  We introduce a new functor category: the category $\mathcal{P}_{d,n}$ of
strict polynomial functors with bounded by $n$ domain of degree $d$ over a
field of characteristic $p>0$. It is equivalent to the category of finite
dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to
apply the tools available in functor categories to representations of the
algebraic group $\operatorname{GL}_n$. We investigate in detail the homological
algebra in ${\cal P}_{d,n}$ for $d=p$ and establish equivalences between
certain subcategories of ${\cal P}_{d,n}$'s which resemble the
Spanier-Whitehead duality in stable homotopy theory.
