Path-Connected Components of Affine Schemes and Algebraic K-Theory

We introduce a functor $\mathfrak{M}:\mathbf{A}\mathbf{l}\mathbf{g}\times {\mathbf{A}\mathbf{l}\mathbf{g}}^{\mathrm{o}\mathrm{p}}\to \mathrm{p}\mathrm{r}\mathrm{o}\text{-}\mathbf{A}\mathbf{l}\mathbf{g}$ constructed from representations of ${\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbf{A}\mathbf{l}\mathbf{g}}(A,B\otimes ?)$. As applications, the following items are introduced and studied: (i) Analogue of the functor ${\pi }_0$ for algebras and affine schemes. (ii) Cotype of Weibel’s concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $\mathbf{A}\mathbf{l}\mathbf{g}$ that are enriched over idempotent semigroups. (v) Classifying homotopy pro-algebras for Corti~{n}as-Thom’s KK-groups and Weibel’s homotopy K-groups.