In the theory of ``motives", algebraic cycles are central objects. For instance, the so-called ``motivic cohomology groups”, that give the universal bigraded ordinary cohomology on smooth varieties, are obtained from a complex of abelian groups consisting of certain algebraic cycles. In this talk, we discuss how one can go beyond it, and we show that an infinitesimal version of the above complex of abelian groups of algebraic cycles can be identified with the big de Rham-Witt complexes after a suitable Zariski sheafification. This in a sense implies that the crystalline cohomology theory admits a description in terms of algebraic cycles, going back to a result of S. Bloch and L. Illusie in the 1970s. This is based on a joint work with Amalendu Krishna.