This talk is motivated by the recent work of Morton and
Samuelson which states that the Turaev skein algebra for torus is
isomorphic to a specialization of the elliptic Hall algebra. In this
talk we introduce the category $B_q$ which is an $\mathbb{F}_1$-analogue
of the category of coherent sheaves over an elliptic curve. Although our
category is not an abelian category, even nor an additive category, it
is an example of so-called belian and quasi-exact category in the sense
of Deitmar. Then we can consider the Hall algebra $U_q$ associated to
$B_{q}$ using Szczesny's construction of Hall algebra for monoid
representations. The main statement is that $U_{q}$ is isomorphic
to the Turaev skein algebra of torus. Thus our construction gives the
`B-side counter-part' of the torus skein algebra directly, not replying
on an `bi-hand' specialization process on Hall algebra.