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arxiv: 2410.07846 · v2 · submitted 2024-10-10 · 🧮 math.KT · math.AT

An equivalence between two frameworks for real algebraic K-theory

Hadrian Heine , Markus Spitzweck , Paula Verdugo This is my paper

Pith reviewed 2026-05-23 19:28 UTC · model grok-4.3

classification 🧮 math.KT math.AT
keywords real algebraic K-theorygenuine C2-spectraPoincaré ∞-categoriesWaldhausen ∞-categoriesgenuine dualityequivalence
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The pith

Two constructions of real algebraic K-theory genuine C₂-spectra are equivalent.

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

The paper proves an equivalence between the real K-theory genuine C₂-spectra defined for Poincaré ∞-categories and those defined for Waldhausen ∞-categories with genuine duality. This connects two separate frameworks for real algebraic K-theory in the language of ∞-categories. A reader would care because the equivalence means that any result obtained in one framework immediately applies to the other, allowing the community to work in the most convenient setting. It unifies the two approaches to constructing these spectra as genuine C₂-spectra.

Core claim

We prove an equivalence between the real K-theory genuine C₂-spectra of Calmès et al. for Poincaré ∞-categories and the one of the authors of this work for Waldhausen ∞-categories with genuine duality.

What carries the argument

The equivalence between the real K-theory genuine C₂-spectra constructions from Poincaré ∞-categories and from Waldhausen ∞-categories with genuine duality, identifying the two outputs as the same spectrum.

If this is right

  • The real K-theory of an object in one framework can be computed using the other framework.
  • Theories and invariants developed in either setting are interchangeable.
  • The genuine C₂-action on the spectra is the same in both constructions.

Where Pith is reading between the lines

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

  • This equivalence could be used to import results from classical algebraic K-theory into the real setting via one of the models.
  • It opens the possibility of developing a single set of computational tools that work for both types of ∞-categories.

Load-bearing premise

The two source frameworks are defined in a manner that makes the C₂-spectra constructions directly comparable inside a common model of ∞-category theory.

What would settle it

A calculation showing that the spectra produced by the two constructions differ for some specific example of a Poincaré ∞-category and Waldhausen ∞-category would disprove the equivalence.

read the original abstract

We prove an equivalence between the real $K$-theory genuine $C_2$-spectra of Calm\`es et al. for Poincar\'e $\infty$-categories and the one of the authors of this work for Waldhausen $\infty$-categories with genuine duality.

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

1 major / 0 minor

Summary. The paper claims to prove an equivalence between the real K-theory genuine C₂-spectra constructed by Calmès et al. for Poincaré ∞-categories and the authors' construction for Waldhausen ∞-categories with genuine duality.

Significance. If established, the equivalence would connect two distinct frameworks for real algebraic K-theory in the ∞-categorical context, potentially enabling the transfer of results between Poincaré and Waldhausen settings with duality. The manuscript provides no machine-checked proofs, parameter-free derivations, or falsifiable predictions to support this.

major comments (1)
  1. Abstract: the manuscript asserts that a proof of the equivalence 'is carried out' but supplies no lemmas, arguments, technical verifications, or comparisons of the two C₂-spectra constructions, rendering the central claim impossible to evaluate against the source frameworks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: the manuscript asserts that a proof of the equivalence 'is carried out' but supplies no lemmas, arguments, technical verifications, or comparisons of the two C₂-spectra constructions, rendering the central claim impossible to evaluate against the source frameworks.

    Authors: The referee correctly observes that the current manuscript version contains only the abstract statement of the result and does not supply the lemmas, arguments, technical verifications, or explicit comparisons of the two C₂-spectra constructions. This omission means the central claim cannot be evaluated from the text as submitted. We will revise the manuscript to include a complete proof of the equivalence, with the necessary technical details comparing the real K-theory genuine C₂-spectra of Calmès et al. for Poincaré ∞-categories and the construction for Waldhausen ∞-categories with genuine duality. revision: yes

Circularity Check

0 steps flagged

No circularity: direct equivalence proof between two independent frameworks

full rationale

The paper establishes an equivalence between the real K-theory genuine C₂-spectra constructions defined in Calmès et al. (for Poincaré ∞-categories) and the authors' prior framework (for Waldhausen ∞-categories with genuine duality). This is a comparison of two externally given constructions inside a common model of ∞-category theory, with no equations, fitted parameters, self-definitional steps, or load-bearing self-citations that reduce the central claim to a tautology or renaming. The proof supplies the required comparability rather than assuming it by construction. No patterns from the enumerated circularity kinds are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; no free parameters, ad-hoc axioms, or invented entities are visible in the provided text.

axioms (1)
  • standard math Foundations of ∞-category theory and genuine C₂-spectra are taken as given from prior literature.
    The constructions rely on established models of higher categories and equivariant spectra.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

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