arXiv subjects

Jonte Gödicke

Publications and source records attributed to Jonte Gödicke.

4 recordsLinked to original sources

arXiv2026

The $(\infty,\infty)$-category of spans

In this paper, we construct the $(\infty,\infty)$-category $\mathsf{Span}_\infty(\mathcal{C})$ of spans, also known as correspondences, in any given $(\infty,1)$-category $\mathcal{C}$ with finite limits. This yields new models for the span $(\infty,n)$-categories for $n \in \mathbb{N} \cup \{\infty\}$. We characterize the mapping $(\infty, n-1)$-categories in these $(\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., $\mathsf{E}_k$-algebras in $(\infty, n)$-categories for $n, k>1$.

math.CT↗

arXiv2026

Sphericalization and the Universal Spherical Adjunction

For every adjunction of stable $\infty$-categories -- or more generally, in any locally stable $(\infty,2)$-category -- we give a simple procedure for inverting the twist and cotwist functors associated to this adjunction. As a consequence, we obtain an explicit construction for a left and right adjoint to the inclusion of the $(\infty,2)$-category of spherical adjunctions of stable $\infty$-categories into all adjunctions. We utilize these adjoints to give a description of the walking spherical adjunction, a locally stable $(\infty,2)$-category which classifies spherical adjunctions, and to provide a synthetic proof of the fact that every spherical functor admits infinitely many left and right adjoints.

math.CT↗

arXiv2025

An $\infty$-Category of 2-Segal Spaces

Algebra objects in $\infty$-categories of spans admit a description in terms of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects and extend this correspondence to an equivalence of $\infty$-categories. Additionally, for every $\infty$-category with finite limits $\mathcal{C}$, we introduce a notion of a birelative $2$-Segal object in $\mathcal{C}$ and establish a similar equivalence with the $\infty$-category of bimodule objects in spans. Examples of these concepts arise from algebraic and hermitian K-theory through the corresponding Waldhausen $S_{\bullet}$-construction. Apart from their categorical relevance, these concepts can be used to construct homotopy coherent representations of Hall algebras.

math.AT↗

arXiv2024

Simons Lectures on Categorical Symmetries

Global Categorical Symmetries are a powerful new tool for analyzing quantum field theories. This volume compiles lecture notes from the 2022 and 2023 summer schools on Global Categorical Symmetries, held at the Perimeter Institute for Theoretical Physics and at the Swiss Map Research Station in Les Diableret. Specifically, this volume collects the lectures: * An introduction to symmetries in quantum field theory, Kantaro Ohmori * Introduction to anomalies in quantum field theory, Clay Córdova * Symmetry Categories 101, Michele Del Zotto * Applied Cobordism Hypothesis, David Jordan * Finite symmetry in QFT, Daniel S. Freed These volumes are devoted to interested newcomers: we only assume (basic) knowledge of quantum field theory (QFT) and some relevant maths. We try to give appropriate references for non-standard materials that are not covered. Our aim in this first volume is to illustrate some of the main questions and ideas together with some of the methods and the techniques necessary to begin exploring global categorical symmetries of QFTs.

math-ph↗