An \(\infty\)-category is called zero-pointed if it has an initial and a terminal object and if the unique morphism from the initial to the terminal object is an isomorphism. In this case, we call the initial/terminal object a zero object. A zero-pointed \(\infty\)-category is called semi-additive if it furthermore has finite products and finite coproducts and if the canonical morphism \(x\sqcup y \rightarrow x \times y\) is an isomorphism. In this case, we write the product/coproduct as \(x\oplus y\) and refer to it as a direct sum. A semi-additive \(\infty\)-category is called additive if furthermore the shear map \(( \pi_1, \nabla)\colon x\oplus x \rightarrow x \oplus x\) is an isomorphism, where \(\pi_1\colon x \oplus x \rightarrow x\) denotes the projection to the first factor (using that \(x \oplus x\) is a product) and \(\nabla\colon x \oplus x \rightarrow x\) is the fold map (using that \(x \oplus x\) is a coproduct). A functor between additive \(\infty\)-categories is called additive if it preserves finite coproducts. We denote the \(\infty\)-category of additive functors between two additive \(\infty\)-categories \(\mathcal A, \mathcal B\) by \(\mathrm{Fun}^{\sqcup}(\mathcal A,\mathcal B)\). This \(\infty\)-category is itself an additive \(\infty\)-category [GGN15, Cor. 2.9]. The notion of an additive \(\infty\)-category is a direct generalization of the ordinary \(1\)-categorical notion, and indeed an ordinary \(1\)-category is additive in the usual sense if and only if it(s nerve) is additive in the \(\infty\)-categorical sense. Conversely, a semi-additive \(\infty\)-category \(\mathcal C\) is additive if and only if its homotopy category \(h_1\mathcal C\) is additive as an ordinary \(1\)-category [GGN15, Prop. 2.8].
A zero-pointed \(\infty\)-category is called stable if it admits finite colimits and if any square is a pullback square if and only if it is a pushout square. A functor between stable \(\infty\)-categories is called exact if it preserves finite colimits. Given two stable \(\infty\)-categories \(\mathcal C, \mathcal D\), we denote the \(\infty\)-category of exact functors between them by \(\mathrm{Fun}^{\mathrm{ex}}(\mathcal C,\mathcal D)\). This \(\infty\)-category \(\mathrm{Fun}^{\mathrm{ex}}(\mathcal C,\mathcal D)\) is itself stable since it is a full subcategory of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) (which is stable by [Lur17, Prop. 1.1.3.1]) that contains the zero object and is stable under forming fibers and cofibers, as a straightforward computation shows. In what follows, we will only consider idempotent complete stable categories.
We use the following notation:
\(\mathrm{add}\) for the full \(\infty\)-subcategory of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) consisting of additive, idempotent complete, small \(\infty\)-categories.
\(\mathrm{st}\) for the full \(\infty\)-subcategory of \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) consisting of stable, idempotent complete, small \(\infty\)-categories.
Since stable \(\infty\)-categories are additive and exact functors preserve finite coproducts, there is a forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\).
All additive and stable \(\infty\)-categories will be implicitly assumed to be idempotent complete. In particular, we have defined \(\mathrm{add}\) and \(\mathrm{st}\) as full subcategories of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\).
As in subsection 3.1 (in particular proposition 3.2.8), it will be useful to characterize small additive or stable \(\infty\)-categories in terms of projectively resp. compactly generated presentable \(\infty\)-categories.
Following definition 3.2.3, we use the following notations:
\({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) for the full subcategory of \(\mathrm{Pr}^\mathrm{L}\) on the stable, presentable \(\infty\)-categories and \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) for the full subcategory on the additive, presentable \(\infty\)-categories.
\(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) for the full subcategory of \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) on the stable presentable \(\infty\)-categories which are compactly generated as \(\infty\)-categories, and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\) for the full subcategory of \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) on the additive presentable \(\infty\)-categories which are projectively generated as \(\infty\)-categories.
The following hold.
The equivalence \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) restricts to an equivalence between full subcategories \[\mathcal P^{\Sigma}\colon\mathrm{add}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}.\] Its inverse is \((-)^{\mathrm{cp}}\) which takes a projectively generated additive presentable \(\infty\)-category \(\mathcal C\) to its full subcategory \(\mathcal C^{\mathrm{cp}}\) on the compact-projective objects.
The equivalence \(\operatorname{Ind}\colon \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) restricts to an equivalence between full subcategories \[\operatorname{Ind}\colon\mathrm{st}\xrightarrow{\simeq}\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}.\] Its inverse is \((-)^{\mathrm{c}}\) which takes a compactly generated stable presentable \(\infty\)-category \(\mathcal C\) to its full subcategory \(\mathcal C^{\mathrm{c}}\) on the compact objects.
Proof.
We will prove part ([004S]), the proof of part ([004T]) is entirely analogous and can for example be found in [BGT13, Lem. 2.20]. Recall from proposition 3.2.8 that \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is an equivalence, whose inverse is \((-)^{\mathrm{cp}}\). To prove statement (1), it therefore suffices to show that the essential image of the composite \(\mathrm{add}\hookrightarrow \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\simeq \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is the full subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\).
If \(\mathcal C\) is a small additive \(\infty\)-category, then \(\mathcal P^{\Sigma}(\mathcal C) \simeq \mathrm{Fun}^{\sqcup}(\mathcal C^{\mathrm{op}}, \mathcal S)\) is additive by [GGN15, Cor. 2.9]. On the other hand, if \(\mathcal D\) is any projectively generated additive presentable category, then the full subcategory on its compact-projective objects is closed under finite coproducts and hence is again additive. Therefore, \(\mathcal D\) is in the image of \(\mathrm{add}\hookrightarrow \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2