An essential aspect of $Coh(X^{\vee})$ is that new bundles can be constructed from extensions of known bundles $E_{1},E_{2}$, namely

$$0\to E_{1}\to E\to E_{2}\to 0.$$

Such extension sequences are classified by the complex vector space $\text{Ext}^{1}(E_{2},E_{1})$. Due to the $\mathbb{C}^{*}$-scaling, the choice of $E$ is parametrised by the projective space $\mathbb{P}(\text{Ext}^{1}(E_{2},E_{1}))$. In general, extensions are not symmetric in $E_{1}$ and $E_{2}$. The extensions

$$0\to E_{2}\to E\to E_{1}\to 0$$

are classified by $\text{Ext}^{1}(E_{1},E_{2})$, which is a quite different space. Extensions can also be viewed as distinguished triangles in $D^{b}Coh(X^{\vee})$.

In the mirror picture, exact sequences do not make a priori sense, but one can talk about distinguished triangles, whose geometric sources are the graded Lagrangian connected sums $L_{1}\#L_{2}$, fitting into a distinguished triangle

$$L_{1}\to L_{1}\#L_{2}\to L_{2}\to L_{1}[1].$$

Such distinguished triangles are classified by $HF^{1}(L_{2},L_{1})$.³²³² 32 The caveat is that unlike bundles, the neck length of the Lagrangian connected sum cannot be arbitrarily large. Again there is a scaling symmetry related to the neck size of the Lagrangian connected sum, and there is an asymmetry between $L_{1}\#L_{2}$ and $L_{2}\#L_{1}$.

This analogy is a prime example of homological mirror symmetry. On either side, only pure complex geometry/pure symplectic geometry appears.