ScalingStacks

Proof. We denote the $\infty$-category $\mathcal{D}\times_{\mathcal{C}}\mathcal{C}_{X/}$ by $\mathcal{D}_{X/}$. Let $\eta\colon X\to RL\left(X\right)$ be the $X$-component of the unit of the adjunction $L\dashv R$. By T.2.1.2.1, the projections $p_{0}\colon\mathcal{C}_{\eta/}\to\mathcal{C}_{X/}$ and $p_{1}\colon\mathcal{C}_{\eta/}\to\mathcal{C}_{RL\left(X\right)/}$ are left fibrations. Moreover, since $\Delta^{\left\{1\right\}}\hookrightarrow\Delta^{1}$ is right anodyne, the map $p_{1}$ is an equivalence of $\infty$-categories. By T.2.2.3.3, we can choose an inverse $p_{1}^{-1}\colon\mathcal{C}_{RL\left(X\right)/}\to\mathcal{C}_{\eta/}$ to $p_{1}$ that strictly commutes with the projections to $\mathcal{C}$. We obtain a commutative diagram of simplicial sets

There is an induced map from the upper left corner to the pullback of the outer rectangle without the upper left corner, which is another commutative diagram of simplicial sets

Since left fibrations are closed under base change (T.2.1.2.1), the vertical maps are left fibrations over $\mathcal{D}$. Hence, to show that the top map is an equivalence it is enough to show that the induced map on fibers is a homotopy equivalence (T.2.2.3.3). For every $Y\in\mathcal{D}$ we get a map

which is by construction obtained by applying the functor $R$ and pre-composing with the unit $\eta\colon X\to RL\left(X\right)$. By the universal property of the unit map this is a homotopy equivalence for all $Y\in\mathcal{D}$ and therefore the map $\mathcal{D}_{L\left(X\right)/}\to\mathcal{D}_{X/}$ is an equivalence of $\infty$-categories. ∎

Proof. Let $p\colon\mathcal{M}\to\Delta^{1}$ be the coCartesian fibration associated with the functor $L$ (which is also Cartesian, since $L$ has a right adjoint). We can assume that we have a commutative diagram

such that $s|_{\Delta^{\left\{0\right\}}\times\mathcal{C}}=\operatorname{Id}$, $s|_{\Delta^{\left\{1\right\}}\times\mathcal{C}}=L$ and $s|_{\Delta^{1}\times\left\{X\right\}}$ is a coCartesian edge of $\mathcal{M}$ for every $X\in\mathcal{C}$ (combine T.5.2.1.1 and T.5.2.1.3). It is clear from T.1.2.9.2 that for any pair of $\infty$-categories with objects $X\in\mathcal{C}$ and $Y\in\mathcal{D}$ there is a canonical isomorphism

Hence, we get an induced commutative diagram

The functor $p_{X}$ is a Cartesian and coCartesian fibration by the duals of T.2.4.3.1(1) and T.2.4.3.2(1). Moreover, an edge in $\mathcal{M}_{X/}$ is (co)Cartesian if and only if its projection to $\mathcal{M}$ is (co)Cartesian by the duals of T.2.4.3.1(2) and T.2.4.3.2(2), which shows that the functor $L_{X}$ is associated with $p_{X}$. It follows that $L_{X}$ has a right adjoint $R_{X}$.

Assuming that $R$ is fully faithful, we will show that $R_{X}$ is fully faithful by showing that the counit of the adjunction $L_{X}\dashv R_{X}$ is an equivalence. For every object, the counit map is an edge of $\mathcal{M}_{X/}$. Since the projection $\mathcal{M}_{X/}\to\mathcal{M}$ is conservative, it is enough to show that the counit map of $L_{X}\dashv R_{X}$ is mapped to the counit map of $L\dashv R$. Indeed, for an object $Y\in\mathcal{D}\simeq\mathcal{M}|_{\Delta^{\left\{1\right\}}}$, we choose a Cartesian edge $e\colon R\left(Y\right)\to Y$ and a coCartesian edge $d\colon R\left(Y\right)\to L\left(R\left(Y\right)\right)$, and combine them into a commutative diagram of the form:

where $f|_{\Delta^{\left\{0,1\right\}}}=d$ and $f|_{\Delta^{\left\{0,2\right\}}}=e$. Since $d$ is coCartesian, there exists a lift $\overline{f}\colon\Delta^{2}\to\mathcal{M}$ that gives an edge

that is isomorphic to the counit map of the adjunction $L\dashv R$ at $Y$ in the homotopy category $h\mathcal{D}$. We can similarly construct the counit map for an object of $\mathcal{M}_{X/}$. The assertion now follows from the above characterization of (co)Cartesian edges in $\mathcal{M}_{X/}$. ∎

Proof. Consider the commutative square

The vertical functors and the bottom horizontal functor preserve pullbacks. The right vertical functor is conservative. It follows that the top horizontal functor preserves pullbacks as well. ∎

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. ∎

Proof. Follows from 2.1.5 and the dual of T.5.2.7.8. ∎