Covariant functors and shapes in the category of compacts

In this paper, we consider covariant functors $F:Comp\to Comp$ acting in category of shape-preserving compact sets [2], infinite compact sets, and shape equivalence [9]. Also we study action of compact functors and shape properties of the compact space $X$ consisting of connected components $\square X $ of the compact $X$ as well as shape identity $ShX=ShY$ of infinite compacts $X$ and $Y$ for the space $P(X)$ of probability measures and its subspaces.