arxiv: 2605.05195 · v1 · submitted 2026-05-06 · 🧮 math.AT · math.CT

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Stable homotopy theory of higher categories

Hadrian Heine

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Pith reviewed 2026-05-08 15:55 UTC · model grok-4.3

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keywords stable homotopy theoryhigher categoriesBrown representabilitycategorical spectrahomological algebraderived categoryrig

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The pith

Inverting endomorphism categories creates a stable homotopy theory of higher categories in which higher categories act as spaces and categorical spectra represent their homology theories.

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

The paper demonstrates that inverting endomorphism categories extends the core principle of stable homotopy theory to higher categories. In this extended theory, higher categories take the place of spaces, and categorical spectra serve as the objects that represent homology theories on them. This is captured by a categorical Brown representability theorem. If the claim holds, it allows the construction of long exact sequences for categorical homology theories and the development of homological algebra in higher categorical settings, including the derived category of a rig. Readers would care as it bridges classical stable homotopy with the world of higher categories.

Core claim

Inverting endomorphism categories leads to a stable homotopy theory of higher categories, in which higher categories play the role of spaces and categorical spectra represent homology theories of higher categories. The main result is a categorical Brown representability theorem classifying categorical homology theories by categorical spectra. Classical stable homotopy theory is recovered by inverting morphisms. Stabilization is realized by spectrum objects, and the passage from unstable to stable is controlled within a stable range. Categorical homology theories give rise to long exact sequences and support a higher-categorical homological algebra, with an application to the derived category

What carries the argument

Inversion of endomorphism categories, which stabilizes the homotopy theory and enables categorical spectra to classify homology theories via a Brown representability theorem.

Load-bearing premise

That inverting endomorphism categories produces a setting where the classical stable homotopy principles, like Brown representability, hold for higher categories.

What would settle it

Finding a categorical homology theory on higher categories that cannot be represented by a categorical spectrum would falsify the Brown representability theorem.

read the original abstract

Stable homotopy theory is governed by the principle that after inverting loop spaces, homotopy types become the representing objects for homology theories. We show that this principle extends to higher category theory: inverting endomorphism categories leads to a stable homotopy theory of higher categories, in which higher categories play the role of spaces and categorical spectra represent homology theories of higher categories. Classical stable homotopy theory is recovered by inverting morphisms. While several fundamental features of classical stable homotopy theory persist in this setting, new phenomena arise from categorical dimension. In particular, stabilization is realized by spectrum objects, and the passage from unstable to stable homotopy theory is controlled within a stable range. Our main result is a categorical Brown representability theorem classifying categorical homology theories by categorical spectra. As a consequence, categorical homology theories give rise to long exact sequences and support a higher-categorical homological algebra. As an application, we construct the derived category of a rig, extending homological algebra beyond additive contexts.

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.

Desk Editor's Note private letter to a colleague

The abstract lays out a coherent extension of stable homotopy to higher categories via inverting endomorphism categories, yielding a categorical Brown representability theorem, but the absence of any proofs leaves the central claims uncheckable.

read the letter

The main point is that Heine inverts endomorphism categories instead of morphisms to produce a stable homotopy theory where higher categories act like spaces and categorical spectra represent their homology theories. Classical stable homotopy appears as the special case of inverting morphisms, and the setup is claimed to deliver long exact sequences plus homological algebra for rigs. The abstract presents this as recovering known features while adding new behavior from categorical dimension, including a stable range for stabilization via spectrum objects. The main result is the categorical Brown representability theorem that classifies the homology theories by these spectra. If the details work, the rig application would genuinely push homological algebra past additive categories. The construction is motivated directly from the classical principle, and the abstract keeps the analogy tight without obvious overreach. The serious limitation is that only the abstract is available. No definitions of the inversion functor, no construction of the spectrum objects, and no proof of the representability theorem can be examined, so it is impossible to check for gaps, circularity, or whether the stable range actually holds as stated. The soundness assessment stays low until the full arguments are visible. This is aimed at people already working in higher category theory and stable homotopy who want a framework for homology theories on non-additive structures. A reader who knows the classical Brown theorem and wants to see the categorical lift would get a clear outline of the idea and the claimed consequences. The paper deserves peer review because the claims are substantial enough that a referee should check whether the technical steps deliver what the abstract promises.

Referee Report

1 major / 0 minor

Summary. The paper extends the principle of stable homotopy theory to higher category theory by inverting endomorphism categories, yielding a stable homotopy theory in which higher categories play the role of spaces and categorical spectra represent homology theories of higher categories. Classical stable homotopy theory is recovered by inverting morphisms. New phenomena arise from categorical dimension, with stabilization realized by spectrum objects and the unstable-to-stable passage controlled in a stable range. The main result is a categorical Brown representability theorem classifying categorical homology theories by categorical spectra. Consequences include long exact sequences and higher-categorical homological algebra, with an application to the derived category of a rig.

Significance. If the constructions and the categorical Brown representability theorem hold, the work would provide a foundational generalization of stable homotopy theory to higher categories, enabling classification of homology theories via spectra and extending homological algebra to non-additive contexts. The recovery of classical theory as a special case and the discussion of dimension-dependent phenomena suggest a coherent framework with potential for further applications in categorical homotopy theory.

major comments (1)

Only the abstract is available for review; no proofs, derivations, technical constructions (such as the inversion of endomorphism categories or the definition of categorical spectra), or detailed arguments for the Brown representability theorem can be examined. This prevents verification of soundness, consistency with the paper's own equations, or identification of any gaps in the central claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary of the paper and for recognizing its potential to extend stable homotopy theory to higher categories. We address the single major comment below.

read point-by-point responses

Referee: Only the abstract is available for review; no proofs, derivations, technical constructions (such as the inversion of endomorphism categories or the definition of categorical spectra), or detailed arguments for the Brown representability theorem can be examined. This prevents verification of soundness, consistency with the paper's own equations, or identification of any gaps in the central claims.

Authors: The full manuscript, including the complete technical constructions for inverting endomorphism categories, the definition of categorical spectra, all derivations, and the detailed proof of the categorical Brown representability theorem, is available on arXiv as 2605.05195. The abstract served only as a summary for the initial overview; the submitted review package contains the entire paper with the relevant sections, equations, and arguments. We are prepared to supply any specific excerpts or additional explanations if a particular part of the argument requires closer inspection. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract outlines an extension of stable homotopy theory to higher categories via inversion of endomorphism categories, leading to a categorical Brown representability theorem. No equations, derivations, or self-citations are provided in the available text that would allow inspection for reductions by construction, fitted inputs renamed as predictions, or load-bearing self-referential steps. The main result is presented as a classification theorem with consequences for homological algebra, without evident self-definition or smuggling of ansatzes. Per the hard rules, absence of inspectable load-bearing steps that reduce to inputs requires a score of 0 and empty steps list.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim appears to rest on the domain assumption that inverting endomorphism categories yields a stable homotopy theory, together with the introduction of categorical spectra as representing objects. No explicit free parameters or invented entities with independent evidence are stated in the abstract.

axioms (1)

domain assumption Inverting endomorphism categories produces a stable homotopy theory of higher categories in which higher categories play the role of spaces
This is the governing principle stated in the abstract as the extension of classical stable homotopy theory.

invented entities (1)

categorical spectra no independent evidence
purpose: Represent homology theories of higher categories
Introduced in the abstract as the stable objects that classify categorical homology theories via the Brown representability theorem.

pith-pipeline@v0.9.0 · 5444 in / 1459 out tokens · 37020 ms · 2026-05-08T15:55:00.493349+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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