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Some properties of the A$_{\infty}$-nerve

Mattia Ornaghi

TL;DR

The paper proves that the ${A}_{ abla}$-nerve sends quasi-equivalences between strictly unital ${A}_{ abla}$-categories to weak equivalences in the Joyal model structure and shows that the nerve of a pretriangulated ${A}_{ abla}$-category is a stable $"infty$-category, with the induced homotopy-category functor being triangulated. It analyzes the relationship between ${A}_{ abla}$-categories and dg-categories via the DK adjunction, and clarifies how pretriangulated envelopes and idempotent completions behave under the nerve construction. The work also discusses model-categorical aspects, including Morita-type considerations and fibrancy in the field case, and provides explicit constructions (homotopy pullbacks/pushouts) to extend results to general commutative rings. Overall, it connects algebraic ${A}_{ abla}$-enhancements with stable $"infty$-categories, clarifying when nerve constructions preserve triangulated and idempotent-complete structures, and raises questions about monoidal compatibility of the nerve functor.

Abstract

The aim of this paper is to prove that the A$_{\infty}$-nerve of two quasi-equivalent A$_{\infty}$-categories (linear over a commutative ring) are weak-equivalent in the Joyal model structure. As a consequence we prove that the A$_{\infty}$-nerve of a pretriangulated A$_{\infty}$-category is a stable $\infty$-category.

Some properties of the A$_{\infty}$-nerve

TL;DR

The paper proves that the -nerve sends quasi-equivalences between strictly unital -categories to weak equivalences in the Joyal model structure and shows that the nerve of a pretriangulated -category is a stable -category, with the induced homotopy-category functor being triangulated. It analyzes the relationship between -categories and dg-categories via the DK adjunction, and clarifies how pretriangulated envelopes and idempotent completions behave under the nerve construction. The work also discusses model-categorical aspects, including Morita-type considerations and fibrancy in the field case, and provides explicit constructions (homotopy pullbacks/pushouts) to extend results to general commutative rings. Overall, it connects algebraic -enhancements with stable -categories, clarifying when nerve constructions preserve triangulated and idempotent-complete structures, and raises questions about monoidal compatibility of the nerve functor.

Abstract

The aim of this paper is to prove that the A-nerve of two quasi-equivalent A-categories (linear over a commutative ring) are weak-equivalent in the Joyal model structure. As a consequence we prove that the A-nerve of a pretriangulated A-category is a stable -category.

Paper Structure

This paper contains 14 sections, 17 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. $\hbox{A}_{\infty}$-modules, quasi-equivalences and pretriangulated $\hbox{A}_{\infty}$-categories
  3. Brief background on $\hbox{A}_{\infty}$-categories
  4. Quasi-equivalences between $\hbox{A}_{\infty}$-categories
  5. Pretriangulated $\hbox{A}_{\infty}$-categories
  6. $\hbox{A}_{\infty}$-nerve
  7. Brief background on $\infty$-categories
  8. Stable $\infty$-categories
  9. $\hbox{A}_{\infty}$-nerve
  10. Properties of the $\hbox{A}_{\infty}$-nerves
  11. Simplicial Objects and DK-correspondence
  12. Model structures
  13. Main results
  14. Stable $\infty$-categories vs pretriangulated $\hbox{A}_{\infty}$-categories

Key Result

Theorem 1

The $\hbox{A}_{\infty}$-nerve sends quasi-equivalences of (strictly unital) $\hbox{A}_{\infty}$-categories in weak-equivalences of $\infty$-categories.

Theorems & Definitions (77)

  • Theorem : \ref{['M']}
  • Theorem : \ref{['Ttr']}
  • Definition 2.1: $\hbox{A}_\infty$-category
  • Example 2.1
  • Definition 2.2: Unit
  • Definition 2.3: Strictly unital $\hbox{A}_{\infty}$-functor
  • Definition 2.4: Homotopy category
  • Definition 2.5: Quasi-equivalence
  • Example 2.2
  • Theorem 2.1: COS, COS2
  • ...and 67 more