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Pure t-motives were introduced by G. Anderson as higher dimensionalgeneralizations of Drinfeld modules, and as the appropriate analogs of abelianvarieties in the arithmetic of function fields. In order to construct modulispaces for pure t-motives the second author has previously introduced theconcept of abelian \tau-sheaf. In this article we clarify the relation betweenpure t-motives and abelian \tau-sheaves. We obtain an equivalence of therespective quasi-isogeny categories. Furthermore, we develop the elementarytheory of both structures regarding morphisms, isogenies, Tate modules, andlocal shtukas. The later are the analogs of p-divisible groups.

Pure Anderson Motives and Abelian \tau-Sheaves