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On orthogonal factorization systems and double categories

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Pith's one-line read This paper proves that the ∞-category of orthogonal factorization systems is equivalent to the ∞-category of factorization double categories, via an explicit pair of functors Fact and Cnr, and derives an (un)straightening equivalence for…

desk verdict A credible, genuinely new dictionary between factorization systems and double ∞-categories, with one load-bearing extension claim that needs proof before acceptance. read the letter →

arxiv 2501.01363 v1 pith:4QEZFCHV submitted 2025-01-02 math.CT math.AT

Branko Juran This is my paper · ORCID
classification math.CTmath.AT MSC 18N4518N40
keywords orthogonalfactorizationsystemsdouble∞-categoriesSegalspacesfibrationsstraighteningspancategoryadequatetriples
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a full and faithful embedding of orthogonal factorization systems into double $\infty$-categories, and proves the embedding is an equivalence onto the subcategory of 'factorization double categories'—those double categories in which every bottom-left corner has a unique square filler. This makes precise the classical intuition that double categories generalize orthogonal factorization systems: the 'wrong order' composition of an egressive and an ingressive morphism is uniquely rewritten in the correct order, which is exactly the lifting property of a factorization system. The author also proves an (un)straightening equivalence for double $\infty$-categories, showing that certain fibrations over a double category are classified by maps into a specific double category, and recovers the span category construction of adequate factorization systems as the horizontal opposite.

What carries the argument

The load-bearing object is the functor $\mathrm{Fact}$ (Construction 3.3), defined by restricting the Yoneda embedding along the bicosimplicial object $([m],[n]) \mapsto [m] \bar{\times} [n]$, where the factorization system on $[m] \times [n]$ has egressive morphisms those constant in the second factor and ingressive those constant in the first. The inverse functor $\mathrm{Cnr}$ (Construction 3.14) sends a factorization double category to its 'category of corners': the complete Segal space $\mathrm{cnr}(C)$ with $C(-,0)$ and $C(0,-)$ as the two subcategories. The equivalence is proved by showing the unit (Proposition 3.18) and counit (Proposition 3.16) are natural equivalences; the counit uses the Segal and factorization conditions, while the unit relies on the equivalence $\mathrm{map}_{\mathrm{OFS}}(\mathrm{Ar}([n]), C^{\dagger}) \simeq \mathrm{map}_{\mathrm{Cat}_1}([n], C)$ imported from [HHLN23b, Prop. 4.8]. The cartesian condition of Proposition 3.1—that $C(1,1)$ is the pullback of $C(1,0)$ and $C(0,1)$ over $C(0,0)$—characterizes the essential image.

What would settle it

A direct computation of the unit map for a non-adequate factorization system would settle the matter: if for some system $C^\dagger$ and some $n \ge 1$ the map $\mathrm{map}_{\mathrm{OFS}}(\mathrm{Ar}([n]), C^\dagger) \to \mathrm{map}_{\mathrm{Cat}_1}([n], C)$ is not a homotopy equivalence, then Proposition 3.18 fails and Theorem 3.19 is false. The paper gives no example, so testing any non-adequate system, such as a category admitting an ambigressive square that is not a pullback, would resolve the gap.

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Extended reading notes

Core claim

The central claim is Theorem 3.19: the functor $\mathrm{Fact} : \mathrm{OFS} \to \mathrm{DCat}$, sending a factorization system $C^{\dagger} = (C, C_{\mathrm{eg}}, C_{\mathrm{in}})$ to the double $\infty$-category with horizontal morphisms the egressive maps and vertical morphisms the ingressive maps, induces an equivalence of $\infty$-categories $\mathrm{OFS} \simeq \mathrm{DCat}_{\mathrm{OF}}$, where $\mathrm{DCat}_{\mathrm{OF}}$ is the full subcategory of double $\infty$-categories satisfying the cartesian condition of Proposition 3.1—equivalently, double categories in which every 'corner' has a unique square filling it. The inverse $\mathrm{Cnr}$ sends a factorization double category to its category of corners, with the vertical and horizontal morphisms as the two classes. The paper further shows that curved orthofibrations and op-Gray fibrations over $C^{\dagger}$ correspond exactly to (cocart,right)- and (cart,right)-fibrations over $\mathrm{Fact}(C^{\dagger})$, and that the (un)straightening equivalence of Theorem 4.6 restricts to these classes. Finally, the automorphism group of the $\infty$-category of adequate factorization systems is $\mathbb{Z}/2\mathbb{Z}$, generated by the span category construction, which is shown to coincide with taking the horizontal opposite of $\mathrm{Fact}(C^{\dagger})$.

Load-bearing premise

The paper assumes, without proving it, that a certain comparison between mapping spaces that was previously proved for 'adequate' factorization systems also holds for all factorization systems; this step is what makes the inverse construction a true inverse, and if it fails the whole equivalence collapses.

Editorial extensions

If this is right

  • Every orthogonal factorization system is completely determined by its factorization double category, so the two theories are equivalent and factorization systems can be studied through the well-developed machinery of double $\infty$-categories.
  • Curved orthofibrations and op-Gray fibrations over a factorization system are exactly (cocart,right)- and (cart,right)-fibrations over the associated double category, so fibrations of double categories subsume the earlier notion.
  • The (un)straightening equivalence of Theorem 4.6 classifies (cocart,right)-fibrations over a double category by maps into the large double category $\mathrm{Sq}^{\mathrm{oplax}}(\mathrm{Cat}_1^{(2)})$, giving a uniform straightening statement.
  • For adequate factorization systems, the span category functor is the same as taking the horizontal opposite of the associated double category, and the only other automorphism is the identity, so there are no hidden symmetries of the adequate theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the unproved extension of [HHLN23b, Prop. 4.8] fails, Theorem 3.19 would reduce to a statement about adequate systems; the paper itself only needs the general statement for Theorem A, so checking that extension is the most direct test of the main result.
  • The corner-filling condition suggests a concrete recipe for building factorization systems from double categories: given any double $\infty$-category, take the subcategory of 'corners' that are uniquely fillable and see whether the two morphism classes are closed under composition.
  • The automorphism computation for adequate systems may generalize to other full subcategories of factorization systems defined by closure properties, where the horizontal-opposite operation would still provide an involution.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper constructs a functor Fact from the ∞-category of orthogonal factorization systems to the ∞-category of double ∞-categories, and proves (Theorem 3.19) that it restricts to an equivalence OFS ≃ DCat_OF, where DCat_OF is the full subcategory of factorization double categories satisfying the cartesian condition of Proposition 3.1. The paper then proves an (un)straightening equivalence for (cocart,right)-fibrations of double categories (Theorem 4.6), uses it to identify curved orthofibrations and op-Gray fibrations of factorization systems with (cocart,right)- and (cart,right)-fibrations over the associated double categories (Proposition 4.5), and finally computes the automorphism group of the ∞-category of adequate factorization systems (Theorem 5.5).

Significance. If the main theorem is correct, the paper provides a clean and elegant equivalence between orthogonal factorization systems and a natural class of double ∞-categories, making precise a long-standing intuition and giving a functorial embedding that respects the ∞-categorical structure. The straightening theorem for (cocart,right)-fibrations and its specialization to orthofibrations of factorization systems is a useful contribution, and the computation of the automorphism group of the category of adequate factorization systems is a nice application. The paper is well organized, builds on a substantial body of prior work, and offers several explicitly constructed functors that will likely be reusable. The main weakness is a load-bearing unproven extension of a cited result from [HHLN23b] beyond its stated hypotheses.

major comments (3)
  1. [Construction 3.17 and Proposition 3.18] The equivalence map_OFS(Ar([n]), C†) ≃ map_Cat1([n], C) is taken from [HHLN23b, Prop. 4.8], a result stated for orthogonal adequate triples. The paper extends it to all factorization systems by asserting that the proof 'makes no essential use of the adequate triple property', but no verification of this extension is provided. This equivalence is precisely the input that makes the natural transformation id =⇒ Cnr∘Fact an equivalence in Proposition 3.18, and it is therefore load-bearing for Theorem 3.19. If the extension fails for some non-adequate factorization system, the unit of the adjunction is not invertible and the main equivalence does not follow. Please supply a full proof that map_OFS(Ar([n]), C†) ≃ map_Cat1([n], C) holds for arbitrary factorization systems, or restrict the statement of Theorem 3.19 to the class for which the cited result is verified.
  2. [Lemma 3.12] Lemma 3.12 asserts that the saturation of the single morphism (2) agrees with the spine inclusions (3). This claim is essential for Proposition 3.13, which establishes that cnr(C) is a complete Segal space exactly when C is a factorization double category, and hence is needed for the construction of the inverse functor Cnr. The proof leaves the crucial combinatorial verification to the reader: 'We leave the details to the reader' for the retract direction, and later 'an explicit combinatorial argument shows ... We leave the details to the reader' for the pushout square. Since the claim is nontrivial and load-bearing, please include a detailed proof of the saturation equality or provide a reference that contains it.
  3. [Proposition 3.16] Proposition 3.16, which proves that the counit of the adjunction is an equivalence, is dispatched by reducing to the cases m,n ≤ 1, excluding (1,1), and then saying that 'in the remaining three cases, one verifies that the map is an equivalence by unwinding the definitions.' These cases are the entire content of the counit equivalence in Theorem 3.19, and at least the case (m,n) = (1,0) is used again in the proof of Proposition 3.18. Please spell out these cases explicitly, or give a conceptual argument that covers them uniformly.
minor comments (4)
  1. [Proposition 3.13] In the proof of Proposition 3.13, the sentence 'Similarly, one can show that it has a left inverse and so does f' appears garbled; presumably the intended statement is that g has a left inverse and so does f, or something analogous.
  2. [Definition 5.3] In Definition 5.3, the line 'We denote the full subcategory of adequate factorization double categories by DCat ⊥ ⊂ DCat⊥' seems to contain a typo; the intended inclusion is probably DCat⊥ ⊂ DCat.
  3. [References] The reference [Ště23] spells the author's name as 'Miloslac ˇStˇ ep´ an' in the bibliography; please check and correct the spelling.
  4. [Construction 3.17] In Construction 3.17, the phrase 'embedding [n] into Ar([n]) by sending each element to the identity arrow' would benefit from a precise description of the functor; presumably it sends i to the identity morphism id_i of i.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equivalence OFS ~= DCat_OF is built from definitions, direct constructions, and independent cited results, not from fitted inputs or self-citation.

full rationale

The paper's central claim, Theorem 3.19, is a structural equivalence theorem. The functor Fact is defined by restricted Yoneda from factorization systems to double categories (Construction 3.3), and the subcategory DCat_OF is characterized independently by the cartesian condition in Proposition 3.1. Lemma 3.4 proves that Fact lands in DCat_OF using the ordinary composition equivalence of Proposition 2.15, not by assuming the theorem. The inverse functor Cnr is constructed explicitly from mapping spaces out of arrow categories (Constructions 3.10 and 3.14), and the unit and counit are checked in Propositions 3.16 and 3.18. The one load-bearing citation, Construction 3.17's use of [HHLN23b, Prop. 4.8], is an external result by other authors; it is not a self-citation, and the paper explicitly states that the cited proof does not use the adequate-triple property. Whether that extension is correct is a proof-verification question, not circularity. Theorem B is proved directly by Kan-extension arguments and standard citations, and Theorem C's automorphism computation is an independent mapping-space argument; the identification of the nontrivial automorphism with Barwick's Span uses the independent result [HHLN23b, Thm. 4.12]. Some details are left to the reader, e.g. in Lemma 3.12 and parts of Proposition 3.16, but these are omitted combinatorial checks, not steps where a conclusion is identified with its own hypothesis. There is no fitting of parameters, no prediction that reduces to a definition, and no load-bearing self-referential chain. The derivation is self-contained in the sense required here; any unresolved cited extension would be a correctness gap, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on standard theorems in infinity-category theory, on the chosen model of double categories, and on one unverified extension of a cited result. The central claim is not circular.

assumptions (4)
  • ad hoc to paper The equivalence map_OFS(Ar([n]), C†) ~= map_Cat1([n], C) from [HHLN23b, Prop. 4.8] holds for all factorization systems, not just adequate triples.
    Stated in Construction 3.17 without proof. The paper says the cited proof makes no essential use of the adequate triple property, but this extension is load-bearing for the unit of the adjunction in Theorem 3.19.
  • domain assumption The model of double infinity-categories as bisimplicial spaces whose vertical and horizontal slices are complete Segal spaces, with a single object space, is the right setting for the comparison.
    Remark 2.6 notes this definition is not the direct higher analog of classical double categories; the paper follows [Nui24]. This choice is not derived but assumed.
  • standard math The Gray tensor product of [m] and [n] used in Construction 2.11 agrees with the one in [HHLN23a].
    Ensured by [HHLN23a, Prop. 5.1.9]; invoked to define the oplax square functor.
  • standard math Background theorems in infinity-category theory (Joyal-Tierney nerve, Rezk completeness, Lurie's HTT) are valid as stated.
    Used throughout Sections 2 and 4 without proof, following standard citations.

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Pith. "Pith review of On orthogonal factorization systems and double categories." pith.science (2026). https://pith.science/paper/4QEZFCHV

@misc{pith2026250101363,
  author       = {Pith},
  title        = {Pith review of: On orthogonal factorization systems and double categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QEZFCHV}},
  note         = {Machine review of arXiv:2501.01363}
}
abstract

We prove that the $\infty$-category of orthogonal factorization systems embeds fully faithfully into the $\infty$-category of double $\infty$-categories. Moreover, we prove an (un)straightening equivalence for double $\infty$-categories, which restricts to an (un)straightening equivalence for op-Gray fibrations and curved orthofibrations of orthogonal factorization systems.

Figures

Figures reproduced from arXiv: 2501.01363 by the authors.

Figure 1
Figure 1. Ar ([3]), the squares are labeled by the the smallest bicosimplicial subspace P i,j in which they are contained Again using that pushouts in PSh(∆ × ∆) are computed pointwise, one verifies that the following square is a pushout for all 1 ≤ j + 1 ≤ i ≤ n − 1: ([0] ⊠ [1]) ∪[0]⊠[0] ([1] ⊠ [0]) P i,j+1 [1] ⊠ [1] P i,j where the lower map is the map classifying the element in P i,j (1, 1) ⊂ Ar ([n]) (1, 1) = map([1] ⋆ [1… view at source ↗

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Forward citations

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Works this paper leans on

10 extracted references · 6 canonical work pages · cited by 1 Pith paper

  1. [3]

    Homotopy-coherent al gebra via Segal conditions

    arXiv: 2211.02576 [math.AT] . [CH21] Hongyi Chu and Rune Haugseng. “Homotopy-coherent al gebra via Segal conditions”. In: Advances in Mathematics 385 (2021), p. 107733. [Ehr63] Charles Ehresmann. “Cat´ egories doubles et cat´ eg ories structur´ ees”. In:C. R. Acad. Sci. Paris 256 (1963), pp. 1198–1201. [GKT18] Imma G´ alvez-Carrillo, Joachim Kock, and And...

    arXiv 2021

  2. [10]

    arXiv: 2305.06714 [math.AT] . 27

    work page Pith review arXiv

  3. [170]

    Princeton University Press, Princeton, NJ, 2009, pp

    Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009, pp. xviii +925. [Lur17] Jacob Lurie. Higher Algebra

    work page 2009

  4. [431]

    Con- temp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 277–

    work page 2007

  5. [2013]

    Lax monoidal adjunctions, two-variable fibrations and the calc ulus of mates

    [HHLN23a] Rune Haugseng, Fabian Hebestreit, Sil Linskens, and Joost Nuiten. “Lax monoidal adjunctions, two-variable fibrations and the calc ulus of mates”. In: Proc. Lond. Math. Soc. (3) 127.4 (2023), pp. 889–957. [HHLN23b] Rune Haugseng, Fabian Hebestreit, Sil Linskens, and Joost Nuiten. “Two- variable fibrations, factorisation systems and ∞-categories of ...

    work page 2023

  6. [2017]

    Duality for groups

    url: https://people.math.harvard.edu/~lurie/papers/HA.pdf. [Mac50] Saunders MacLane. “Duality for groups”. In: Bull. Amer. Math. Soc. 56 (1950), pp. 485–516. [Nui24] Joost Nuiten. “On straightening for Segal spaces”. In: Compositio Mathe- matica 160.3 (2024), pp. 586–656. [Rez01] Charles Rezk. “A model for the homotopy theory of hom otopy theory”. In: Tra...

    work page 1950

  7. [2020]

    Fibrations of $\infty$-categories

    arXiv: 1702.02681 [math.CT] . [AGH24] Fernando Abell´ an, Andrea Gagna, and Rune Haugseng . Straightening for lax transformations and adjunctions of (∞, 2)-categories

    work page Pith review arXiv

  8. [2022]

    [ˇStˇ e23] Miloslac ˇStˇ ep´ an.Factorization systems and double categories

    arXiv: 2207.09244 [math.AT] . [ˇStˇ e23] Miloslac ˇStˇ ep´ an.Factorization systems and double categories

    arXiv

Show all 10 references
  1. [2023]

    [Joy08] Andr´ e Joyal

    arXiv: 2312.09889 [math.CT] . [Joy08] Andr´ e Joyal. Notes on Quasi-categories

    arXiv

  2. [2024]

    Spectral Mackey functors and equiv ariant algebraic K- theory (I)

    arXiv: 2404.03971 [math.CT] . [Bar17] Clark Barwick. “Spectral Mackey functors and equiv ariant algebraic K- theory (I)”. In: Adv. Math. 304 (2017). 25 [Bri18] Pedro Boavida de Brito. “Segal objects and the Groth endieck construc- tion”. In: 708 (2018), pp. 19–44. [BS24] Shaul...

    2017 arXiv

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