Given an ordinary additive \(1\)-category \(\mathcal A\), the \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) can be defined, see [Lur17, § 1.3.1], using the technology of dg nerves as follows.
Definition 3.4.1. ([Lur17, Cons. 1.3.1.6 and Rem. 1.3.2.2]).
For an ordinary additive \(1\)-category \(\mathcal A\), we let \({\mathbf K}^b(\mathcal A):=N_{\mathrm{dg}}(\mathrm{Ch}^b(\mathcal A))\) denote the dg nerve of the dg category of bounded chain complexes in \(\mathcal A\).
Viewing \(\mathcal A\) as an additive \(\infty\)-category, there is a canonical additive functor \(\mathcal A\rightarrow{\mathbf K}^b(\mathcal A)\) induced from the functor that interprets objects of \(\mathcal A\) as chain complexes concentrated in degree zero.
For an ordinary additive \(1\)-category \(\mathcal A\), the dg nerve \({\mathbf K}^b(\mathcal A)= N_{\mathrm{dg}}(\mathrm{Ch}^b(\mathcal A))\) is a stable \(\infty\)-category.
Proof.
The dg nerve of the dg category of (unbounded) chain complexes is an \(\infty\)-category by [Lur17, Prop. 1.3.1.10] and stable by [Lur17, Prop. 1.3.2.10]. The full dg subcategory of bounded chain complexes is closed under shifts and formation of mapping cones, and thus its dg nerve \({\mathbf K}^b(\mathcal A)\) is itself a stable \(\infty\)-category, by [Lur17, Lem. 1.1.3.3] and the discussion after [Lur17, Proof of Prop. 1.3.2.10]. ◻
By [Lur17, Rem. 1.3.1.11], the homotopy 1-category \(h_1{\mathbf K}^b(\mathcal A)\), recalled in subsection A.2.1, is the chain homotopy category \(\mathrm{K}^b(\mathcal A)\) in the sense of definition 2.2.2. Note that, unlike the notion of derived category, which can only be defined for abelian categories, the stable \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) is defined for any (possibly non-abelian) additive category \(\mathcal A\).
In corollary 3.4.10, we will prove that \({\mathbf K}^b(\mathcal A)\) is the universal stable \(\infty\)-category associated to \(\mathcal A\). To do so, we express \({\mathbf K}^b\) in terms of the functors constructed in the previous sections.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2