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arxiv: 2606.02478 · v1 · pith:SRZQPJPYnew · submitted 2026-06-01 · 🧮 math.AT · math.CT

Regular clock map and trace space

Philippe Gaucher This is my paper

Pith reviewed 2026-06-28 11:36 UTC · model grok-4.3

classification 🧮 math.AT math.CT
keywords directed spacesclock mapsregular mapssmall-orthogonality classprecubical setstransverse setshomotopy equivalencetrace space
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The pith

Regular clock maps form a small-orthogonality class inside clock maps, so their category is locally presentable, and the path-to-trace quotient is always a homotopy equivalence.

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

The paper defines a regular clock map as a regular map from a saturated directed space to the directed circle. It proves that the full subcategory of regular clock maps is a small-orthogonality class in the category of all clock maps, which immediately implies local presentability. Geometric realizations of both precubical sets and transverse sets automatically produce regular clock maps. On the underlying directed space of any regular clock map the canonical map from directed paths to traces is a homotopy equivalence. These facts give a concrete, well-behaved class of maps to the circle inside the broader theory of directed spaces.

Core claim

A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. The category of regular clock maps is a small-orthogonality class of the category of clock maps and is therefore locally presentable. Every geometric realization of a precubical set or a transverse set yields a regular clock map. For any regular clock map the canonical quotient from directed paths to traces on the underlying directed space is a homotopy equivalence.

What carries the argument

The regular clock map, a regular map from a saturated directed space to the directed circle, which carries the orthogonality argument that produces local presentability and the homotopy equivalence of path-to-trace quotients.

If this is right

  • The category of regular clock maps is locally presentable.
  • Geometric realizations of precubical sets and transverse sets are regular clock maps.
  • The path-to-trace quotient is a homotopy equivalence on the underlying directed space of every regular clock map.
  • Small-orthogonality classes inside clock maps inherit the good categorical properties needed for further constructions in directed homotopy theory.

Where Pith is reading between the lines

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

  • The local presentability may allow the construction of model structures or adjunctions that are unavailable for arbitrary clock maps.
  • The homotopy equivalence property could be tested on concrete examples coming from cubical or simplicial models of directed circles.
  • One could ask whether the same orthogonality and homotopy conclusions hold when the target is replaced by other directed spaces with circle-like homotopy type.

Load-bearing premise

The notions of regular map, saturated directed space, and clock map are already well-defined on directed spaces, and geometric realizations of precubical and transverse sets satisfy the required regularity condition.

What would settle it

An explicit regular clock map for which the canonical map from directed paths to traces fails to be a homotopy equivalence, or a clock map that cannot be obtained by small-orthogonality from a regular one.

read the original abstract

A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. We prove that the category of regular clock maps is a small-orthogonality class of the category of clock maps. Hence it is locally presentable. Any geometric realization of precubical sets and of transverse sets gives rise to a regular clock map. Finally, we prove that for the underlying directed space of a regular clock map, the canonical quotient from directed paths to traces is always a homotopy equivalence.

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 paper defines a regular clock map as a regular map of directed spaces from a saturated directed space to the directed circle. It proves that the category of regular clock maps is a small-orthogonality class in the category of clock maps and is therefore locally presentable. It further shows that geometric realizations of precubical sets and transverse sets yield regular clock maps, and that the canonical quotient map from directed paths to traces is a homotopy equivalence on the underlying directed space of any regular clock map.

Significance. If the stated results hold, the work supplies a categorical characterization of regular clock maps that inherits local presentability from standard orthogonality-class arguments, connects these objects to the geometric realizations of two standard combinatorial models (precubical and transverse sets), and establishes a homotopy-equivalence property that may simplify computations involving trace spaces. The reliance on the general fact that small orthogonality classes in locally presentable categories remain locally presentable is a strength when the regularity and saturation hypotheses are verified by construction.

minor comments (2)
  1. The abstract states several theorems without indicating the sections or lemmas that contain their proofs; adding forward references to the relevant statements (e.g., “see Theorem 3.4”) would improve readability.
  2. The definition of “regular map” and “saturated directed space” is presupposed from the ambient literature; a short paragraph recalling the precise axioms used in this paper would make the manuscript more self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central claims are established via standard category-theoretic arguments: the small-orthogonality class property follows from the general theorem that small orthogonality classes in locally presentable categories are locally presentable, and the geometric realization and homotopy equivalence statements are derived directly from the given definitions of regularity, saturation, and clock maps without any reduction to fitted parameters, self-referential equations, or load-bearing self-citations. The derivation chain is self-contained against external benchmarks in category theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger records the background notions the claims rest upon and the new definition introduced.

axioms (2)
  • domain assumption The category of clock maps is equipped with the structure required for small-orthogonality classes to be well-defined.
    Invoked when the paper asserts that regular clock maps form such a class.
  • domain assumption Geometric realizations of precubical sets and transverse sets produce saturated directed spaces admitting regular clock maps.
    Stated directly in the abstract as a fact used to generate examples.
invented entities (1)
  • regular clock map no independent evidence
    purpose: A restricted class of clock maps whose category inherits local presentability and good homotopy properties.
    Newly defined in the paper; no independent evidence supplied beyond the definition itself.

pith-pipeline@v0.9.1-grok · 5590 in / 1435 out tokens · 36818 ms · 2026-06-28T11:36:46.693597+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

17 extracted references · 7 canonical work pages · 1 internal anchor

  1. [1]

    Adámek, H

    J. Adámek, H. Herrlich, and G. E. Strecker. Abstract and concrete categories: the joy of cats.Repr. Theory Appl. Categ., (17):1–507, 2006. Reprint of the 1990 original [Wiley, New York; MR1051419]

    2006

  2. [2]

    Adámek and J

    J. Adámek and J. Rosický.Locally presentable and accessible categories. Cambridge University Press, Cambridge, 1994.https://doi.org/10.1017/cbo9780511600579. 004

    work page doi:10.1017/cbo9780511600579 1994

  3. [3]

    Fahrenberg and M

    U. Fahrenberg and M. Raussen. Reparametrizations of continuous paths.J. Homotopy Relat. Struct., 2(2):93–117, 2007

    2007

  4. [4]

    Fajstrup, E

    L. Fajstrup, E. Goubault, E. Haucourt, S. Mimram, and M. Raussen.Directed algebraic topology and concurrency. With a foreword by Maurice Herlihy and a preface by Samuel Mimram. SpringerBriefs Appl. Sci. Technol. Springer, 2016. https://doi.org/10.1007/978-3-319-15398-8

    work page doi:10.1007/978-3-319-15398-8 2016

  5. [5]

    Fajstrup and J

    L. Fajstrup and J. Rosický. A convenient category for directed homotopy.Theory Appl. Categ., 21:7–20, 2008

    2008

  6. [6]

    P. Gaucher. Homotopical interpretation of globular complex by multipointed d-space. Theory Appl. Categ., 22(22):588–621, 2009

    2009

  7. [7]

    P. Gaucher. Left properness of flows.Theory Appl. Categ., 37(19):562–612, 2021

    2021

  8. [8]

    P. Gaucher. Directed degeneracy maps for precubical sets.Theory Appl. Categ., 41(7):194–237, 2024

    2024

  9. [9]

    P. Gaucher. Homotopy theory of Moore flows (III).North-West. Eur. J. Math., 10:55–113, 2024

    2024

  10. [10]

    P. Gaucher. Natural homotopy of multipointed d-spaces.Math. J. Okayama Univ., 68:13–62, 2026

    2026

  11. [11]

    P. Gaucher. Towards a theory of natural directed paths.Compositionality, 7(6), 2026. https://doi.org/10.46298/compositionality-7-6

    work page doi:10.46298/compositionality-7-6 2026

  12. [12]

    M. Grandis. Directed homotopy theory. I.Cah. Topol. Géom. Différ. Catég., 44(4):281– 316, 2003. 16

    2003

  13. [13]

    S. Henry. Minimal model structures, 2020. https://doi.org/10.48550/arXiv. 2011.13408

    work page internal anchor Pith review doi:10.48550/arxiv 2020

  14. [14]

    Hirschowitz, M

    A. Hirschowitz, M. Hirschowitz, and T. Hirschowitz. Saturating directed spaces. J. Homotopy Relat. Struct., 9(2):273–283, 2014. https://doi.org/10.1007/ s40062-013-0025-8

    2014

  15. [15]

    Krishnan

    S. Krishnan. A convenient category of locally preordered spaces.Appl. Categ. Structures, 17(5):445–466, 2009.https://doi.org/10.1007/s10485-008-9140-9

    work page doi:10.1007/s10485-008-9140-9 2009

  16. [16]

    M. Raussen. Trace spaces in a pre-cubical complex.Topology Appl., 156(9):1718–1728, 2009.https://doi.org/10.1016/j.topol.2009.02.003

    work page doi:10.1016/j.topol.2009.02.003 2009

  17. [17]

    Ziemiański

    K. Ziemiański. Categories of directed spaces.Fund. Math., 217(1):55–71, 2012. https://doi.org/10.4064/fm217-1-5. Université Paris Cité, CNRS, IRIF, F-75013, Paris, France URL:http://www.irif.fr/˜gaucher 17

    work page doi:10.4064/fm217-1-5 2012