$K$-theory on the category of infinite cell complexes

The main purpose of the article is to compute cohomology operations in $K$-theory $\operatorname{mod}p$ and operations from the usual cohomology theory into $K$-theory. The proposed method is based on the extension of $K$-theory to a category of infinite complexes, which makes it possible to use spectral sequences of the “bar-construction” type.