A constructive approach to Fourier-Mukai transforms for projective spaces via $A_{\infty}$-functors between pretriangulated dg categories

We discuss the following problem: how can an arbitrary Fourier-Mukai transform $\phi: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow \mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories of two projective spaces of dimensions $a$ and $b$ be expressed in explicit terms as an exact functor between the homotopy categories $\mathrm{K}^{\mathrm{b}}( \mathbb{B}^a ) \rightarrow \mathrm{K}^{\mathrm{b}}( \mathbb{B}^b )$ generated by the full strong exceptional sequences of the line bundles $\mathbb{B}^a = \{\mathcal{O}(-a), \dots, \mathcal{O}\}$ and $\mathbb{B}^b = \{\mathcal{O}(-b), \dots, \mathcal{O}\}$? We show that this problem can be reduced to the following task which is independent of any prescribed Fourier-Mukai kernel: finding an $A_{\infty}$-functor $P$ in explicit terms whose induced functor on homotopy categories yields the embedding of $\{ \mathcal{O}(i) \boxtimes \mathcal{O}(j) \mid i = -2a, \dots, 0, ~~j = -2b, \dots 0 \}$ into $\mathrm{D}^{\mathrm{b}}( \mathbb{P}^{a} \times \mathbb{P}^{b})$. As our main technical tool we provide an explicit formula for the lift of an $A_{\infty}$-functor $F: \mathbf{A} \rightarrow \mathbf{B}$ between a dg category $\mathbf{A}$ and a pretriangulated dg category $\mathbf{B}$ to the pretriangulated hull of $\mathbf{A}$ given by the universal property of pretriangulated hulls. As a further application of this tool, we provide a simple example of two non-isomorphic exact functors between triangulated categories that coincide on the full subcategory generated by a full strong exceptional sequence.