Lemma 2.5 (0N2Z) · ScalingStacks

Lemma 2.5. Let $\mathcal{S}$ be an $\infty$ -category and let $S\in\mathcal{S}$ be an object.

(1)

For each morphism $S^{\prime}\to S$ in $\mathcal{S}$ , the canonical functor among over $\infty$ -categories $(\mathcal{S}_{/S})_{/(S^{\prime}\to S)}\to\mathcal{S}_{/S^{\prime}}$ is an equivalence.
(2)

Should $\mathcal{S}$ admit finite coproducts, the over $\infty$ -category $\mathcal{S}_{/S}$ admits finite coproducts and they are preserved by the projection functor $\mathcal{S}_{/S}\to\mathcal{S}$ .

In addition, the $\infty$ -category of symmetric monoidal $\infty$ -categories ${\sf Cat}_{\infty}^{\otimes}$ admits limits and they are preserved by the forgetful functor ${\sf Cat}_{\infty}^{\otimes}\to\Cat$ .

0N30

Proof. Through the defining adjunctions for over $\infty$ -categories, the first assertion follows because, for each $\infty$ -category $\mathcal{K}$ , the canonical diagram among $\infty$ -categories

is a pushout; here, for $\mathcal{K}$ and $\mathcal{J}$ $\infty$ -categories,

\mathcal{K}\star\mathcal{I}~:=~\mathcal{K}\underset{\mathcal{K}\times\{0\}\times\mathcal{I}}{\coprod}\mathcal{K}\times\{0<1\}\times\mathcal{I}\underset{\mathcal{K}\times\{1\}\times\mathcal{I}}{\coprod}\mathcal{I}

denotes the join of $\infty$ -categories. The second assertion follows directly from the universal property of coproducts. The final assertion follows from Proposition 3.2.2.1 of [Lu2], which in particular gives that, for each Cartesian closed presentable $\infty$ -category $\mathcal{C}$ , the forgetful functor from commutative algebras $\Alg_{\sf Com}(\mathcal{C}^{\times})\to\mathcal{C}$ preserves and creates limits. Apply this result to the case $\mathcal{C}=\Cat$ .

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