An object \(c\) of an \(\infty\)-category \(\mathcal C\) with geometric realizations (i.e. colimits indexed by \(\Delta^{\mathrm{op}}\)) is called projective [Lur09, Def. 5.5.8.18] if \(\mathrm{Hom}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathcal S\) preserves geometric realizations.
An object \(c\) of an \(\infty\)-category \(\mathcal C\) with sifted colimits [Lur09, Def. 5.5.8.1] is called compact-projective if \(\mathrm{Hom}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathcal S\) preserves sifted colimits [Lur09, Rem. 5.5.8.20].
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2