arXiv subjects

Christian Schlichtkrull

Publications and source records attributed to Christian Schlichtkrull.

14 recordsLinked to original sources

arXiv2026

Localization sequences for logarithmic topological cyclic homology

We introduce the notion of an E_k-ring with prelogarithmic structure, define logarithmic topological Hochschild homology and logarithmic topological cyclic homology in this context, and establish localization sequences for these theories. Our approach is based on Thom R-algebras. It extends and strengthens our earlier work on the subject in several regards. Our examples include the fraction field of topological K-theory, the existence of which was suggested by calculations by Ausoni and the first author. To illustrate the computational accessibility of log THH and log TC, we determine these for non-negative even periodic sphere spectra, with their canonical prelogarithmic structures.

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arXiv2026

Combinatorial and homotopical aspects of $E_n$-operads

We show that a certain class of categorical operads give rise to $E_n$-operads after geometric realization. The main arguments are purely combinatorial and avoid the technical topological assumptions otherwise found in the literature.

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arXiv2020

Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding infinity-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted K-theory.

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arXiv2018

Virtual vector bundles and graded Thom spectra

We introduce a convenient framework for constructing and analyzing orthogonal Thom spectra arising from virtual vector bundles. This framework enables us to set up a theory of orientations and graded Thom isomorphisms with good multiplicative properties. The theory is applied to the analysis of logarithmic structures on commutative ring spectra.

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arXiv2017

Generalized Thom spectra and their topological Hochschild homology

We develop a theory of R-module Thom spectra for a commutative symmetric ring spectrum R and we analyze their multiplicative properties. As an interesting source of examples, we show that R-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on R. We apply the general theory to obtain a description of the R-based topological Hochschild homology associated to an R-algebra Thom spectrum.

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arXiv2016

Logarithmic topological Hochschild homology of topological K-theory spectra

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand into the p-local complex connective K-theory spectrum, equipped with suitable log structures, is a formally log THH-etale map, and compute the V(1)-homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the V(1)-homotopy of the ordinary THH of ku.

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arXiv2015

Braided injections and double loop spaces

We consider a framework for representing double loop spaces (and more generally E-2 spaces) as commutative monoids. There are analogous commutative rectifications of braided monoidal structures and we use this framework to define iterated double deloopings. We also consider commutative rectifications of E-infinity spaces and symmetric monoidal categories and we relate this to the category of symmetric spectra.

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arXiv2015

Localization sequences for logarithmic topological Hochschild homology

We study the logarithmic topological Hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory. Our results apply, for example, to connective covers of periodic ring spectra like real and complex topological K-theory.

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arXiv2012

Group completion and units in I-spaces

The category of I-spaces is the diagram category of spaces indexed by finite sets and injections. This is a symmetric monoidal category whose commutative monoids model all E-infinity spaces. Working in the category of I-spaces enables us to simplify and strengthen previous work on group completion and units of E-infinity spaces. As an application we clarify the relation to Gamma-spaces and show how the spectrum of units associated with a commutative symmetric ring spectrum arises through a chain of Quillen adjunctions.

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arXiv2012

Diagram spaces and symmetric spectra

We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the I-spaces, which are diagrams indexed by finite sets and injections, and J-spaces, which are diagrams indexed by the Grayson-Quillen construction on the category of finite sets and bijections. We show that the category of I-spaces provides a convenient model for the homotopy category of spaces in which every E-infinity space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J-spaces model graded E-infinity spaces. Using the theory of J-spaces we introduce the graded units of a symmetric ring spectrum. The graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures.

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arXiv2010

Higher topological Hochschild homology of Thom spectra

In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology.

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arXiv2010

Thom spectra that are symmetric spectra

We analyze the functorial and multiplicative properties of the Thom spectrum functor in the setting of symmetric spectra, and we establish the relevant homotopy invariance.

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arXiv2009

The cyclotomic trace for symmetric ring spectra

The purpose of this paper is to present a simple and explicit construction of the Bokstedt-Hsiang-Madsen cyclotomic trace relating algebraic K-theory and topological cyclic homology. Our construction also incorporates Goodwillie's idea of a global cyclotomic trace.

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arXiv2004

Units of ring spectra and their traces in algebraic K-theory

Let GL_1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition BGL_1(R)->K(R)->THH(R)->Ω^{\infty}(R), where K(R) is the algebraic K-theory and THH(R) the topological Hochschild homology of R. As a corollary we show that classes in π_{i-1}(R) not annihilated by the stable Hopf map give rise to non-trivial classes in K_i(R) for i\geq 3.

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