AI 中文总结
本研究建立可对偶加法范畴的完备理论，给出等价刻画与殆数学联系，证明其由平坦对象生成并等价于紧装配加法范畴，应用于解析几何刻画连通核模范畴，还构造泛有限稳定局部化不变量并建立其与非连通代数$K$-理论的关系。
AI 中文摘要
我们建立了可对偶加法范畴的完备理论，给出了若干等价刻画， notably将其识别为满足$\text{AB4}^*$和$\text{AB6}$公理的分离格罗滕迪克预稳定范畴。我们通过证明这类范畴恰好是连通$\text{E}_1$-环上的连通殆模范畴，建立了其与殆数学的联系。此外，我们证明可对偶加法范畴由平坦对象生成，且取平坦对象的操作给出了可对偶加法范畴与紧装配加法范畴之间的等价。作为解析几何中的主要应用，我们通过泛性质刻画了adic $\text{E}_\text{∞}$-环$R$上的连通核模范畴$\text{Nuc}(R)_{\text{≥}0}$（Clausen–Scholze意义下），将其识别为连通完备$R$-模范畴的加性刚性化。最后，我们构造了可对偶加法范畴的泛有限稳定局部化不变量，即可表现稳定预稳定动机范畴$\text{Mot}_\text{pst}$，并证明其单位余表示非连通代数$K$-理论，且小加法范畴与可对偶加法范畴的动机生成同一个可表现稳定子范畴。
英文摘要
We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the $\mathrm{AB4}^*$ and $\mathrm{AB6}$ axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective $\mathbb{E}_1$-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category $\mathrm{Nuc}(R)_{\geq 0}$ of connective nuclear modules (in the sense of Clausen--Scholze) over an adic $\mathbb{E}_\infty$-ring $R$ via a universal property, identifying it as the additive rigidification of the category of connective complete $R$-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category $\mathcal{M}\mathrm{ot}_{\mathrm{pst}}$ of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic $K$-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.
Comments97 pages; comments welcome!
