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

Tobias Dyckerhoff

TL;DR

The work investigates higher Segal spaces at the crossroads of cyclic polytopes, Street's orientals, and higher correspondences, coupling geometric and higher-categorical perspectives. It shows that cyclic polytopes provide a geometric model for orientals and that the resulting $\omega$-categories align with Street's free $\omega$-category on the simplex, with Rambau's lemma governing their combinatorics. The authors develop a lax-monad viewpoint, identifying higher Segal objects with lax monads in higher correspondence categories, and establish path-space and excision criteria that elevate lower/upper $d$-Segal objects to full $k$-Segal structures for larger $k$. They formulate higher correspondences via barycentric subdivision, introducing truncations $\operatorname{co}_n(\mathcal{C})$, $\operatorname{co}_n^l(\mathcal{C})$, $\operatorname{co}_n^u(\mathcal{C})$ and connecting Segal conditions to monads in these settings. The paper also sketches future directions, including coherent Pachner moves and categorified Dold–Kan phenomena in 2-categorical contexts, hinting at applications to higher algebraic K-theory and related combinatorial structures.

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.

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

TL;DR

The work investigates higher Segal spaces at the crossroads of cyclic polytopes, Street's orientals, and higher correspondences, coupling geometric and higher-categorical perspectives. It shows that cyclic polytopes provide a geometric model for orientals and that the resulting -categories align with Street's free -category on the simplex, with Rambau's lemma governing their combinatorics. The authors develop a lax-monad viewpoint, identifying higher Segal objects with lax monads in higher correspondence categories, and establish path-space and excision criteria that elevate lower/upper -Segal objects to full -Segal structures for larger . They formulate higher correspondences via barycentric subdivision, introducing truncations , , and connecting Segal conditions to monads in these settings. The paper also sketches future directions, including coherent Pachner moves and categorified Dold–Kan phenomena in 2-categorical contexts, hinting at applications to higher algebraic K-theory and related combinatorial structures.

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.
Paper Structure (11 sections, 18 theorems, 73 equations, 2 figures)

This paper contains 11 sections, 18 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cyclic polytopes
  3. Higher Segal objects
  4. Orientals
  5. The interplay of higher Segal conditions
  6. Higher correspondences
  7. Monads
  8. Higher excision
  9. Further perspectives
  10. Coherent Pachner moves.
  11. Additive coefficients and $2$-categorical methods.

Key Result

Proposition 2.5

Let $n,d \ge 1$.

Figures (2)

  • Figure 1: The cyclic polytopes $C([5],3)$, $C([5],2)$, and $C([5],1)$, related via the projection maps $\pi$.
  • Figure 2: The two triangulations of $C([4],3)$ corresponding to the upper boundary ${\EuScript U}([4],3)$ and lower boundary $\mathcal{L}([4],3)$ of $|\Delta^4|$, respectively.

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Example 2.7
  • Lemma 2.8
  • proof
  • Remark 3.1
  • ...and 48 more