Abstract: this talk, based on joint work in progress with S. Arkhipov,will describe a class of infinite-dimensional algebro-geometric objects (we call them Tate ind-schemes) for which we develop a theory of homology based on "semiinfinite" cycles. Among examples of Tate ind-schemes are projectivizations of Tate vector spaces such as k((t)), Givental models for loop spaces of projective varieties and algebro-geometric versions of semiinfinite flag varieties in the spirit of Feigin-Frenkel.