Stable moduli spaces of hermitian forms

We prove that Grothendieck-Witt spaces of Poincar\'e categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubi's classical hermitian and quadratic K-groups with the genuine Grothendieck-Witt groups from our joint work with Calm\`es, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our solution of several conjectures in hermitian K-theory. The method of proof is abstracted from work of Galatius and Randal-Williams on cobordism categories of manifolds using the identification of the Grothendieck-Witt space of a Poincar\'e category as the homotopy type of the associated cobordism category.Comment: 94 pages, v2: New appendix by Y.Harpaz that gives a simplified proof of the main result in a special case, furthermore added Section 8.4 about real algebraic K-spectra and made miscellaneous minor improvement

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 ∞-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.publishedVersio

The localisation theorem for the K -theory of stable ∞-categories

Funding Information: We heartily thank Dustin Clausen, from whom we first learned a clean proof of the localisation theorem, for several very helpful discussions. We further thank Ferdinand Wagner for the permission to use some of his pretty TikZ diagrams in this note. During the preparation of this manuscript, F. H. was a member of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG) and furthermore a member of the cluster ‘Mathematics Münster: Dynamics-Geometry-Structure’ at the University of Münster (DFG grant nos. EXC 2047 390685813 and EXC 2044 390685587, respectively). F. H. acknowledgesthe Mittag-Leffler Institute for its hospitality during the research programme ‘Higher algebraic structures in algebra, topology and geometry’, supported by the Swedish Research Council under grant no. 2016-06596. A. L. was supported by the Research Training Group ‘Algebro-Geometric Methods in Algebra, Arithmetic and Topology’ at the University of Wuppertal (DFG grant no. GRK 2240) and W. S. by the priority programme ‘Geometry at Infinity’ (DFG grant no. SPP 2026) at the University of Augsburg. Publisher Copyright: Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.Peer reviewedPublisher PD

A note on quadratic forms

For a field extension $L/K$ we consider maps that are quadratic over $L$ but whose polarisation is only bilinear over $K$. Our main result is that all such are automatically quadratic forms over $L$ in the usual sense if and only if $L/K$ is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.Comment: 12 pages, comments very welcome (in particular regarding previously known results!

The localisation theorem for the K\mathrm{K}-theory of stable ∞\infty-categories

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\mathrm{K}$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\mathrm{K}$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and demonstrate that the cofinality theorem can be derived from the universal property alone.Comment: 27 pages, v2: minor revision following a referee report; to appear in Proceedings of the RSE, Ranicki memorial issu

Twisted spin cobordism

In der vorliegenden Arbeit wird eine parametrisierte Variante des Spin-Bordismus untersucht. Als 'twists' werden hierbei nur die klassischen Bündeltwists zugelassen. Die Atiyah-Bott-Shapiro-Orientierung überträgt, diese in die K-Theorie. Es wird ein Spaltungssatz für diese Homologietheorien bewiesen. Im Anschluss wird die ordinäre Homologie der zugrunde liegenden parametrisierten Spektren berechnet, insbesondere im Hinblick auf die Zusammenhänge zu positiver Skalarkrümmung, welche ganz zu Beginn der Arbeit als Motivation dargestellt sind

Orthofibrations and monoidal adjunctions

We study various types of fibrations over a product of two $\infty$-categories, and show how they can be dualised over one of the two factors via an explicit construction in terms of spans. Among other things, we use this to prove that given an adjunction between monoidal $\infty$-categories, there is an equivalence between lax monoidal structures on the right adjoint and oplax monoidal structures on the left adjoint functor.Comment: 48 page

Two-variable fibrations, factorisation systems and categories of spans

Open Access via the CUP agreement Funding statement During the preparation of this text, FH and SL were members of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG), and FH was furthermore a member of the cluster ‘Mathematics Münster: Dynamics-Geometry-Structure’ at the University of Münster under grant nos. EXC 2047 and EXC 2044, respectively. FH would also like to thank the Mittag-Leffler Institute for its hospitality during the research program ‘Higher Algebraic Structures in Algebra, Topology and Geometry’, supported by the Swedish Research Council (VR) under grant no. 2016-06596. FH and JN were further supported by the European Research Council (ERC) through the grants ‘Moduli Spaces, Manifolds and Arithmetic’, grant no. 682922, and ‘Derived Symplectic Geometry and Applications’, grant no. 768679, respectively. SL was supported by the DFG Schwerpunktprogramm 1786 ‘Homotopy Theory and Algebraic Geometry’ (project ID SCHW 860/1-1).Peer reviewedPublisher PD