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arxiv: 2607.01058 · v1 · pith:ACX4ZEPJnew · submitted 2026-07-01 · 🧮 math.RT · math.CT· math.RA

Large silting mutation in extriangulated categories

Diego Alberto Barcel\'o Nieves This is my paper

Pith reviewed 2026-07-02 02:46 UTC · model grok-4.3

classification 🧮 math.RT math.CTmath.RA
keywords silting mutationextriangulated categoriesn-cosilting complexestilting modulesmutation theoryrepresentation theory
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The pith

Silting mutation extends to infinite dimensions in extriangulated categories that admit all set-indexed products and coproducts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an infinite-dimensional analog of silting mutation inside extriangulated categories equipped with all set-indexed products and coproducts. It then uses this construction to define mutation operations on n-cosilting complexes over an arbitrary ring and on infinite-dimensional n-(co)tilting modules over rings of finite global dimension. The same construction is shown to recover an earlier notion of cosilting mutation. A reader would care because the result moves mutation techniques from the compact or finite-dimensional setting into the general module categories that arise in representation theory.

Core claim

The authors introduce large silting mutation in extriangulated categories with set-indexed (co)products and apply it to obtain a theory of mutation for n-cosilting complexes over an arbitrary ring, as well as for infinite-dimensional n-(co)tilting modules over a ring of finite global dimension. The former theory is also shown to reinterpret the cosilting mutation introduced in an earlier paper.

What carries the argument

Large silting mutation, the infinite-dimensional analog of silting mutation defined via set-indexed products and coproducts in an extriangulated category.

If this is right

  • A mutation theory for n-cosilting complexes over arbitrary rings is obtained.
  • A mutation theory for infinite-dimensional n-(co)tilting modules over rings of finite global dimension is obtained.
  • The new construction reinterprets an existing cosilting mutation theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow mutation to be studied in module categories without compactness or finite-generation hypotheses.
  • It suggests a route to unify cluster-tilting and tau-tilting mutation in unbounded or infinite-dimensional settings.
  • The same mechanism could be tested on derived categories whose objects are unbounded complexes.

Load-bearing premise

The extriangulated category admits all set-indexed products and coproducts.

What would settle it

An explicit extriangulated category with all set-indexed products and coproducts in which the defined large mutation of an n-cosilting complex fails to yield another n-cosilting complex.

read the original abstract

Silting mutation in triangulated categories, both at the level of objects and of subcategories, was introduced in arXiv:1009.3370, and later generalized to extriangulated categories in arXiv:2303.08125. It simultaneously encompasses the mutation theories of cluster-tilting objects in cluster theory and of compact 2-term silting complexes and support $\tau$-tilting modules in $\tau$-tilting theory. In this article, we develop an infinite-dimensional analog of silting mutation in extriangulated categories with set-indexed (co)products, which we then apply to obtain a theory of mutation for $n$-cosilting complexes over an arbitrary ring, as well as for infinite-dimensional $n$-(co)tilting modules over a ring of finite global dimension. The former theory is also shown to reinterpret the cosilting mutation introduced in arXiv:2201.02147.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops an infinite-dimensional analog of silting mutation for extriangulated categories that admit all set-indexed products and coproducts. The construction is applied to obtain mutation theories for n-cosilting complexes over arbitrary rings and for infinite-dimensional n-(co)tilting modules over rings of finite global dimension; it is also shown to reinterpret the cosilting mutation of arXiv:2201.02147.

Significance. If the central constructions hold, the work supplies a uniform framework that extends finite-dimensional silting-mutation results to large settings while preserving the unification of cluster-tilting and support au-tilting theories. The explicit hypothesis on set-indexed (co)products makes the scope of applicability transparent and directly satisfied in the target categories of complexes and modules.

minor comments (2)
  1. [Main construction] The statement of the main theorem (presumably in §3 or §4) should explicitly record the precise functoriality of the large mutation operation with respect to the extriangulated structure, to avoid any ambiguity when the construction is applied to the category of complexes.
  2. [Introduction] A short remark comparing the new large mutation with the finite case of arXiv:2303.08125 would help readers track which axioms are used only in the infinite setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the scope of the work on large silting mutation in extriangulated categories and its applications to n-cosilting complexes and infinite-dimensional (co)tilting modules.

Circularity Check

0 steps flagged

Minor self-citation for finite case; new infinite-dimensional construction independent

full rationale

The paper explicitly constructs large silting mutation in extriangulated categories admitting set-indexed (co)products, then applies it to n-cosilting complexes and infinite-dimensional modules. Prior finite-dimensional results are cited for context (arXiv:1009.3370, arXiv:2303.08125) and one existing theory is reinterpreted (arXiv:2201.02147), but these citations are not load-bearing for the central definitions or proofs. The hypothesis of set-indexed (co)products is stated upfront and directly verified in the target categories; no step reduces by definition or self-citation chain to its own inputs. This is the expected non-circular outcome for an extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the extriangulated category possesses all set-indexed (co)products; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The extriangulated category admits all set-indexed products and coproducts.
    This is the structural hypothesis required for the large mutation to be defined and for the applications to n-cosilting complexes and infinite-dimensional modules.

pith-pipeline@v0.9.1-grok · 5690 in / 1241 out tokens · 45036 ms · 2026-07-02T02:46:02.796849+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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