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Snigdhayan Mahanta

Publications and source records attributed to Snigdhayan Mahanta.

At least 19 recordsLinked to original sources
arXiv2019

$C^*$-algebraic drawings of dendroidal sets

In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. 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 combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable $\infty$-categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant K-theory. Finally, a method to analyse graph algebras in terms of trees is sketched.

math.OA↗
arXiv2017

Noncommutative stable homotopy and stable infinity categories

The noncommutative stable homotopy category $\mathtt{NSH}$ is a triangulated category that is the universal receptacle for triangulated homology theories on separable $C^*$-algebras. We show that the triangulated category $\mathtt{NSH}$ is topological as defined by Schwede using the formalism of (stable) infinity categories. More precisely, we construct a stable presentable infinity category of noncommutative spectra and show that $\mathtt{NSH}^{op}$ sits inside its homotopy category as a full triangulated subcategory, from which the above result can be deduced. We also introduce a presentable infinity category of noncommutative pointed spaces that subsumes $C^*$-algebras and define the noncommutative stable (co)homotopy groups of such noncommutative spaces generalizing earlier definitions for separable $C^*$-algebras. The triangulated homotopy category of noncommutative spectra admits (co)products and satisfies Brown representability. These properties enable us to analyse neatly the behaviour of the noncommutative stable (co)homotopy groups with respect to certain (co)limits. Along the way we obtain infinity categorical models for some well-known bivariant homology theories like $\mathrm{KK}$-theory, $\mathrm{E}$-theory, and connective $\mathrm{E}$-theory via suitable (co)localizations. The stable infinity category of noncommutative spectra can also be used to produce new examples of generalized (co)homology theories for noncommutative spaces.

math.OA↗
arXiv2017

On the Generating Hypothesis in Noncommutative Stable Homotopy

Freyd's Generating Hypothesis is an important problem in topology with deep structural consequences for finite stable homotopy. Due to its complexity some recent work has examined analogous questions in various other triangulated categories. In this short note we analyze the question in noncommutative stable homotopy, which is a canonical generalization of finite stable homotopy. Along the way we also discuss Spanier--Whitehead duality in this extended setup.

math.OA↗
arXiv2017

$\mathrm{G}$-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem

We develop a version of $\mathrm{G}$-theory for an $\mathbb{F}_1$-algebra (i.e., the $\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$) and establish its first properties. We construct a Cartan assembly map to compare the Chu--Morava $\mathrm{K}$-theory for finite pointed groups with our $\mathrm{G}$-theory. We compute the $\mathrm{G}$-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday--Whitehead groups over $\mathbb{F}_1$ that admit functorial maps into classical Whitehead groups under some reasonable hypotheses. We initiate a conjectural formalism using combinatorial Grayson operations to address the Equivariant Nishida Problem - it asks whether $\mathbb{S}^G$ admits operations that endow $\oplus_nπ_{2n}(\mathbb{S}^G)$ with a pre-$λ$-ring structure, where $G$ is a finite group and $\mathbb{S}^G$ is the $G$-fixed point spectrum of the equivariant sphere spectrum.

math.AT↗
arXiv2017

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 subcategory, the category of pro-$C^*$-algebras that are cofiltered limits of separable $C^*$-algebras. By stabilizing our model category we produce a general model categorical formalism for triangulated and bivariant homology theories of $C^*$-algebras (or, more generally, that of pointed noncommutative spaces), whose stable $\infty$-categorical counterparts were constructed earlier by the third author. Finally, we use our model structure to develop a bivariant $\mathrm{K}$-theory for all projective systems of separable $C^*$-algebras generalizing the construction of Bonkat and show that our theory naturally agrees with that of Bonkat under some reasonable assumptions.

math.KT↗
arXiv2016

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 connective E-theory and compute the connective E-theory groups of $\mathcal{O}_\infty$-stable $C^*$-algebras. We also introduce and study the nonconnective version of Quillen's nonunital K'-theory in the framework of stable $\infty$-categories. This is done in order to promote our earlier result relating topological $\mathbb{T}$-duality to noncommutative motives to the $\infty$-categorical setup. Finally, we carry out some computations in the case of stable and $\mathcal{O}_\infty$-stable $C^*$-algebras.

math.KT↗
arXiv2015

Colocalizations of noncommutative spectra and bootstrap categories

We construct a compactly generated and closed symmetric monoidal stable $\infty$-category $\mathtt{NSp'}$ and show that $\mathtt{hNSp'}^{op}$ contains the suspension stable homotopy category of separable $C^*$-algebras $\mathtt{ΣHo^{C^*}}$ constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of $\mathtt{NSp'}$, namely, $\mathtt{NSp'}[\mathbb{K}^{-1}]$ and $\mathtt{NSp'}[\mathcal{Z}^{-1}]$, both of which are shown to be compactly generated and closed symmetric monoidal. We prove that Kasparov $KK$-category of separable $C^*$-algebras sits inside the homotopy category of $\mathtt{KK_\infty} := \mathtt{NSp'}[\mathbb{K}^{-1}]^{op}$ as a fully faithful triangulated subcategory. Hence $\mathtt{KK_\infty}$ should be viewed as the stable $\infty$-categorical incarnation of Kasparov $KK$-category for arbitrary pointed noncommutative spaces (including nonseparable $C^*$-algebras). As an application we find that the bootstrap category in $\mathtt{hNSp'}[\mathbb{K}^{-1}]$ admits a completely algebraic description. We also construct a $K$-theoretic bootstrap category in $\mathtt{hKK_\infty}$ that extends the construction of the UCT class by Rosenberg-Schochet. Motivated by the algebraization problem we finally analyse a couple of equivalence relations on separable $C^*$-algebras that are introduced via the bootstrap categories in various colocalizations of $\mathtt{NSp'}$.

math.KT↗
arXiv2015

Algebraic K-theory, K-regularity, and T-duality of $\mathcal{O}_\infty$-stable $C^*$-algebras

We develop an algebraic formalism for topological $\mathbb{T}$-duality. More precisely, we show that topological $\mathbb{T}$-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an $E_\infty$-operad starting from any strongly self-absorbing $C^*$-algebra $\mathcal{D}$. Then we show that there is a functorial topological K-theory symmetric spectrum construction ${\bf K}_Σ^{top}(-)$ on the category of separable $C^*$-algebras, such that ${\bf K}_Σ^{top}(\mathcal{D})$ is an algebra over this operad; moreover, ${\bf K}_Σ^{top}(A\hat{\otimes}\mathcal{D})$ is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of $C^*$-algebras. We also show that $\mathcal{O}_\infty$-stable $C^*$-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of $ax+b$-semigroup $C^*$-algebras coming from number theory and that of $\mathcal{O}_\infty$-stabilized noncommutative tori.

math.KT↗
arXiv2015

Higher nonunital Quillen K'-theory, KK-dualities and applications to topological $\mathbb{T}$-dualities

Quillen introduced a new $K'_0$-theory of nonunital rings and showed that, under some assumptions (weaker than the existence of unity), this new theory agrees with the usual algebraic $K^{alg}_0$-theory. For a field $k$ of characteristic $0$, we introduce higher nonunital $K$-theory of $k$-algebras, denoted $KQ$, which extends Quillen's original definition of the $K'_0$ functor. We show that the $KQ$-theory is Morita invariant and satisfies excision connectively, in a suitable sense, on the category of idempotent $k$-algebras. Using these two properties we show that the $KQ$-theory agrees with the topological $K$-theory of stable $C^*$-algebras. The machinery enables us to produce a DG categorical formalism of topological homological $\mathbb{T}$-duality using bivariant $K$-theory classes. A connection with strong deformations of $C^*$-algebras and some other potential applications to topological field theories are discussed towards the end.

math.KT↗
arXiv2013

Twisted K-theory, K-homology and bivariant Chern-Connes type character of some infinite dimensional spaces

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate the twisted periodic cyclic homology via locally convex algebras and the local cyclic homology via C^*-algebras (in the compact case).

math.KT↗
arXiv2013

Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite coefficients. For all torsion free Gromov hyperbolic groups G, we demonstrate that the canonical algebra homomorphism \cpt[G]\map C^*_r(G)\hat{\otimes}\cpt induces an isomorphism between their algebraic K-theory groups.

math.KT↗
arXiv2012

Continuous homotopy invariance of bivariant local cyclic homology for σ-C^*-algebras

We establish the continuous homotopy invariance of bivariant local cyclic homology on the category of all σ-C^*-algebras. The argument relies vitally on an isomorphism between the smooth and continuous cylinder constructions using a technical criterion due to Meyer. As a consequence we compute the local cyclic homology of the infinite sphere.

math.OA↗
arXiv2011

Operator algebra quantum groups of universal gauge groups

In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra setting. More precisely, we propose a concise definition of sigma-C*-quantum groups and explain the concept here. At the same time, we put this definition in the mathematical context of countably compactly generated groups as well as C*-compact quantum groups.

math.QA↗
arXiv2011

Operator algebra quantum homogeneous spaces of universal gauge groups

In this paper, we quantize universal gauge groups such as SU(\infty), as well as their homogeneous spaces, in the sigma-C*-algebra setting. More precisely, we propose concise definitions of sigma-C*-quantum groups and sigma-C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute it for the quantum homogeneous spaces associated to the universal gauge group SU(\infty).

math.QA↗
arXiv2009

Noncommutative correspondence categories, simplicial sets and pro $C^*$-algebras

We show that a $KK$-equivalence between two unital $C^*$-algebras produces a correspondence between their DG categories of finitely generated projective modules which is a $\mathbf{K}_*$-equivalence, where $\mathbf{K}_*$ is Waldhausen's $K$-theory. We discuss some connections with strong deformations of $C^*$-algebras and homological dualities. Motivated by a construction of Cuntz we associate a pro $C^*$-algebra to any simplicial set. We show that this construction is functorial with respect to proper maps of simplicial sets, that we define, and also respects proper homotopy equivalences. We propose to develop a noncommutative proper homotopy theory in the context of topological algebras.

math.KT↗
arXiv2008

Noncommutative tori and the Riemann-Hilbert correspondence

We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on $\mathbb{C}^*$. We functorially relate this category to the category of holomorphic vector bundles on noncommutative tori as introduced by Polishchuk and Schwarz and study the induced map between the corresponding K-theories. In addition, there is a forgetful functor to the category of noncommutative elliptic curves of Soibelman and Vologodsky, as well as a forgetful functor to the category of vector bundles on $\mathbb{C}^*$ with regular singular connections. The category that we consider has the nice property of being a Tannakian category, hence it is equivalent to the category of representations of an affine group scheme. Via an equivariant version of the Riemann-Hilbert correspondence we determine this group scheme to be (the algebraic hull of) $\mathbb{Z}^2$. We also obtain a full subcategory of the category of holomorphic bundles of the noncommutative torus, which is equivalent to the category of representations of $\mathbb{Z}$. This group is the proposed topological fundamental group of the noncommutative torus (understood as a degenerate elliptic curve) and we study Nori's notion of étale fundamental group in this context.

math.QA↗
arXiv2008

Noncommutative Geometry in the Framework of Differential Graded Categories

In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.

math.AG↗