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

Barnea, Ilan; Joachim, Michael; Mahanta, Snigdhayan

doi:10.48550/arxiv.1508.04283

Cited by 3 publications

(4 citation statements)

References 39 publications

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“…Remark 3.13. Actually Proposition 3.19 of [2] proves a pointed version of the above Lemma. The desired result can be shown using similar methods and hence its proof is omitted.…”

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confidence: 71%

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C∗–algebraic drawings of dendroidal sets

Mahanta

2019

Algebr. Geom. Topol.

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In recent years the theory of dendroidal sets has emerged as an important framework for combinatorial topology. In this article we introduce the concept of a C *algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on C * -algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable ∞-categories. As a consequence we obtain a new homotopy theory for C * -algebras that is well-adapted to the notion of weak operadic equivalences. Finally, a method to analyse graph algebras in terms of trees is sketched. Catwhere the vertical arrow N (resp. N d ) denotes the nerve (resp. dendroidal nerve) functor. Cisinski-Moerdijk constructed a cofibrantly generated model structure on dSet [10], such that the fibrant objects are precisely the ∞-operads [28]. Over the last decade the theory of dendroidal sets has reached an advanced stage, subsuming several aspects of the theory of operads and that of simplicial sets [11,12]. This article is motivated by both practical and philosophical considerations, namely, from a practical standpoint an interaction between combinatorial topology (dendroidal sets) and combinatorial noncommutative topology (graph algebras) seems worthwhile; from a philosophical viewpoint a link between the two predominant paradigms of noncommutative geometry is certainly desirable. Noncommutative geometryà la Connes has produced over the last three decades striking applications to problems in topology, analysis, mathematical physics, and several other areas of mathematics [13,14]. The basic objects in this setup are C * -algebras that are generalized by the ∞-category of noncommutative spaces [33]. The other form of noncommutative geometry has emerged through the works of Drinfeld, Keller, Kontsevich, Lurie, Manin, Tabuada, Toën, and several others [34,24,25,28,44] with remarkable applications to problems in algebra, algebraic geometry, representation theory, and K-theory (a nonexhaustive list). In its current state the basic objects of this setup are differential graded categories or stable ∞-categories and they can all be subsumed in the world of ∞-operads. Dendroidal sets provide a convenient model for ∞-operads (see [20] for a comparison with Lurie's model [28] for ∞-operads without constants). It has been a challenge to reconcile the two paradigms of noncommutative geometry. In view of the disparate nature of the ingredients of the two paradigms a bridge between the basic objects of the two worlds in the form ∞-categorical adjunctions seems to be a reasonable target to begin with. While connecting two different viewpoints on (arguably) the same topic an ∞-categorical adjunction is admittedly the second best option; an equivalence of ∞-categories would be the best outcome but we believe such an expectation to be unrealistic in this context. ...

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“…Remark 3.13. Actually Proposition 3.19 of [2] proves a pointed version of the above Lemma. The desired result can be shown using similar methods and hence its proof is omitted.…”

mentioning

confidence: 71%

“…The following result is proven in [2] using the formalism of weak (co)fibration categories [3]. Lemma 3.12.…”

mentioning

confidence: 99%

C∗–algebraic drawings of dendroidal sets

Mahanta

2019

Algebr. Geom. Topol.

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“…op is an equivalence (see Proposition 3.17 of [1]), from which the assertion follows (see Proposition 5.3.5.11 of [27]).…”

Section: Remark 41mentioning

confidence: 92%

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

Mahanta

2017

J. Noncommut. Geom.

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Abstract. Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal ∞-categorical models for separable C * -algebras SC * ∞ and noncommutative spectra NSp using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of SC * ∞ and NSp with respect to strongly self-absorbing C * -algebras. We analyse the homotopy categories of the localizations of SC * ∞ and give universal characterizations thereof. We construct a stable ∞-categorical model for bivariant connective E-theory and compute the connective E-theory groups of O ∞ -stable C * -algebras. We also introduce and study the nonconnective version of Quillen's nonunital K ′ -theory in the framework of stable ∞-categories. This is done in order to promote our earlier result relating topological T-duality to noncommutative motives to the ∞-categorical setup. Finally, we carry out some computations in the case of stable and O ∞ -stable C * -algebras.

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“…There are various construction of such a stable ∞-category [Mah15], [BJM15], [JJ07]. Note that we do not expect (6.21) to be true for general separable C * -algebras in place of C, i.e., the direct sum of separable C * -algebras does not represent the coproduct in KK.…”

Section: Analytic Locally Finite K-homologymentioning

confidence: 99%