For a representation theoretic point of view on (classical) weight structures in the context of Soergel bimodules we refer to [ES22]. Soergel bimodules appear in there as Springer motives attached to Bott–Samelson resolutions of Schubert varieties in the full flag variety and form an additive idempotent complete coheart, see [ES22, Ex. 2.2] and compare with example 3.4.8.
3.4.3 \({\mathbf K}^b\) as a left adjoint[005E]
We now show that for an ordinary additive idempotent-complete \(1\)-category \(\mathcal A\), the universal stable \(\infty\)-category \(\mathcal A^{\mathrm{fin}}\) from proposition 3.4.5 is equivalent to \({\mathbf K}^b(\mathcal A)\).
Consider the symmetric monoidal left adjoint \((-)^{\mathrm{fin}} \colon \mathrm{add}\rightarrow\mathrm{st}\) of the forgetful functor from proposition 3.4.5. Categories in the image of \((-)^{\mathrm{fin}}\) carry so-called weight structures, which were originally introduced independently by Bondarko in [Bon10] and (under the name of co-t-structures) by Pauksztello in [Pau08], and afterwards adopted to the \(\infty\)-categorical setting by Elmanto and Sosnilo [ES22], whose exposition we closely follow.
By [ES22, Def. 2.2.1] a weight structure on an idempotent complete stable \(\infty\)-category \(\mathcal D\) is a pair \((\mathcal D_{\leq 0}, \mathcal D_{\geq 0})\) of two full idempotent complete subcategories fulfilling the following conditions:
\(\Sigma \mathcal D_{\geq 0} \subset \mathcal D_{\geq 0}, \Sigma^{-1} \mathcal D_{\leq 0} \subset \mathcal D_{\leq 0}\). We write \(\mathcal D_{\geq n} = \Sigma^n \mathcal D_{\geq 0}\), \(\mathcal D_{\leq n} = \Sigma^n \mathcal D_{\leq 0}\).
For \(x \in \mathcal D_{\leq 0}\) and \(y \in \mathcal D_{\geq 1}\), we have \(\pi_0(\mathrm{Hom}_{\mathcal D}(x,y)) \simeq 0.\)
For any object \(x\), there is a fiber sequence \(x_{\leq 0} \rightarrow x \rightarrow x_{\geq 1}\) with \(x_{\leq 0} \in \mathcal D_{\leq 0}, x_{\geq 1} \in \mathcal D_{\geq 1}\).
Note that condition (3) merely requires the existence of such a fiber sequence, neither is it unique nor functorially associated to \(x\). A weight structure is called bounded if \(\mathcal D= \bigcup_{n}(\mathcal D_{\geq -n} \cap \mathcal D_{\leq n})\). For any weight structure, the weight heart \(\mathcal D^{\heartsuit} \coloneqq \mathcal D_{\geq 0} \cap \mathcal D_{\leq 0}\) is additive and idempotent complete.
Just like t-structures, weight structures only depend on and may be constructed in terms of the underlying (triangulated) homotopy category of \(\mathcal D\).
Given an ordinary additive, idempotent complete \(1\)-category \(\mathcal A\), the \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) has a canonical bounded weight structure with \({\mathbf K}^b(\mathcal A)_{\geq 0}\) the (dg nerve on the) full subcategory of chain complexes supported in non-negative homological degrees. The weight heart \({\mathbf K}^b(\mathcal A)^\heartsuit \simeq \mathcal A\) recovers the original additive \(1\)-category and its inclusion is the canonical functor \(\mathcal A\rightarrow{\mathbf K}^b(\mathcal A)\).
A functor \(F\colon\mathcal C\rightarrow\mathcal D\) between stable, idempotent-complete \(\infty\)-categories with weight structure is weight exact if it is exact and the restriction of \(F\) to the full subcategory \(\mathcal C_{\geq 0} \subseteq \mathcal C\) factors through the full subcategory \(\mathcal D_{\geq 0} \subseteq \mathcal D\) and the restriction of \(F\) to \(\mathcal C_{\leq 0} \subseteq \mathcal C\) factors through \(\mathcal D_{\leq 0} \subseteq \mathcal D\). Weight exact functors \(F \colon \mathcal C\rightarrow\mathcal D\) restrict to additive functors \(F^{\heartsuit}\colon \mathcal C^{\heartsuit} \rightarrow\mathcal D^{\heartsuit}\) between the weight hearts.
Let \(\mathrm{st}^{bw}\) denote the \(\infty\)-category of idempotent complete stable categories equipped with bounded weight structures and weight exact functors.
A key result of [ES22] is that \((-)^{\mathrm{fin}}\colon\mathrm{add}\rightarrow\mathrm{st}\) factors as an equivalence \(\mathrm{add}\rightarrow\mathrm{st}^{bw}\) followed by the functor \(\mathrm{st}^{bw} \rightarrow\mathrm{st}\) which forgets the weight structure, see [ES22, Const. 2.2.7]. An inverse of this equivalence \(\mathrm{add}\rightarrow \mathrm{st}^{bw}\) is given by the functor \((-)^{\heartsuit}\colon \mathrm{st}^{bw} \rightarrow\mathrm{add}\) taking the weight heart, see [ES22, Theorem 2.2.9]. The following corollary is a consequence of this theorem:
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. ◻
Extending corollary 3.4.10, we think of \((-)^{\mathrm{fin}}:\mathrm{add}\rightarrow\mathrm{st}\) as the correct generalization of \({\mathbf K}^b(\mathcal A)\) from ordinary additive, idempotent-complete \(1\)-categories to additive, idempotent-complete \(\infty\)-categories \(\mathcal A\).
Abusing notation, we will henceforth write \({\mathbf K}^b(-) \coloneqq (-)^{\mathrm{fin}}\colon \mathrm{add}\rightarrow\mathrm{st}\) for the left adjoint to the forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\), even when applied to additive \(\infty\)-categories.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2