(also nonabelian homological algebra)

**Context**

**Basic definitions**

**Stable homotopy theory notions**

**Constructions**

**Lemmas**

**Homology theories**

**Theorems**

An unbounded chain complex $X_\bullet \in Ch_\bullet$

$\cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} X_{-1} \stackrel{\partial}{\to} X_{-2} \stackrel{\partial}{\to} \cdots$

is called *connective* if it has no nontrivial homology groups in negative degree. Often one means more strictly that it is connective if it is concentrated in non-negative degree, hence if $X_{-n} \simeq 0$ for all $n \geq 1$, hence if it is of the form

$\cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} \cdots
\,.$

Such connective chain complexes are the connective objects in the $\infty$-category of chain complexes. Rearding under the stable Dold-Kan correspondence as $H R$-module spectra they are the *connective spectra*.

Hence a connective chain complex is in particular a bounded chain complex, bounded from below.

- The singular chain complex of a topological space is connective. This is directly related to the fact that the homotopy groups of topological spaces are in non-negative degree, as opposed to their stabilization by spectra, which may have homotopy groups also in negative degrees.

The Dold-Kan correspondence asserts that connective chain complexes of abelian groups are equivalent to abelian simplicial groups.

Last revised on April 20, 2023 at 07:25:43. See the history of this page for a list of all contributions to it.