In the paper weak sufficient conditions for the reduction of the chain complex of a twisted cartesian product $F{\times }_{\tau }B$ to a chain complex of free finitely generated abelian groups are found.

We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X:ℐ\to \text{sSet}$ such that each simplicial set $X\left(i\right)$ has effective homology, we present an algorithm computing the homotopy colimit $\text{hocolim}X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\text{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations....

Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $𝐊\left(R\text{-}\mathrm{Proj}\right)$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is ${\aleph }_1$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion $j_{!}^{}:𝐊\left(R\text{-}\mathrm{Proj}\right)\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ and the orthogonal subcategory $𝒮={𝐊\left(R\text{-}\mathrm{Proj}\right)}^{\perp }$. And we even showed that the inclusion $𝒮\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ has a right adjoint; this forces some natural map to be an equivalence...

We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra $𝔤$ by another hom-Lie algebra $𝔥$ and discuss the case where $𝔥$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction...