Lemma 4.4.2 (05Z0) · ScalingStacks

[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. ∎