Subsection 3.10. (0MT5) · ScalingStacks

[05U4]

Proposition 3.11. Let $C$ and $D$ be categories, and $W\subset D$ a subcategory such that $\iso D\subset W$ . Then there are natural isomorphisms

N(D^{C},\we(D^{C}))\approx N(D,W)^{NC}\approx N(D,W)^{\discnerve C}.

[05U5]

Proof. We must show that for each $m,n\geq 0$ the natural maps $N(D^{C},\we(D^{C}))\rightarrow N(D,W)^{NC}\rightarrow N(D,W)^{\discnerve C}$ induce one-to-one correspondences amongst the sets of

(1)

functors $[m]\times[n]\rightarrow D^{C}$ which carry “vertical” maps into $\we(D^{C})$ ,
(2)

maps $F(m)\times\Delta[n]\rightarrow N(D,W)^{NC}$ of simplicial spaces, and
(3)

maps $F(m)\times\Delta[n]\rightarrow N(D,W)^{\discnerve C}$ of simplicial spaces.

By (3.8) and (3.12) it will suffice to show this in the case $m=n=0$ , in which case the result becomes a straightforward computation. ∎

[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 $D$ . ∎

3.10. Classification diagrams of functor categories[0MT5]