Lemma 4.4.1 (05YY) · ScalingStacks

[05YY]

Lemma 4.4.1. Let $F\colon\mathcal{C}\leftrightarrows\mathcal{D}\penalty\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muU$ be a monadic adjunction between presentable $\infty$ -categories. If the monad $T=U\circ F$ preserves $n$ -connected morphisms, then $U$ detects $n$ -connected morphisms. Namely, given a morphism $f\colon A\to B$ in $\mathcal{D}$ , if $U\left(f\right)$ is $n$ -connected for some $n\geq-2$ , then $f$ is $n$ -connected.

[05YZ]

Proof. Given a morphism $f\colon A\to B$ in $\mathcal{D}$ , using the canonical simplicial resolution provided by the proof of A.4.7.3.13, we can express it as a colimit of the simplicial diagram of morphisms:

\operatorname*{colim}\limits_{\Delta^{op}}\left(T^{n+1}\left(A\right)\to T^{n+1}\left(B\right)\right),

which one can write as

\operatorname*{colim}\limits_{\Delta^{op}}\left(FT^{n}U\left(A\right)\to FT^{n}U\left(B\right)\right).

If $U\left(f\right)$ is $n$ -connected as in the statement, then since $T$ preserves $n$ -connected morphisms by assumption and $F$ preserves $n$ -connected morphisms by being left adjoint, it follows that all the maps in the diagram are $n$ -connected. By T.5.2.8.6(7), the map $f$ is also $n$ -connected. ∎