  In this paper, we define a family of categories for each Liouville manifold,
which is an enhanced version of the category first introduced by Tamarkin.
Using our categories, for any (possibly non-exact immersed) Lagrangian brane,
we develop a theory of sheaf quantization generalizing the previous researches.
In particular, our theory involves the notion of a sheaf-theoretic bounding
cochain, which is a conjectural counterpart of the theory of
Fukaya--Oh--Ohta--Ono. We also study several structures of our categories for
sufficiently Weinstein manifolds and properties known in the classical Tamarkin
category; intersection points estimates, interleaving distances, energy
stability with respect to Guillermou--Kashiwara--Schapira autoequivalence, and
the completeness of the distance. We conjecture that our category is equivalent
to a Fukaya category defined over the Novikov ring.
