ScalingStacks

Proof. First, we show that $Y^{\operatorname{\scriptsize{red}}}$ is in fact reduced. Applying $L$ to the defining diagram of $Y^{\operatorname{\scriptsize{red}}}$ and using the fact that $L$ preserves pullbacks, we see that the map $L\left(Y^{\operatorname{\scriptsize{red}}}\right)\to LR\left(\varnothing\right)$ is the pullback of the map $L\left(Y\right)\to LRL\left(Y\right)$, which is an equivalence (from the fact that the counit $LR\left(Y\right)\to Y$ is an equivalence, the zig-zag identities and the 2-out-of-3 property). It follows that the map $L\left(Y^{\operatorname{\scriptsize{red}}}\right)\to LR\left(\varnothing\right)$ is an equivalence, but $LR\left(\varnothing\right)\to\varnothing$ is an equivalence as well (since $R$ is fully faithful) and we are done.

Now, we show that $\rho$ is a co-localization. Let $Z$ be a reduced object. We have a homotopy pullback diagram of spaces

and we note that the space of maps from a reduced object to any object in the essential image of $R$ is contractible. ∎