一个稳定的∞-范畴用于同调K-理论
A stable $\infty$-category for equivariant $KK$-theory
AI总结：
本文构造了一个稳定的∞-范畴用于同调K-理论，并探讨了其在群作用下的变化和扩展。
摘要：
对于可数群G，我们构造了一个小的、幂等完备的、对称单向的、稳定的∞-范畴KK^G_{sep}，其同调范畴恢复了分离的G-C*代数的三角同调K-理论，并展示了其普遍性质。同样，我们考虑了一个相关的可呈现的、对称单向的、稳定的∞-范畴KK^G，它接收了一个从可能非分离的G-C*代数出发的对称单向函子kk^G，并讨论了其普遍性质。除了对称单向结构外，我们还构建了各种改变群函子，将这些KK范畴联系起来，对于不同的G。我们使用这一点来定义并建立一个（谱值的）同调K-理论，在恰当且局部紧致的G-拓扑空间上，允许在任意的G-C*代数中使用系数。最后，我们将函子kk^G从G-C*代数扩展到G-C*范畴。这些构造在一篇关于同调Paschke对偶性和组装映射的配套论文中起关键作用。
英文摘要：
For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable $G$-$C^*$-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}$ which receives a symmetric monoidal functor $\mathrm{kk}^{G}$ from possibly non-separable $G$-$C^*$-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying $G$. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite $K$-homology theory on proper and locally compact $G$-topological spaces, allowing for coefficients in arbitrary $G$-$C^*$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from $G$-$C^*$-algebras to $G$-$C^*$-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
