Searcharxiv⌕ Search

arXiv subjects

Steffen Sagave

Publications and source records attributed to Steffen Sagave.

At least 19 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.

math.AT↗
arXiv2020

Homotopical and operator algebraic twisted K-theory

Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real and complex case. We also improve several comparison results about twisted K-theory of C*-algebras to include multiplicative structures. Our results can also be interpreted in the infinity-categorical setup for parametrized spectra.

math.AT↗
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.

math.AT↗
arXiv2020

A strictly commutative model for the cochain algebra of a space

The commutative differential graded algebra $A_{\mathrm{PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal{I}}(X)$ of $A_{\mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal{I}$ to model $E_{\infty}$ differential graded algebras by strictly commutative objects, called commutative $\mathcal{I}$-dgas. We define a functor $A^{\mathcal{I}}$ from simplicial sets to commutative $\mathcal{I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty}$ dga of cochains. The functor $A^{\mathcal{I}}$ shares many properties of $A_{\mathrm{PL}}$, and can be viewed as a generalization of $A_{\mathrm{PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal{I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

math.AT↗
arXiv2020

Homotopy invariance of convolution products

The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the first author is fully homotopical. This implies that every presentably symmetric monoidal $\infty$-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.

math.AT↗
arXiv2019

Comparing cyclotomic structures on different models for topological Hochschild homology

The topological Hochschild homology $THH(A)$ of an orthogonal ring spectrum $A$ can be defined by evaluating the cyclic bar construction on $A$ or by applying Bökstedt's original definition of $THH$ to $A$. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for $THH(A)$. This implies that the two versions of topological cyclic homology resulting from these variants of $THH(A)$ are equivalent.

math.AT↗
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.

math.AT↗
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.

math.AT↗
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.

math.AT↗
arXiv2015

Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity. Our approach builds on the graded variant of E-infinity spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E-infinity spaces and use it to exhibit our spectrum of units functor as right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures.

math.AT↗
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.

math.AT↗
arXiv2014

Strictly commutative models for E-infinity quasi-categories

In this short note we show that E-infinity quasi-categories can be replaced by strictly commutative objects in the larger category of diagrams of simplicial sets indexed by finite sets and injections. This complements earlier work on diagram spaces by Christian Schlichtkrull and the second author.

math.AT↗
arXiv2014

Derived Logarithmic Geometry I

In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log étale maps and use this to define derived log stacks.

math.AG↗
arXiv2013

Logarithmic structures on topological K-theory spectra

We study a modified version of Rognes' logarithmic structures on structured ring spectra. In our setup, we obtain canonical logarithmic structures on connective K-theory spectra which approximate the respective periodic spectra. The inclusion of the p-complete Adams summand into the p-complete connective complex K-theory spectrum is compatible with these logarithmic structures. The vanishing of appropriate logarithmic topological Andre-Quillen homology groups confirms that the inclusion of the Adams summand should be viewed as a tamely ramified extension of ring spectra.

math.AT↗
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.

math.AT↗
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.

math.AT↗