  In recent years different aspects of categorification of the boson-fermion
correspondence have been studied. In this paper we propose a categorification
of the boson-fermion correspondence based on the category of tensor modules of
the Lie algebra $sl(\infty)$ of finitary infinite matrices. By $\mathbb T^+$ we
denote the category of "polynomial" tensor $sl(\infty)$-modules. There is a
natural "creation" functor $\mathcal T_N: \mathbb T^+\to \mathbb T^+$,
$M\mapsto N\otimes M,\quad M,N\in \mathbb T^+$. The key idea of the paper is to
employ the entire category $\mathbb T$ of tensor $sl(\infty)$-modules in order
to define the "annihilation" functor $\mathcal D_N: \mathbb T^+\to \mathbb T^+$
corresponding to $\mathcal T_N$. We show that the relations allowing to express
fermions via bosons arise from relations in the cohomology of complexes of
linear endofunctors on $\mathbb T^+$.
