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Free colimit completion in $\infty$-categories

Charles Rezk

Abstract

We show how several useful properties of Ind-constructions in $\infty$-categories extend to arbitrary free colimit completion constructions.

Free colimit completion in $\infty$-categories

Abstract

We show how several useful properties of Ind-constructions in -categories extend to arbitrary free colimit completion constructions.
Paper Structure (27 sections, 27 theorems, 26 equations)

This paper contains 27 sections, 27 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Basic $\infty$-categorical notions
  3. Universes
  4. Colimits
  5. Presheaves
  6. Slices of presheaves
  7. Free colimit completion
  8. Regular closure
  9. Regular classes
  10. Explicit description of free colimit completion
  11. Cofinality
  12. Regular classes and $\infty$-groupoids
  13. Constructing regular classes
  14. Filtration and cofiltration classes
  15. A recognition principle for free colimit completion
  16. ...and 12 more sections

Key Result

Theorem 3.1

lurie-higher-topos*5.1.5.6 For any cocomplete $\infty$-category $A$, restriction along $\rho$ induces an equivalence from the category of colimit preserving functors $\operatorname{PSh}(C)\rightarrow A$ to the category of functors $C\rightarrow A$. In particular, any functor $f\colon C\rightarrow A$ admits an essentially unique extension $\widehat{f}\colon \operatorname{PSh}(C)\rightarrow A$ to a

Theorems & Definitions (55)

  • Remark 2.5
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4: Embedding theorem
  • proof : Sketch proof
  • Proposition 3.5
  • proof
  • Remark 4.1
  • ...and 45 more