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Excellent metrics on triangulated categories, and the involutivity of the map taking $\mathcal{S}$ to $\mathfrak{S}({\mathcal{S})^{\mathrm{op}}}$

Amnon Neeman

TL;DR

The article advances the theory of metrics on triangulated categories by isolating the class of excellent metrics and analyzing how these metrics behave under the S-construction and Yoneda functors. It proves that excellence is compatible with the involutive passage to opposites and to the S-construction, yielding equivalences between L- and S- completions and establishing that strong-triangle structures are preserved under the relevant functors. The work also clarifies how good extensions interact with very good and excellent metrics, showing that such extensions preserve triangulated structure and enable two-step periodic behavior. A broad supply of examples is provided, especially from bounded-above t-structures, demonstrating that very good (and thus excellent) metrics are abundant in natural settings like derived categories, schemes, and spectra. The results lay groundwork for proving uniqueness of enhancements in a sequel, highlighting substantial structural control over enhancements via metric-exact categorical frameworks.

Abstract

In the article arXiv:1806.06471 we defined good metrics on triangulated categories, and then studied the construction, that began with a triangulated category $\mathcal{S}$ together with a good metric $\{\mathcal{M}_i,\,i\in\mathbb{N}\}$, and out of it cooked up another triangulated category $\mathfrak(\mathcal{S})$. We went on to study examples, and produced many for which the construction is involutive. By this we mean that, if you let $\mathcal{T}=\mathfrak{S}(\mathcal{S})^{\mathrm{op}}$, then there is a choice of metric on $\mathcal{T}$ for which $\mathcal{S}=\mathfrak{S}(\mathcal{T})^{\mathrm{op}}$. In this article we study this phenomenon much more carefully, with the focus being on understanding the metrics for which involutivity occurs. As it turns out there is a large class of them, the excellent metrics on triangulated categories. At the end we will produce a few new examples of excellent metrics. And our reason for going to all this trouble is that the results of this article will permit us to prove new and surprising statements about uniqueness of enhancements. Those results will come in a sequel to this article, which is joint with Canonaco and Stellari.

Excellent metrics on triangulated categories, and the involutivity of the map taking $\mathcal{S}$ to $\mathfrak{S}({\mathcal{S})^{\mathrm{op}}}$

TL;DR

The article advances the theory of metrics on triangulated categories by isolating the class of excellent metrics and analyzing how these metrics behave under the S-construction and Yoneda functors. It proves that excellence is compatible with the involutive passage to opposites and to the S-construction, yielding equivalences between L- and S- completions and establishing that strong-triangle structures are preserved under the relevant functors. The work also clarifies how good extensions interact with very good and excellent metrics, showing that such extensions preserve triangulated structure and enable two-step periodic behavior. A broad supply of examples is provided, especially from bounded-above t-structures, demonstrating that very good (and thus excellent) metrics are abundant in natural settings like derived categories, schemes, and spectra. The results lay groundwork for proving uniqueness of enhancements in a sequel, highlighting substantial structural control over enhancements via metric-exact categorical frameworks.

Abstract

In the article arXiv:1806.06471 we defined good metrics on triangulated categories, and then studied the construction, that began with a triangulated category together with a good metric , and out of it cooked up another triangulated category . We went on to study examples, and produced many for which the construction is involutive. By this we mean that, if you let , then there is a choice of metric on for which . In this article we study this phenomenon much more carefully, with the focus being on understanding the metrics for which involutivity occurs. As it turns out there is a large class of them, the excellent metrics on triangulated categories. At the end we will produce a few new examples of excellent metrics. And our reason for going to all this trouble is that the results of this article will permit us to prove new and surprising statements about uniqueness of enhancements. Those results will come in a sequel to this article, which is joint with Canonaco and Stellari.
Paper Structure (9 sections, 58 theorems, 98 equations)

This paper contains 9 sections, 58 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The metrics on $\mathfrak{L}({\mathscr S})$ and $\mathfrak{S}({\mathscr S})$
  3. Excellent metrics and the equivalence $\widehat{Y}:\mathfrak{L}({\mathscr S})^{\hbox{\rm\tiny op}}\longrightarrow\mathfrak{L}[\mathfrak{S}({\mathscr S})^{\hbox{\rm\tiny op}}]$
  4. Strong triangles in $\mathfrak{L}({\mathscr S})^{\hbox{\rm\tiny op}}\cong\mathfrak{L}[\mathfrak{S}({\mathscr S})^{\hbox{\rm\tiny op}}]$
  5. The excellence of $\mathfrak{S}({\mathscr S})^{\hbox{\rm\tiny op}}$, and the inclusion ${\mathscr S}^{\hbox{\rm\tiny op}}\subset\mathfrak{S}[\mathfrak{S}({\mathscr S})^{\hbox{\rm\tiny op}}]$
  6. Good extensions of very good metrics
  7. Good extensions of excellent metrics
  8. An abundant source of examples of very good metrics
  9. The situation in which the metrics of Section \ref{['S95']} are excellent

Key Result

Proposition 7

Let ${\mathscr S}$ be a triangulated category, and suppose we are given on ${\mathscr S}$ a good metric $\{{\mathscr M}_i\subset{\mathscr S},\, i\in{\mathbb N}\}$. Then in Definition D3.-3 we introduce a string of subcategories $\{{\mathscr N}_i\subset\mathfrak{S}({\mathscr S}),\,i\in{\mathbb N}\}$,

Theorems & Definitions (154)

  • Remark 4
  • Remark 6
  • Proposition 7
  • Theorem 9
  • Remark 13
  • Lemma 14
  • Remark 15
  • Theorem 16
  • Example 17
  • Example 23
  • ...and 144 more