In our context, the Novikov exponent amounts to $(f_{L}-f_{L^{\prime}})(p)$ at $p\in CF^{0}(L,L^{\prime})$ and $(f_{L^{\prime}}-f_{L})(q)$ at $q\in CF^{0}(L^{\prime},L)$. The quantity

$$-\min\{\text{Novikov exponents among all intersection points of $\alpha,\beta$}\}$$

provides a candidate notion of distance $d(L,L^{\prime})$ between Lagrangian branes. Notice this distance bounds the energy of the holomorphic discs with boundary on $L,L^{\prime}$. Given three objects $L,L^{\prime},L^{\prime\prime}$, by considering the composition of the generators, it is easy to deduce $d(L,L^{\prime\prime})\leq d(L,L^{\prime})+d(L^{\prime},L^{\prime\prime})$.

Does this notion of distance have any precompactness property? Namely, given a sequence of sufficiently smooth Lagrangian objects $L_{i}\in\mathcal{L}$, (eg. a minimizing sequence for the Solomon functional), and assuming the Lagrangian potentials are uniformly bounded, then up to making gauge equivalent choices of local systems and bounding cochains, when can we extract a Cauchy subsequence?

One may also ask if the weak Lagrangian branes agree with ordinary Lagrangian branes in the case of smooth immersed Lagrangians: