Let \(c\) be an object in an ordinary \(1\)-category \(\mathcal C\) with small colimits. Then, \(c\) is called
compact, if \(\mathrm{Hom}_{\mathcal C}(c, -) \colon \mathcal C\rightarrow\mathrm{Set}\) preserves filtered colimits;
\(1\)-projective, if \(\mathrm{Hom}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathrm{Set}\) preserves geometric realizations (equivalently, reflective coequalizers);
compact \(1\)-projective if \(\mathrm{Hom}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathrm{Set}\) preserves sifted colimits, or equivalently if \(c\) is compact and \(1\)-projective.
We say that \(\mathcal C\) is compactly generated (resp. \(1\)-projectively generated) if there is a small set of compact (resp. compact 1-projective) objects which generate \(\mathcal C\) under small colimits. We denote the full subcategory of compact, resp. compact \(1\)-projective, objects in \(\mathcal C\) by \(\mathcal C^{\mathrm{c}}\), resp. \(\mathcal C^{\mathrm{c}1\mathrm{p}}\).