DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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The augmented unnormalized homogeneous bar complex

Definition

For $n \geq 0$ , let $B_{n} (G)$ be the free abelian group on $(g_{0}, \dots, g_{n}) \in G^{n + 1}$ , with diagonal left action $h (g_{0}, \dots, g_{n}) = (h g_{0}, \dots, h g_{n})$ . For $n \geq 1$ , put $d_{n} = \sum_{i = 0}^{n} (- 1)^{i} d_{i} : B_{n} (G) ⟶ B_{n - 1} (G),$ where $d_{i}$ deletes the $i$ th vertex. In degree zero use the augmentation $ε : B_{0} (G) \to Z$ , $ε (g_{0}) = 1$ , with trivial action on $Z$ . Thus the augmented complex is $\dots \to B_{1} (G) \to B_{0} (G) \overset{ε}{\to} Z \to 0$ ; no undefined object $B_{- 1}$ or differential $d_{0}$ is used.

Depends on

Integral group modules and the trivial module

Used by

Bar augmentation Definition
The normalized homogeneous bar complex Definition
The first bar differentials Example
The bar differential is equivariant and squares to zero Lemma
Homogeneous and inhomogeneous cochains agree Theorem
Normalized and unnormalized bars are homotopy equivalent Theorem
The bar complex is a free resolution of the trivial module Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Weibel, Definition 6.5.1 (standard reference, not scraped)