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Models for rational $(\infty, 1)$-categories

Eleftherios Chatzitheodoridis

TL;DR

The paper develops two equivalent models for rational $(\infty,1)$-categories, namely rational complete Segal spaces and rational Segal categories, by enriching $(\infty,1)$-categories in rational spaces. It constructs these models via left Bousfield localization and left transfer, and proves they are Quillen equivalent while preserving cartesian structure. A key advance is the rationalization of the homotopy theory of spaces with nontrivial fundamental group, enabling a robust framework for rational enrichment. The results pave the way for rational analogs of other $(\infty,1)$-category models and offer tools for applying rational homotopy theory to enriched higher category theory.

Abstract

We introduce rational $(\infty, 1)$-categories, which are $(\infty, 1)$-categories enriched in spaces whose higher homotopy groups are rational vector spaces. We provide two models for rational $(\infty, 1)$-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent.

Models for rational $(\infty, 1)$-categories

TL;DR

The paper develops two equivalent models for rational -categories, namely rational complete Segal spaces and rational Segal categories, by enriching -categories in rational spaces. It constructs these models via left Bousfield localization and left transfer, and proves they are Quillen equivalent while preserving cartesian structure. A key advance is the rationalization of the homotopy theory of spaces with nontrivial fundamental group, enabling a robust framework for rational enrichment. The results pave the way for rational analogs of other -category models and offer tools for applying rational homotopy theory to enriched higher category theory.

Abstract

We introduce rational -categories, which are -categories enriched in spaces whose higher homotopy groups are rational vector spaces. We provide two models for rational -categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent.

Paper Structure

This paper contains 23 sections, 43 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Simplicial sets and model categories
  3. Simplicial sets
  4. Model categories
  5. Left Bousfield localization
  6. Cartesian model categories
  7. Rational homotopy theory of non-simply connected spaces
  8. The construction of local spheres
  9. A characterization of rational spaces
  10. The model category for rational spaces
  11. Simplicial spaces and complete Segal spaces
  12. Simplicial spaces
  13. Complete Segal spaces
  14. Segal categories
  15. Definition of a Segal category
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exist: These model categories are Quillen equivalent.

Theorems & Definitions (83)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8: Hirschhorn2003
  • Definition 2.9
  • ...and 73 more