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Representable triangulated functors in terms of semiorthogonal decompositions

Jonas Frank, Mathias Schulze

TL;DR

This paper extends Bondal and Kapranov's representability results from cohomological functors to triangulated $\\mathsf{S}$-functors within an enriched, monoidal framework. It develops a setting with $\\mathsf{S}$ symmetric closed additive monoidal and $\\mathsf{S}$-triangulated categories, and proves that a triangulated $\\mathsf{S}$-functor $\\tau: \\mathcal{M} \\to \\mathcal{S}$ is $\\mathsf{S}$-representable whenever its restrictions to the components of a semiorthogonal decomposition $(\\mathcal{U}, \\mathcal{V})$ are $\\mathsf{S}$-representable. The proof glues component representations using the enriched Yoneda lemma and homotopy-cartesian arguments, with a variant under an additional SOD $(\\W, \\mathcal{U})$ simplifying certain steps (BK-alt-hyp). Conceptually, the work generalizes the classical BK89 lifting of representations to a setting where hom-objects live in an ambient triangulated category $\\mathsf{S}$, with potential applications to lifting Serre functors in derived categories.

Abstract

A theorem of Bondal and Kapranov lifts representations of cohomological functors from semiorthogonal decompositions of triangulated categories. We present a version of this result for triangulated functors. To this end, we introduce suitable terminology in enriched category theory and transfer the original proof.

Representable triangulated functors in terms of semiorthogonal decompositions

TL;DR

This paper extends Bondal and Kapranov's representability results from cohomological functors to triangulated -functors within an enriched, monoidal framework. It develops a setting with symmetric closed additive monoidal and -triangulated categories, and proves that a triangulated -functor is -representable whenever its restrictions to the components of a semiorthogonal decomposition are -representable. The proof glues component representations using the enriched Yoneda lemma and homotopy-cartesian arguments, with a variant under an additional SOD simplifying certain steps (BK-alt-hyp). Conceptually, the work generalizes the classical BK89 lifting of representations to a setting where hom-objects live in an ambient triangulated category , with potential applications to lifting Serre functors in derived categories.

Abstract

A theorem of Bondal and Kapranov lifts representations of cohomological functors from semiorthogonal decompositions of triangulated categories. We present a version of this result for triangulated functors. To this end, we introduce suitable terminology in enriched category theory and transfer the original proof.
Paper Structure (4 sections, 13 theorems, 11 equations)

This paper contains 4 sections, 13 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Monoidal and enriched categories
  3. Enriched representability and Yoneda lemma
  4. Bondal--Kapranov for triangulated functors

Key Result

Proposition 1

Let $\mathcal{M}$ be a triangulated category, linear over a field, with finite dimensional hom-vector spaces. Suppose that $(\mathcal{U}, \mathcal{V})$ is a semiorthogonal decomposition of $\mathcal{M}$. Then $\mathcal{M}$ admits a Serre functor if both $\mathcal{U}$ and $\mathcal{V}$ do so.

Theorems & Definitions (36)

  • Proposition : Bondal--Kapranov
  • Theorem
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 26 more