  Biracks and biquandles, which are useful for studying the knot theory, are
special families of solutions of the set-theoretic Yang-Baxter equation. A
homology theory for the set-theoretic Yang-Baxter equation was developed by
Carter, Elhamdadi, and Saito in order to construct knot invariants. In this
paper, we construct a normalized (co)homology theory of a set-theoretic
solution of the Yang-Baxter equation. We obtain some concrete examples of
non-trivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its
geometric realization $BX$ is discussed, which has the potential to build
invariants of links and knotted surfaces. In particular, we demonstrate that
the second homotopy group of $BX$ is finitely generated if the biquandle $X$ is
finite.
