Lemma 3.12 (05U6) · ScalingStacks

[05U6]

Lemma 3.12. Let $C$ a category, and $W$ a subcategory with ${\operatorname{ob}}W={\operatorname{ob}}C$ . Then there is a natural isomorphism

N(\widetilde{C^{[n]}},\we(\widetilde{C^{[n]}}))\approx(NC)^{\Delta[n]},

where $\widetilde{C^{[n]}}\subset C^{[n]}$ denotes the full subcategory whose objects are those functors $[n]\rightarrow C$ which factor through $W\subset C$ , and $\we(\widetilde{C^{[n]}})=\we(C^{[n]})\cap\widetilde{C^{[n]}}$ .

[05U7]

Proof. For any pair $(D,W)$ of category $D$ and subcategory $W$ , we have that $\we(\widetilde{D^{[n]}})=W^{[n]}$ , and that $\nerve(W^{[n]})=(\nerve W)^{\Delta[n]}$ . We obtain the result by substituting $C^{[m]}$ for $D$ . ∎