Lemma 4.4.4 (05Z3) · ScalingStacks

[05Z3]

Lemma 4.4.4. Let $\mathcal{P}$ be a reduced $\infty$ -operad and let $\mathcal{C}$ be a presentably symmetric monoidal $\infty$ -category. The free-forgetful adjunction

F\colon\mathcal{C}\leftrightarrows\operatorname{Alg}_{\mathcal{P}}\left(\mathcal{C}\right)\penalty\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muU

is monadic and the associated monad $T=U\circ F$ preserves $n$ -connected morphisms.

[05Z4]

Proof. By A.4.7.3.11, the adjunction $F\dashv U$ is monadic. Hence, given a morphism $A\to B$ in $\mathcal{C}$ , by 2.4.6 the morphism $T\left(A\right)\to T\left(B\right)$ can be expressed as

\coprod_{n\geq 0}\left(P\left(n\right)\otimes A^{\otimes n}\right)_{h\Sigma_{n}}\to\coprod_{n\geq 0}\left(P\left(n\right)\otimes B^{\otimes n}\right)_{h\Sigma_{n}}.

By 4.4.2, $n$ -connected morphisms are closed under $\otimes$ and, by T.5.2.8.6, they are closed under colimits. Hence, we obtain that $T\left(A\right)\to T\left(B\right)$ is $n$ -connected as well. ∎