Subsection 4.4 (0M3I) · ScalingStacks

4.4 Connectedness in Algebras[0M3I]

We begin with the following general fact:

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. ∎

We want to apply the above to the free-forgetful adjunction between a symmetric monoidal $\infty$ -category $\mathcal{C}$ and the category of $\mathcal{P}$ -algebras in $\mathcal{C}$ , where $\mathcal{P}$ is a reduced $\infty$ -operad. For this, we need some compatibility between the notion of $n$ -connectedness and the symmetric monoidal structure:

[05Z0]

Lemma 4.4.2. Let $\mathcal{C}$ be a presentably symmetric monoidal $\infty$ -category. For every integer $n\geq-2$ , the class of $n$ -connected morphisms in $\mathcal{C}$ is closed under tensor products.

[05Z1]

Proof. Since $\mathcal{C}$ is presentable and the tensor product commutes with colimits separately in each variable, for each object $X\in\mathcal{C}$ the functor $Y\mapsto X\otimes Y$ is a left adjoint and therefore preserves $n$ -connected morphisms by 4.1.4. Hence, given two $n$ -connected morphisms $f\colon A_{1}\to B_{1}$ and $g\colon A_{2}\to B_{2}$ , the composition

A_{1}\otimes B_{1}\xrightarrow{A_{1}\otimes g}A_{1}\otimes B_{2}\xrightarrow{f\otimes B_{2}}A_{2}\otimes B_{2}

is $n$ -connected as a composition of two $n$ -connected morphisms. ∎

[05Z2]

Example 4.4.3. For every $m$ -topos (with $-1\leq m\leq\infty$ ) and $n\geq-2$ , the class of $n$ -connected morphisms is closed under Cartesian products. In particular, this applies to $\mathcal{S}_{\leq m}^{K}$ for every simplicial set $K$ .

[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. ∎

[05Z5]

Proposition 4.4.5. Let $\mathcal{P}$ be a reduced $\infty$ -operad and let $\mathcal{C}$ be a presentably symmetric monoidal $\infty$ -category. Given a morphism $f\colon A\to B$ in $\operatorname{Alg}_{\mathcal{P}}\left(\mathcal{C}\right)$ , if the underlying map $U\left(f\right)$ is $n$ -connected for some $n\geq-2$ , then $f$ is $n$ -connected.

[05Z6]

Proof. Combine 4.4.4 and 4.4.1. ∎

Tomer Schlank, Lior Yanovski

Original source: arXiv:1808.06006v3

Original source · 1808.06006v3