  The category of abelian varieties over $\mathbb{F}_q$ is shown to be
anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a
non-commutative pro-ring of endomorphisms of a suitably chosen direct system of
abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a
finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence
is represented by what we call $w$-locally projective abelian varieties.
