What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?

By differential abelian group I mean an abelian group $A$ with a self-map $\partial:A\to A$ with $\partial^2=0$ (or equivalently, a $\mathbb{Z}[\partial]/\partial^2$-module). Under …

Consider a $\mathbb{Z}$-graded chain complex $A^{\bullet}$, I know that a bounded below complex is one such that $A^i = 0$ for $i$ sufficiently small, and a bounded above complex i …

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes. On the topological side, …

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 …

Let $A^\bullet$ be the complex: $\cdots \rightarrow A^{n-2} \xrightarrow{d^{n-2}} A^{n-1} \xrightarrow{d^{n-1}} A^{n} \xrightarrow{d^{n}} A^{n+1} \xrightarrow{d^{n+1}} A^{n+2} \xr …

I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and A …

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a wea …

Let $R$ be a commutative ring. For awhile I have been trying to motivate to myself more fully the definition of and various structures on the category $\text{Ch}(R)$ of chain compl …

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes. Has anyo …

Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bou …

Recall that there is an equivalence of categories (Dold-Kan) $$N:\mathrm{s}\mathbf{Ab}\simeq \operatorname{Ch}_{\geq 0}(\mathbf{Ab}):\Gamma$$ between simplicial abelian groups and …

There is a functor from bicomplexes to chain complexes sending a bicomplex to its associated total chain complex. Does this functor have a left adjoint, and if so, what is it?

Main Question Let $R$ be a graded ring, graded by the nonnegative integers. Denote by $\mathrm{gr}R-\mathrm{Mod}$ the category of $\mathbb{Z}$-graded left $R$-modules with morphi …

Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such …