Categorical aspects of bivariant K-theory

Excision in bivariant periodic cyclic cohomology: a categorical approach

We extend Cuntz-Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.The excision theorems for the bivariant periodic cyclic cohomology of discrete,topological and bornological algebras and pro-algebras follow from this.

实C*代数的分次双变量K理论的高阶范畴方法

A higher categorical approach to graded bivariant $K$-theories for Real $C^*$-algebras

该研究针对局部紧第二可数群G作用下的分次实C*代数，给出经典KK理论与E理论的∞范畴细化，得到连接K理论谱的纤维序列，分析K0群自同态并解答民间问题，证明两类范畴非分次C*代数的Dwyer-Kan局部化。

Noncommutative correspondences, duality and D-branes in bivariant K-theory

We describe a categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of C*-algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper. We explicitly work out our theory for a number of examples of noncommutative manifolds.

Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…

Model structure on projective systems of $C^*$-algebras and bivariant homology theories

Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable $C^*$-algebras $\mathrm{Pro}(\mathtt{SC^*})$. The opposite of this model category models the $\infty$-category of pointed noncommutative spaces $\mathtt{N}\mathcal{S_*}$ defined by the third author. Our model structure on $\mathrm{Pro}(\mathtt{SC^*})$ extends the well-known category of fibrant objects structure on $\mathtt{SC^*}$. We show that the pro-category $\mathrm{Pro}(\mathtt{SC^*})$ also contains, as a full coreflective subcateg…

Categorical and combinatorial aspects of descent theory

There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and outlined for weak n-categories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves non-abelian cohomology, stacks, torsors, homotopy, and higher-dimensional categories. Many of the ideas are scattered through the literature or are folklore; a few a…