If \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\) is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) with its Day convolution monoidal structure is also in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), respectively.
Proof.
The Day convolution monoidal structure was defined by identifying \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) with \(\mathcal C\otimes \mathcal P(J)\). The presheaf category \(\mathcal P(J)\) is an object of \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) (in fact, it is generated by a small set of objects which commute with all small colimits). Hence, if \(\mathcal C\) is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or in the subcategory \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then so is \(\mathcal C\otimes \mathcal P(J)\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2