Metric Completions of Triangulated Categories

• Metric completions are defined as the process of adjoining limits of Cauchy sequences in a triangulated category equipped with a rapidly shrinking good metric.
• This framework preserves the triangulated structure by using filtered colimits and connects to universal localizations in representation theory and algebraic geometry.
• The lattice structure of good metrics enables systematic classification and computational insights into derived categories and thick subcategories.

A metric completion of a triangulated category is a process whereby a triangulated category, equipped with a suitable “good metric” (a rapidly shrinking sequence of full subcategories satisfying stability under extensions and shifts), is “completed” so as to adjoin limits of Cauchy sequences, producing a new triangulated category that encodes the corresponding analytic and algebraic closure properties. This notion—developed from ideas of Lawvere on enriched categories and further extended in the context of triangulated categories—now plays a central role in modern representation theory, algebraic geometry, and the theory of universal localizations. The paper of metric completions is particularly advanced for bounded derived categories of hereditary rings and algebras, where results show a deep interaction with the theory of universal localizations and the lattice theory of thick subcategories (Matoušek, 27 Aug 2025).

1. Good Metrics and Their Structure

A “good metric” on a triangulated category $\mathcal{S}$ is a decreasing sequence of full subcategories

$\mathcal{M} = \{B_n\}_{n \in \mathbb{N}}$

with the properties:

• $B_{n+1} \subset B_n$ for all $n$ (shrinking balls around zero);
• $B_n * B_n = B_n$ (closed under extensions);
• Each $B_n$ is often assumed additive (closed under direct summands);
• Stability under shifts: $\Sigma^{\pm 1} B_{n+1} \subset B_n$ (rapid shrinking property).

These conditions ensure that the process of “approximating” an object (via morphisms whose cones are in smaller and smaller $B_n$) mimics the analytic behavior of Cauchy sequences converging in a metric space.

Given a metric, one defines the “limit” of the metric as the intersection $\mathcal{B} = \bigcap_{n \in \mathbb{N}} B_n$, typically a thick subcategory. The metric is called additive if $B_n$ are closed under direct summands. This additive property ensures that $\mathcal{B}$ retains desirable categorical features such as being thick.

2. Metric Completions: Definition and Main Results

Given a triangulated category $\mathcal{S}$ with a good metric $\mathcal{M}$, the metric completion $\mathfrak{S}(\mathcal{S})$ is constructed as follows:

• One considers Cauchy sequences: sequences in $\mathcal{S}$ where the “error” between consecutive objects (defined via the mapping cone) eventually lies in arbitrarily small $B_n$.
• Each Cauchy sequence determines a filtered colimit in the module category $\mathrm{Mod}\text{-}\mathcal{S}$ or, when a good extension is present, in an ambient triangulated category (such as $\mathrm{D}(R)$).
• The completion $\mathfrak{S}(\mathcal{S})$ is then comprised of “compactly supported” objects obtained as such colimits, with further conditions that ensure compatibility with the triangulated structure (extensions, shifts, etc.).

One of the principal results is that $\mathfrak{S}(\mathcal{S})$ inherits a triangulated structure, with distinguished triangles given as colimits of Cauchy sequences of triangles. This result holds without the need for DG enhancement (Neeman, 2018, Neeman, 2019).

In the presence of a “good extension,” i.e., a fully faithful functor $F : \mathcal{S} \to \mathcal{J}$ into a triangulated category with coproducts such that colimits and homotopy colimits are compatible, there is a concrete description of the completion: objects are homotopy colimits of Cauchy sequences lying in the preimage under the Yoneda functor of compactly supported modules.

3. Universal Localisations and Classification of Completions

A key insight is the identification of metric completions with derived categories of universal localisations. For a hereditary commutative noetherian ring $R$ or a hereditary algebra of tame representation type, one may show that: $\mathfrak{S}'_{\mathcal{M}}(\derived^b(\mathrm{mod}\,R)) \simeq \derived^b(\mathrm{mod}\,R_{\mathcal{B}})$ whenever

• the metric $\mathcal{M}$ is of the form $\mathcal{M}_\infty \vee \mathcal{B}$ (the join of the $t$-structure metric and the metric determined by a thick subcategory), and
• $\mathcal{B}$ is countably generated (Matoušek, 27 Aug 2025).

The ring $R_{\mathcal{B}}$ is the universal localisation of $R$ associated to the thick subcategory $\mathcal{B}$ (for example, inverting a collection of maps or modules whose vanishing describes $\mathcal{B}$). When these conditions are not met—e.g., when $\mathcal{M}$ is not of this decomposable form or if $\mathcal{B}$ is not countably generated—the completion is a proper thick subcategory of the original bounded derived category.

Table: Classification of Metric Completions in the Hereditary Case

Metric structure	Completion $\mathfrak{S}'_{\mathcal{M}}(\mathcal{S})$	Other properties
$\mathcal{M}_\infty \vee \mathcal{B}$, $\mathcal{B}$ countably generated	$\derived^b(\mathrm{mod}\,R_{\mathcal{B}})$	New objects appear
Otherwise	$$\mathcal{B}^{\perp} \cap \mathfrak{T} \subset \mathcal{S}$$	Subcategory of $\mathcal{S}$

Here, $\mathcal{M}_\infty$ is the metric arising from the standard $t$-structure, $\mathcal{B}$ is the limiting thick subcategory, and $\mathfrak{T}$ indicates a “regular” or torsion-theoretic subcategory.

The result generalizes to tame hereditary algebras: the regular part (thick subcategory generated by the regular modules) plays an analogous role, and the completion is the bounded derived category of a finite dimensional universal localisation (Matoušek, 27 Aug 2025).

4. The Lattice of Metrics

The set $\mathscr{M}$ of good metrics (modulo a natural equivalence) on a triangulated category forms a lattice, with meet and join given by

$$\mathcal{M} \wedge \mathcal{N} = \{ A_n \cap C_n \}_{n \in \mathbb{N}}, \quad \mathcal{M} \vee \mathcal{N} = \{ \overline{A_n \cup C_n} \}_{n \in \mathbb{N}}$$

where the overline denotes extension-closure. This lattice structure governs how different choices of subcategories (metrics) interact to produce new completions, and underpins the decompositions—e.g., in the representation-finite (Dynkin) case, every metric factors as a join over metrics associated to indecomposables (Matoušek, 27 Aug 2025).

The existence of this lattice allows for systematic analysis of how metrics “combine” (for instance, metrics coming from $t$-structures or from specific thick subcategories) and how the associated completions behave.

5. Methodological Aspects

The metric completion process is built upon several categorical and homological arguments:

• Construction and manipulation of Cauchy sequences inside $\mathcal{S}$, often via “trivialisation” to avoid nonzero summands from $\mathcal{B}$;
• Use of Quillen’s small object argument and other approximation techniques to reduce to countably generated situations;
• Embedding $\mathcal{S}$ into larger triangulated categories (good extensions) to realize completions concretely;
• Analysis of support and Zariski topology to control when new objects enter the completion;
• Application of the lattice-theoretic framework to classify how completions decompose and when two completions are equivalent.

6. Concrete Examples and Applications

For $\mathcal{S} = \derived^b(\mathrm{mod}\,\mathbb{Z})$ and the constant metric given by the thick subcategory generated by $\mathbb{Z}/2\mathbb{Z}$, the sequence

$$\mathbb{Z} \xrightarrow{\cdot 2} \mathbb{Z} \xrightarrow{\cdot 2} \cdots$$

has as its completion the derived category of $\mathbb{Z}[1/2]$, i.e. $\derived^b(\mathrm{mod}\,\mathbb{Z}[1/2])$. Analogous phenomena occur for more general Dedekind domains, tame hereditary algebras, and with thick subcategories generated by regular or torsion modules.

Moreover, for the Kronecker quiver, Cauchy sequences of preprojectives whose cones are a fixed regular simple module yield, via universal localisation, a new derived category associated to the endomorphism algebra of the homotopy colimit of the sequence.

7. Broader Implications and Future Directions

The robust connection between metric completions of triangulated categories and universal localisation yields a powerful toolkit for understanding the landscape of thick subcategories and derived equivalences in representation theory as well as algebraic geometry. This approach helps classify when completions produce new triangulated categories (i.e., new “derived localisations”) or when they simply recover thick subcategories of the original category.

Further directions include a complete characterization of the “excellent metrics” for which the involutivity of the completion correspondence holds (Neeman, 14 May 2025), algorithmic and lattice-theoretic exploits for controlling families of metrics, and applications to the paper of enhancements, phantom categories, and singularity invariants.

The metric approach thus provides both a conceptual and computational framework, organizing a variety of phenomena across pure representation theory, the theory of localizations, and the algebraic geometry of schemes and their derived categories.