arXiv subjects

Victor Saunier

Publications and source records attributed to Victor Saunier.

5 recordsLinked to original sources

arXiv2025

A model for the assembly map of bordism-invariant functors

We study oplax colimits of stable categories, of hermitian categories and of Poincar\'e categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincar\'e categories which is also a Verdier projection, whose kernel we explicitly describe. As a direct application, we generalize the Shaneson splitting for bordism-invariant functors of Poincar\'e categories proved by Calm\`es-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle to allow for twists. We also show our methods can tackle their general twisted Shaneson splitting of Poincar\'e-Verdier localizing invariants which specifies to a twisted Bass-Heller-Swan decomposition for the underlying stable categories, generalizing part of recent work of Kirstein-Kremer.

math.KT↗

arXiv2025

On exact categories and their stable envelopes

We show that Klemenc's stable envelope of exact $\infty$-categories induces an equivalence between stable $\infty$-categories with a bounded heart structure and weakly idempotent complete exact $\infty$-categories. Moreover, we generalise the Gillet-Waldhausen theorem to the connective algebraic K-theory of exact $\infty$-categories and deduce a universal property of connective algebraic K-theory as an additive invariant on exact $\infty$-categories. A key tool is a generalisation of a theorem due to Keller which provides a sufficient condition for an exact functor to induce a fully faithful functor on stable envelopes.

math.KT↗

arXiv2024

Trace methods for stable categories I: The linear approximation of algebraic K-theory

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace properties. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods.

math.AT↗

arXiv2023

A Theorem of the Heart for K-theory of Endomorphisms

We show that Quillen's resolution theorem for K-theory also applies to exact $\infty$-categories. We introduce heart structures on a stable $\infty$-category, generalizing weight structures, and using resolution ideas, we show that the category of stable $\infty$-categories equipped with a heart structure fully-faithfully embeds into the category of exact $\infty$-categories. Consequently, we show a generalized theorem of the heart for K-theory, which is equivalent to its invariance under passage to the stable envelope of exact $\infty$-category in the image of the heart functor. Finally, leveraging the above, we show that K-theory of endomorphisms satisfies the theorem of the heart for weight structures, even allowing coefficients in a suitable bimodule.

math.KT↗

arXiv2023

The Fundamental Theorem of Localizing Invariants

We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better for Karoubi-localizing functors, the Verdier-localizing invariants which are additionally invariant under idempotent completion. This general fundamental theorem specializes to new formulas in the context of non-connective K-theory, topological Hochschild homology and topological cyclic homology as well as connective K-theory of arbitrary ring spectra, and generalizes several known formulas for algebraic K-theory of spaces or connective K-theory of ordinary rings, schemes and $\mathbb{S}$-algebras.

math.KT↗