Proposition 3.11 (05U4) · 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. ∎