Let \(\mathcal A\) be an ordinary additive, idempotent-complete \(1\)-category. We have an equivalence of \(\infty\)-categories \(\mathcal A^{\mathrm{fin}} \simeq {\mathbf K}^b(\mathcal A)\). In particular, for any stable, idempotent-complete \(\infty\)-category \(\mathcal B\), the inclusion of degree-zero chain complexes \(\mathcal A\rightarrow{\mathbf K}^b(\mathcal A)\) induces an equivalence \[\mathrm{Fun}^{\mathrm{ex}}({\mathbf K}^b(\mathcal A), \mathcal B) \rightarrow\mathrm{Fun}^{\sqcup}(\mathcal A, \mathcal B).\]
Proof.
By [ES22, Thm. 2.2.9], for any additive, idempotent complete \(\infty\)-category \(\mathcal A\), the \(\infty\)-category \(\mathcal A^{\mathrm{fin}}\) is uniquely characterized by being a stable, idempotent-complete \(\infty\)-category with bounded weight structure with weight heart \(\mathcal A\). Since for an ordinary, additive, idempotent-complete \(1\)-category \(\mathcal A\), the \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) is a stable, idempotent-complete \(\infty\)-category with weight structure and weight heart \(\mathcal A\), see proposition 3.4.2, example 3.4.8. The result follows. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2