Almost mathematics, Persistence module, and Tamarkin category

We give a precise unification of three theories that are widely used by symplectic geometers: (Almost) modules over the Novikov ring, Persistence modules, and the Tamarkin category. Our method provides new input in this direction, especially in relation to Vaintrob's Novikov/log-perfectoid mirror symmetry for Novikov toric schemes. The results of this paper can also be treated as a study of persistent homology from a higher algebra point of view. As applications, we establish a version of homological mirror symmetry over the Novikov ring for toric varieties and propose a conjecture for homological mirror symmetry over the Novikov ring for log Calabi-Yau varieties.