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How to enhance categories, and why?

D. Kaledin

TL;DR

This paper develops enhanced category theory as a framework to encode homotopical information beyond ordinary categories, connecting localization, derivator-like behavior, and higher-categorical notions. It builds enhanced categories as fibrations over posets, uses the Grothendieck construction and Yoneda-style machinery, and relates to complete Segal spaces to provide robust representations of homotopy-theoretic data. A central theme is representability and universal properties (e.g., Brown representability) in the enhanced setting, yielding a toolkit for limits, Kan extensions, and functor categories that mirrors the classical theory while accommodating higher coherences. The work aims to lay groundwork for a principled axiomatization of Cat^h and to bridge between model-categorical localization, Segal-space approaches, and derivator-like perspectives for practical applications in higher category theory and homotopy theory.

Abstract

This is a companion overview paper to arXiv:2409.17489: we give all the main definitions, constructions and statements, but no proofs.

How to enhance categories, and why?

TL;DR

This paper develops enhanced category theory as a framework to encode homotopical information beyond ordinary categories, connecting localization, derivator-like behavior, and higher-categorical notions. It builds enhanced categories as fibrations over posets, uses the Grothendieck construction and Yoneda-style machinery, and relates to complete Segal spaces to provide robust representations of homotopy-theoretic data. A central theme is representability and universal properties (e.g., Brown representability) in the enhanced setting, yielding a toolkit for limits, Kan extensions, and functor categories that mirrors the classical theory while accommodating higher coherences. The work aims to lay groundwork for a principled axiomatization of Cat^h and to bridge between model-categorical localization, Segal-space approaches, and derivator-like perspectives for practical applications in higher category theory and homotopy theory.

Abstract

This is a companion overview paper to arXiv:2409.17489: we give all the main definitions, constructions and statements, but no proofs.
Paper Structure (29 sections, 28 theorems, 83 equations)

This paper contains 29 sections, 28 theorems, 83 equations.

Table of Contents

  1. Category theory.
  2. Categories and functors.
  3. Fibrations and cofibrations.
  4. Limits and Kan extensions.
  5. Localization and the Yoneda embeddings.
  6. Accessible categories.
  7. Orders and simplices.
  8. Partially ordered sets.
  9. Standard squares.
  10. Nerves and Segal spaces.
  11. Enhanced categories.
  12. Reflexive families.
  13. Enhanced categories and functors.
  14. Extensions and restrictions.
  15. Examples.
  16. ...and 14 more sections

Key Result

Proposition 3.2

For any reflexive family $\mathcal{C} \to \mathcal{I}$, there exists a truncation functor cartesian over $\mathcal{I}$, such that for any category $\mathcal{E}$ and functor $\gamma:\mathcal{C} \to K(\mathcal{I},\mathcal{E})$ cartesian over $I$, we have a unique isomorphism $\gamma \cong K(\gamma_{\sf pt}) \circ k(\mathcal{C})$ that restricts to $\operatorname{\sf id}$ over ${\sf pt} \in

Theorems & Definitions (58)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • ...and 48 more