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This is the second installment in a series of papers applying descriptive settheoretic techniques to both analyze and enrich classical functors fromhomological algebra and algebraic topology. In it, we show that the \v{C}echcohomology functors $\check{\mathrm{H}}^n$ on the category of locally compactseparable metric spaces each factor into (i) what we term their definableversion, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in thecategory $\mathsf{GPC}$ of groups with a Polish cover (a category firstintroduced in this work's predecessor), followed by (ii) a forgetful functorfrom $\mathsf{GPC}$ to the category of groups. These definable cohomologyfunctors powerfully refine their classical counterparts: we show that they arecomplete invariants, for example, of the homotopy types of mapping telescopesof $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that thereexist uncountable families of pairwise homotopy inequivalent mapping telescopesof either sort on which the classical cohomology functors are constant. We thenapply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to provide strongsolutions to higher-dimensional and equivariant generalizations of a seminalproblem in the development of algebraic topology, namely Borsuk and Eilenberg's1936 problem of classifying, up to homotopy, the maps from a solenoidcomplement $S^3\backslash\Sigma$ to the $2$-sphere.  In the course of this work, we record Borel definable versions of a number ofclassical results bearing on both the combinatorial and homotopicalformulations of \v{C}ech cohomology; in aggregate, this work may be regarded aslaying foundations for the descriptive set theoretic study of the homotopyrelation on the space of maps from a locally compact Polish space to apolyhedron, a relation which embodies a substantial variety of classificationproblems arising throughout mathematics.

The definable content of homological invariants II: \v{C}ech cohomology and homotopy classification