Title: Stabilization and costabilization with respect to an action of a monoidal category Show Chinese title

Title: 关于单子范畴作用的稳定化和协稳定化

Abstract: We study actions of monoidal categories on objects in a suitably enriched $2$-category, and applications in stable homotopy theory. Given a monoidal category $\mathcal{I}$ and an $\mathcal{I}$-object $\mathcal{A}$, the (co)stabilization of $\mathcal{A}$ is obtained by universally forcing the $\mathcal{I}$-action to be reversible so that every object of $\mathcal{I}$ acts on $\mathcal{A}$ by auto-equivalences. We introduce a notion of $\mathcal{I}$-equivariance for morphisms between $\mathcal{I}$-objects and give constructions of stabilization and costabilization in terms of weak ends and coends in an enriched $2$-category of $\mathcal{I}$-objects and $\mathcal{I}$-equivariant morphisms. We observe that the stabilization of a relative category with respect to an action coincides with the usual notion of stabilization in stable homotopy theory when the action is defined by loop space functors. We show that several examples that exist in the literature, including various categories of spectra, fit into our setting after fixing $\mathcal{A}$ and the $\mathcal{I}$-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, parameterized spectra indexed by vector bundles are obtained in terms of weak ends in the $2$-category of relative categories. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version Spanier-Whitehead category. In particular, we establish a form of duality between constructions of stable homotopy categories by spectra and by Spanier-Whitehead like categories. Show Chinese abstract

Abstract: 我们研究单子范畴在适当丰富的$2$-范畴中的对象上的作用，以及在稳定同伦理论中的应用。 给定一个单子范畴$\mathcal{I}$和一个$\mathcal{I}$-对象$\mathcal{A}$，$\mathcal{A}$的 (co)稳定化是通过普遍地强制$\mathcal{I}$-作用变为可逆的，使得$\mathcal{I}$的每个对象都通过自等价作用于$\mathcal{A}$。 我们引入了在$\mathcal{I}$-对象之间的态射的$\mathcal{I}$-等变性概念，并在增强的$2$-范畴中通过弱端和余端构造了稳定化和反稳定化，该范畴包含$\mathcal{I}$-对象和$\mathcal{I}$-等变态射。 我们观察到，相对于一个作用的相对范畴的稳定化与稳定同伦理论中的通常稳定化概念一致，当该作用由环路空间函子定义时。 我们表明文献中的一些例子，包括各种谱范畴，在固定$\mathcal{A}$和其上的$\mathcal{I}$-作用后，符合我们的设定。 特别是，序列谱、坐标自由谱、真实等变谱、由向量丛索引的参数化谱的范畴是通过相对范畴的$2$-范畴中的弱端得到的。 另一方面，相对于一个作用的相对范畴的代价稳定化给出了一个类似于Spanier-Whitehead范畴版本的稳定相对范畴。 特别是，我们建立了通过谱构造的稳定同伦范畴与通过类似Spanier-Whitehead范畴构造之间的对偶形式。