Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group.

(For example, given a short exact sequence $$ 1 \to BG_2 \to \mathbb{G} \to G_1 \to 1 $$ and the fiber sequence: $$ B^2G_2 \to B\mathbb{G} \to BG_1. $$ where $\mathbb{G}$ can be viewed as a group or a 2-group.)

The consideration is the following:

$$G \text{ has one object}.$$

$$BG \text{ has one 0-cell}.$$

$$1\text{-morphisms} = \text{ elements of } \mathbb{G}.$$

$$1\text{-cells} = 1\text{-morphisms of } \mathbb{G}.$$

  • Question: What is the precise relation between the cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$?

Thank you in advance for your help, comments, answers!


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