Subsection 4.2.2 (007X) · ScalingStacks

4.2.2 \(\infty\)-categories enriched in graded modules[007X]

remark 4.1.9 motivates the following terminology:

Let \(\mathcal Z\) be a homotopy coherent abelian monoid, i.e. \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define

the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) of presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}(\mathrm{Pr}^\mathrm{L}) ~;\]
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) of projectively generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) ~.\]

For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define

the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}\) of presentably \(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}\)-enriched \(\infty\)-categories, \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}(\mathrm{Pr}^\mathrm{L}) ~;\]
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) of compactly generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})~.\]

As \(\infty\)-categories of modules of commutative algebras in presentably symmetric monoidal categories, both, \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\), are presentably symmetric monoidal \(\infty\)-categories.

[007Z]

Remark 4.2.9.

As in observation 4.2.4, any presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}\)-enriched \(\infty\)-category is additive and any presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-category is stable, and we have the following equivalences: \[\begin{aligned} \mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}& \coloneqq \mathrm{Mod}_{\mathrm{Mod}^{\geq 0,\mathcal Z}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L}) \simeq \mathrm{Mod}_{\mathrm{Mod}^{\geq 0,\mathcal Z}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}) \\ {\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}& \coloneqq \mathrm{Mod}_{\mathrm{Mod}^{\mathcal Z}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L})\simeq\mathrm{Mod}_{\mathrm{Mod}^{\mathcal Z}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}) \end{aligned}\] Similarly, we have the following equivalences for their small variants, abbreviating \(M=\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\): \[\begin{aligned} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}& \coloneqq\mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \simeq \mathrm{Mod}_{M}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}) \simeq \mathrm{Mod}_{M^{\mathrm{cp}}}(\mathrm{add}) \simeq \mathrm{Mod}_{M^{\mathrm{cp}}}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \\ \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}&\coloneqq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}) \simeq \mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\right)^{\mathrm{c}}}(\mathrm{st})\simeq \mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\right)^{\mathrm{c}}}(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}) \end{aligned}\]

[0080]

Example 4.2.10.

As discussed in example 3.5.15, if \(\mathcal Z\) is a discrete (i.e. ordinary) commutative monoid \(Z\), then \((\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})^{\mathrm{cp}}\) and \((\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})^c\) are the \(\infty\)-categories of finitely supported functors \(\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{CProj}_{\mathbb{K}})\) and \(\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{Perf}_{\mathbb{K}})\), i.e. of functors that vanish on all but finitely many elements of \(Z\). In particular, in the case of grading by a discrete monoid \(Z\), we obtain the following equivalences:\[\begin{aligned} \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, Z}} &\simeq \mathrm{Mod}_{\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{CProj}_{\mathbb{K}})}(\mathrm{add}) \\ \mathrm{Pr}^{\mathrm{L},\mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{ Z}} &\simeq \mathrm{Mod}_{\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{Perf}_{\mathbb{K}})}(\mathrm{st}) \end{aligned}\]

Yu Leon Liu, Aaron Mazel-Gee, David Reutter, Catharina Stroppel, Paul Wedrich

Original source: arXiv:2401.02956v2

Original source · 2401.02956v2