3.6.1 Derived \(\infty\)-categories[006G]
We quickly review the basics of the theory of derived \(\infty\)-categories; we refer the reader to [Lur17, § 1.3] for more details.
Given an abelian \(1\)-category \(\mathcal A\), its (unbounded) derived \(\infty\)-category \(\mathcal D(\mathcal A)\) is the \(\infty\)-categorical localization of the \(\infty\)-category of unbounded chain complexes in \(\mathcal A\) (constructed as the dg nerve [Lur17, § 1.3.1] of the corresponding differential graded category) at the quasi-isomorphisms. In particular, the homotopy 1-category \(h_1\mathcal D(\mathcal A)\) agrees with the ordinary derived 1-category of \(\mathcal A\) in the usual sense.
Let \(\mathcal D(\mathcal A)_{\geq 0}\) denote the full subcategory of \(\mathcal D(\mathcal A)\) on the chain complexes with vanishing homology in negative degrees. When \(\mathcal A\) is particularly well-behaved, the \(\infty\)-categories \(\mathcal D(\mathcal A)_{\geq 0}\) and \(\mathcal D(\mathcal A)\) can be expressed in terms of completions (of the type introduced throughout section 3), as we discuss now.
Recall the following classical analogues of definition 3.2.1:
[006H]
Definition 3.6.1.
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. \(\mathcal C^{\mathrm{c}1\mathrm{p}}\).
[006I]
Example 3.6.2.
If \(\mathcal A\) is an abelian \(1\)-category, an object \(c \in \mathcal A\) is \(1\)-projective if and only if it is projective in the usual sense.
[006J]
Example 3.6.3.
A presentable abelian category \(\mathcal A\) is \(1\)-projectively generated if it is compactly generated and if the full subcategory of compact objects \(\mathcal A^{\mathrm{c}}\) has enough projective objects, i.e. if for every compact object \(a\in \mathcal A\) there exists a compact \(1\)-projective object \(p\) and an epimorphism \(p \twoheadrightarrow a\). In particular, this implies that also \(\mathcal A\) has enough projective objects, i.e. that for every object \(a\in \mathcal A\) there exists a \(1\)-projective \(p\) and an epimorphism \(p \twoheadrightarrow a\).
For example, the abelian category \(\mathrm{mod}_k\) is a \(1\)-projectively generated presentable \(1\)-category with \(\mathrm{mod}_k^{\mathrm{c}}\) the full subcategory of finitely generated modules and \(\mathrm{mod}_k^{\mathrm{c}1\mathrm{p}}\) the full subcategory of finitely generated projective \(k\)-modules.
[006L]
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.
The following statements are well-known and can be gathered from various parts of [Lur17, § 1.3]:
[006M]
Proposition 3.6.6.
Let \(\mathcal A\) be a \(1\)-projectively generated presentable abelian \(1\)-category.
The additive presentable \(\infty\)-category \(\mathcal D(\mathcal A)_{\geq 0}\) is equivalent to \(\mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\).
The stable presentable \(\infty\)-category \(\mathcal D(\mathcal A)\) is equivalent to its stabilization \[\mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}}) \otimes \mathrm{Sp}\simeq \operatorname{Ind}{\mathbf K}^b(\mathcal A^{\mathrm{c}1\mathrm{p}}).\]
[006Q]
Proof.
For the first statement, note that \(\mathcal A\) has enough projective objects (see example 3.6.3) and let \(\mathcal D_-(\mathcal A)\) be the dg-nerve of the differential graded category of bounded-below chain complexes of \(1\)-projective objects (i.e. projective objects in the standard abelian sense). Let \(\mathcal D_{-}(\mathcal A)_{\geq 0}\) be the full subcategory on the chain complexes with vanishing homology in negative degrees. Entirely analogous to the proof of [Lur17, Prop. 1.3.3.14], the Dold-Kan correspondence shows that \(\mathcal D_{-}(\mathcal A)_{\geq 0} \simeq \mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\). Since any \(1\)-projectively generated presentable abelian \(1\)-category is Grothendieck abelian [Lur17, Def. 1.3.5.1], it follows from [Lur17, Prop. 1.3.5.24, Def. 1.3.5.8, Prop. 1.3.5.13] that there is a fully faithful embedding \(\mathcal D_{-}(\mathcal A) \rightarrow\mathcal D(\mathcal A)\) with image the chain complexes with bounded-below homology. In particular, this embedding identifies \(\mathcal D_{-}(\mathcal A)_{\geq 0}\) with \(\mathcal D(\mathcal A)_{\geq 0}\).
For the second statement, since the \(t\)-structure \((\mathcal D(\mathcal A)_{\leq 0}, \mathcal D(\mathcal A)_{\geq 0})\) on \(\mathcal D(\mathcal A)\) is right-complete [Lur17, Prop. 1.3.5.21], it follows that \(\mathcal D(\mathcal A)\) is the stabilization of \(\mathcal D(\mathcal A)_{\geq 0}\); since \(\mathcal D(\mathcal A)_{\geq 0} = \mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\) is presentable this stabilization is given by tensoring with \(\mathrm{Sp}\) by [Lur17, Ex. 4.8.1.23]. The equivalence \(\mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}}) \otimes \mathrm{Sp}\simeq \operatorname{Ind}{\mathbf K}^b(\mathcal A^{\mathrm{c}1\mathrm{p}})\) follows then from the definition of \((-)^{\mathrm{fin}}\) in proposition 3.4.5 and its equivalence with \({\mathbf K}^b\) from corollary 3.4.10. ◻