Proposition 2.1.5 (05WW) · ScalingStacks

[05WW]

Proposition 2.1.5. Let $L\colon\mathcal{C}\leftrightarrows\mathcal{D}\penalty\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muR$ be an adjunction between $\infty$ -categories. Assume that $\mathcal{C}$ admits and $L$ preserves pullbacks, that $\mathcal{D}$ admits an initial object, and that $R$ is fully faithful. For every object $Y\in\mathcal{C}$ we consider the following pullback diagram

where the right vertical map is the unit map of $Y$ and the bottom horizontal map is the image under $R$ of the essentially unique map $\varnothing_{\mathcal{D}}\to L\left(Y\right)$ . The top horizontal map $\rho\colon Y^{\operatorname{\scriptsize{red}}}\to Y$ exhibits $Y^{\operatorname{\scriptsize{red}}}$ as a co-localization of $Y$ with respect to $\mathcal{C}^{\operatorname{\scriptsize{red}}}$ (dual to T.5.2.7.6).

[05WX]

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. ∎