All Questions

The following might be very well known for people who works with model categories, but I do not find the answer. Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...

Let $Y \to X \leftarrow Z$ be a cospan of topological spaces and let $W = Y \times_X^h Z$ be their homotopy fiber product. I am interested in sufficient conditions on the map $Y \to X$ that ensure ...

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$. And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : C_{\ast} ...

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...

Let $G,H$ be abelian groups (denoted additively) with their Pontryagin duals denoted as $G^*$ and $H^*$. (The cases I'm interested in are products of $\mathbb Z_n$, $\mathbb Z$, $\mathbb R$, and $\...

Suppose to have a short exact sequence of chain complexes of $\mathbb{Z}$-modules: $$0\to A^\bullet\to B^\bullet\to C^\bullet\to 0$$ such that $A^k,B^k,C^k$ are non zero for $k=0,1,2$. Moreover, ...