Cyclic polytopes, orientals, and correspondences: some aspects of higher Segal spaces

TL;DR

This paper explores the connections between higher Segal spaces, cyclic polytopes, and orientals, providing geometric models and characterizations that advance understanding in higher category theory and algebraic K-theory.

Contribution

It introduces a new geometric interpretation of orientals via cyclic polytopes and characterizes higher Segal spaces as lax monadic structures in higher correspondence categories.

Findings

Cyclic polytopes serve as geometric models for orientals.

Higher Segal spaces are characterized as lax monadic structures.

Examples from algebraic K-theory illustrate the concepts.

Abstract

We discuss the role of higher Segal spaces at the interface of cyclic polytopes, orientals, and higher correspondences. Along the way we review examples from algebraic K-theory, show how cyclic polytopes provide a geometric model for the definition of orientals, and establish a characterization of higher Segal spaces as lax monadic structures in higher correspondence categories.