Condition for Injectivity of the Canonical Projection

Let C be a finite acyclic category and assume that a finite group G acts on G. We impose the following condition on our action. 

Definition 14.12. LetG be a finite acyclic category and letG be a finite group acting on G. We say that the action is regular if it satisfies Gondition (R). 

The next proposition shows why Condition (R) is the right one for our purposes. 

Proposition 14.13. Let C he a finite acyclic category and G a finite group acting on C. This action satisfies Condition (R) if and only if the canonical surjection A A C)/G — A C/G) is injective on 1-skeletons. 

Assume that A is injective on 1-skeletons. Then we have the following computation  

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