On Kasparov-type homotopy functors for $C^\ast$-algebras (PhD dissertation discussion)

Many constructions involving asymptotic homomorphisms in Connes-Higson $E$-theory can be lifted at the level of natural transformations between endofunctors of $C^\ast$-algebras. For such natural transformations we introduce the notion of homotopy and define a category with objects good enough endofunctors and morphisms homotopy classes of natural transformations. Objects of this category induce generalized homotopies of $\ast$-homomorphisms, which can be used to obtain an unsuspended description of $E$-theory and $E$-theoretical analog of $C^\ast$-algebra extension groups. The machinery can also be applied for computing $K$-homology.