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

Categories of abelian varieties over finite fields II: Abelian varieties over finite fields and Morita equivalence