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Quasi-unitial Inner Kan Spaces

Trygve Poppe Oldervoll

Abstract

We show that semi-simplicial spaces that i) admit inner horn fillers up to homotopy and ii) possess units in a weak sense provide a viable model for $\infty$-categories. The existence of units can be expressed through various quasi-unitality conditions, and we compare the natural generalization of three such conditions found in the literature. This work is motivated by applications in Floer homotopy theory.

Quasi-unitial Inner Kan Spaces

Abstract

We show that semi-simplicial spaces that i) admit inner horn fillers up to homotopy and ii) possess units in a weak sense provide a viable model for -categories. The existence of units can be expressed through various quasi-unitality conditions, and we compare the natural generalization of three such conditions found in the literature. This work is motivated by applications in Floer homotopy theory.
Paper Structure (6 sections, 55 theorems, 130 equations)

This paper contains 6 sections, 55 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Simplicial sets and spaces
  3. Semi-Kan spaces
  4. Marked inner Kan spaces
  5. Slices and functor categories
  6. Comparison of quasi-unitality conditions

Key Result

Theorem 1.1

A semi-simplicial space $X$ is a quasi-unital inner Kan space if and only if $X^{\natural}$ is a marked inner Kan space. A map $F\colon X\to Y$ between quasi-unital inner Kan spaces is quasi-unital if and only if it lifts to a map $F^{\natural}\colon X^{\natural} \to Y^{\natural}$ between marked sem

Theorems & Definitions (148)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.5
  • Lemma 2.8
  • proof
  • Remark 2.9
  • ...and 138 more