This is the second in a series of papers applying descriptive set-theoretic methods to analyze and enhance classical functors from homological algebras and algebraic topologies. Among them, each \v{C}ech cohomology functor $\check{\mathrm{H}}^n$ of the category of locally compact separable metric spaces calls (i) the definable version indicates to factor into . The functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ is a group of categories $\mathsf{GPC }$, (ii) followed by a forgetting functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors are powerful improvements over their classical counterparts. For example, we show that they are perfect invariants. For example, the homotopy-type invariant of the mapping telescope of $d$-spheres or $d$-tori for any $d\geq 1$ , in contrast to any one whose classical cohomology functor is constant. that there are countless families of some kind of pairwise homotopy inequality mapping telescopes; Then we apply the functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ to obtain a high-dimensional and equivariant generalization of an important problem in the development of algebraic topology, i.e. Borsuk and Eilenberg’s 1936 problem classifies the map from the solenoidal complement $S^3\backslash\Sigma$ to the $2$ sphere, up to homotopy.
In the course of this work, we record Borel-defined versions of many classical results related to both combinatorial and homotopic formulations of \v{C}ech cohomology. Overall, this work can be viewed as laying the groundwork for a descriptive set-theoretical study of homotopy relations on spaces of maps from locally compact Polish spaces to polyhedra. This relationship embodies a variety of classification problems that arise through mathematics.
