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Enhancement for categories and homotopical algebra

D. Kaledin

TL;DR

This work proposes an enhanced-category framework inspired by Grothendieck's derivator program to formulate abstract homotopy theory in a model-independent way. By organizing enhancements over partially ordered sets via Grothendieck fibrations, it delivers an axiomatization of homotopy-theoretic data that preserves ordinary category-theoretic reasoning while accounting for higher coherence through semicartesian and epivalence phenomena. The text develops foundational tools—relative functor categories, Kan extensions, cylinders, and Brown representability for families of groupoids—leading to a robust theory of enhanced categories Cat^h and a Brown-representability-based universal construction that connects with complete Segal spaces. It also surveys poset-based homotopy analogues (left-closed embeddings, anodyne maps, barycentric subdivision) to instantiate the derivator-like approach in a combinatorial setting. Overall, the theory aims to provide a flexible, coherent, model-free alternative to ∞-categorical frameworks while enabling practical applications to derived-category enhancements and related homotopical algebraic structures.

Abstract

We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual potential applications, such as e.g. enhancements for derived categories of coherent sheaves, in a way that is as close as possible to usual category theory.

Enhancement for categories and homotopical algebra

TL;DR

This work proposes an enhanced-category framework inspired by Grothendieck's derivator program to formulate abstract homotopy theory in a model-independent way. By organizing enhancements over partially ordered sets via Grothendieck fibrations, it delivers an axiomatization of homotopy-theoretic data that preserves ordinary category-theoretic reasoning while accounting for higher coherence through semicartesian and epivalence phenomena. The text develops foundational tools—relative functor categories, Kan extensions, cylinders, and Brown representability for families of groupoids—leading to a robust theory of enhanced categories Cat^h and a Brown-representability-based universal construction that connects with complete Segal spaces. It also surveys poset-based homotopy analogues (left-closed embeddings, anodyne maps, barycentric subdivision) to instantiate the derivator-like approach in a combinatorial setting. Overall, the theory aims to provide a flexible, coherent, model-free alternative to ∞-categorical frameworks while enabling practical applications to derived-category enhancements and related homotopical algebraic structures.

Abstract

We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual potential applications, such as e.g. enhancements for derived categories of coherent sheaves, in a way that is as close as possible to usual category theory.
Paper Structure (160 sections, 483 theorems, 659 equations)

This paper contains 160 sections, 483 theorems, 659 equations.

Table of Contents

  1. Category theory.
  2. Categories and functors.
  3. Categories.
  4. Functors.
  5. Commutative diagrams.
  6. Categories of functors.
  7. Adjunction and limits.
  8. Generalities on adjunction.
  9. Defining an adjunction.
  10. Admissible subcategories.
  11. Cylinders and comma-categories.
  12. Limits and colimits.
  13. Additive categories.
  14. Fibrations and cofibrations.
  15. The Grothendieck construction.
  16. ...and 145 more sections

Key Result

Lemma 1.1.2.3

Assume given an epivalence $\gamma:{\cal C} \to {\cal C}'$ and a Karoubi-dense full subcategory ${\cal C}'_0 \subset {\cal C}'$. Then ${\cal C}_0=\gamma^{-1}({\cal C}'_0) \subset {\cal C}$ is Karoubi-dense.

Theorems & Definitions (1041)

  • Remark 1
  • Example 1.1.1.1
  • Definition 1.1.1.2
  • Example 1.1.1.3
  • Definition 1.1.1.4
  • Example 1.1.2.1
  • Definition 1.1.2.2
  • Lemma 1.1.2.3
  • Lemma 1.1.2.4
  • Lemma 1.1.3.1
  • ...and 1031 more