ScalingStacks

Warning 3.6.5.

remark 3.6.4 not true projectivity: The condition for an object \(c\in \mathcal C\) to be \(1\)-projective (i.e. \(\mathrm{Hom}_{\mathcal C}(c,-)\colon \mathcal C\rightarrow\mathrm{Set}\) preserving geometric realizations) is different to the condition for it to be projective (i.e. \(\mathrm{Hom}_{\mathcal C}(c,-)\colon \mathcal C\rightarrow\mathrm{Set} \rightarrow\mathcal S\) preserving geometric realizations), simply because the inclusion \(\mathrm{Set} \hookrightarrow \mathcal S\) does not preserve geometric realizations. This difference is at the heart of the process of animation [CS24, § 5.1.4], which takes an ordinary cocomplete category \(\mathcal C\) to \(\mathcal P^{\Sigma}(\mathcal C^{\mathrm{c}1\mathrm{p}})\), i.e. freely making the compact 1-projective objects into compact-projective objects.