Bicategories of fractions revisited: towards small homs and canonical 2-cells

Dorette Pronk; Laura Scull

REVIEW 3 major objections 5 minor 13 references

Bicategories of fractions revisited: towards small homs and canonical 2-cells

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a bicategory of fractions can be constructed from conditions weaker than the classical calculus-of-fractions axioms, and that 2-cells in the resulting localization have canonical representatives when the inverted…

desk verdict A useful weakening of the bicalculus axioms with real new content, but two load-bearing coherence proofs are explicitly sketched rather than given, so the main theorem is not fully verified as written. read the letter →

arxiv 1908.01215 v3 pith:DAN3HUNJ submitted 2019-08-03 math.CT

Dorette Pronk , Laura Scull This is my paper · ORCID

classification math.CT MSC 18D0518E35

keywords bicategoriesoffractionscalculuslocalizationweaklyinitialsubclassessmallhom-categoriescanonicalrepresentativespseudopullbacksorbispaces

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

This paper claims that the standard calculus of fractions for bicategories, the tool for formally inverting a class of arrows, can be run from weaker axioms than the classical ones. In place of the requirement that the class of inverted arrows be closed under composition, the paper proposes condition [WB2]: given composable arrows in the class, one can precompose to get a composite that again lies in the class. With that weakening, a bicategory of fractions $B(W^{-1})$ can still be built from single spans and still satisfies the universal property of a localization (Theorem 3.6). The payoff is that one may invert a smaller generating subclass that is small over each object and yet get an equivalent localization, so the hom-categories can be small. The paper further establishes conditions—fullness and co-fullness of the inverted arrows—under which the equivalence classes defining 2-cells collapse to canonical representatives, making the universal map 2-fully faithful and horizontal composition tractable via pseudo pullbacks.

What carries the argument

The load-bearing object is a 2-cell diagram: a span together with an invertible left-hand 2-cell and a right-hand 2-cell, taken up to the paper's equivalence relation. The new axiom [WB2] is the mechanism that lets the construction work without composition-closure: whenever a composite of two W-arrows is needed, it supplies a further precomposing arrow so that the relevant composite returns to W; Notation 3.2 packages these choices and uses them to define composition, whiskering, and the associativity cells. A second mechanism is the notion of a weakly initial subclass (Definition 4.1): if W is weakly initial in a larger class V and satisfies [WB1] and [WB5], it automatically inherits the rest of the axioms and yields an equivalent localization. The third mechanism is the co-full (respectively co-fully faithful) property of an arrow: post-composition with such an arrow induces a full (respectively faithful) functor between hom-categories, which is exactly what makes representatives of 2-cells canonical.

What would settle it

Take a small finite bicategory and a class W satisfying [WB1]–[WB5] whose [WB2]-precompositions are nontrivial, and check by direct calculation whether every instance of the associativity pentagon from Appendix B commutes; if one instance fails, $B(W^{-1})$ as constructed is not a bicategory and Theorem 3.6 is false.

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Extended reading notes

Core claim

The paper's central claim is that the bicalculus-of-fractions conditions used in [5] are not necessary for localization. Theorem 3.6 proves that if W satisfies [WB1]–[WB5], with [WB2] replacing closure under composition by the requirement that each composable pair $v,w$ in W admit $u$ with $wvu\in W$, then spans $\xleftarrow{w} \xrightarrow{f}$ with $w\in W$ form a bicategory $B(W^{-1})$ whose canonical map $J_W\colon B\to B(W^{-1})$ sends W to internal equivalences and satisfies the universal property of the bicategory of fractions. Section 4 shows that the closure of W under composition and 2-isomorphism satisfies the original BF1–BF5 conditions, so $B(W^{-1})$ is biequivalent to the classical localization; moreover any weakly initial subclass satisfying [WB1] and [WB5] yields an equivalent bicategory of fractions. Sections 5–6 show that when the arrows of W are co-full, every 2-cell has a representative with any specified left-hand invertible 2-cell; when they are co-ff this representative is unique, the universal map is 2-full and 2-faithful, and pseudo pullbacks can be used as canonical left-hand cells to simplify horizontal composition.

Load-bearing premise

The whole construction stands or falls on whether the associativity pentagon for the new composition of spans commutes; the paper only sketches this verification in an appendix and leaves full details to the reader.

Editorial extensions

If this is right

Any class W satisfying [WB1]–[WB5] yields a bicategory of fractions $B(W^{-1})$; its closure under composition and 2-isomorphism satisfies the classical axioms, so the new localization is biequivalent to the classical one and the universal property holds for pseudo, lax, and oplax transformations.
If W is weakly initial in V and satisfies [WB1] and [WB5], the inclusion induces a biequivalence $B(W^{-1})\simeq B(V^{-1})$; when the weakly initial subclass is small over each object, the localization has small hom-categories.
When the arrows of W are co-ff, each 2-cell in the localization has exactly one representative with a prescribed left-hand 2-cell, and the universal map $J_W$ is 2-full and 2-faithful.
When pseudo pullbacks of W-cospans exist, W is pullback closed, and W-arrows are co-ff, every 2-cell has a unique pseudo-pullback representative, and horizontal composition is computed from the universal arrows of the pseudo pullbacks.
For orbigroupoids, essential equivalences satisfy the hypotheses, and the subclass of essential covering maps is locally small and weakly initial, so the localized orbifold bicategory has small hom-categories and canonical 2-cells.

Reading between the lines

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

The same weakly-initial-subclass transfer should apply to other localizations: whenever a large class of weak equivalences contains a small weakly initial subclass, the localization can be built from the small subclass and will be locally small, even if the whole class is not.
The co-full/co-ff conditions identify the exact point at which quotienting 2-cells by equivalence classes becomes unnecessary; this gives a general explanation for why 'faithful fractions' constructions can present a localization as a 2-category with no quotienting at all.
The pseudo-pullback representatives turn horizontal composition into a deterministic construction from universal properties, so hom-categories of such localizations should be computable in explicit examples beyond orbispaces, for instance whenever the inverted arrows admit pseudo pullbacks and are co-ff.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about this paper before reading it closely. First, the core ideas are genuinely good: weakening BF2 to WB2, the weakly-initial-subclass theorem (4.3), and the canonical-representative results in Section 5 are real advances that will make bicategories of fractions easier to construct and use, especially for orbispaces. Second, the central existence theorem 3.6 is not fully proved in the manuscript: the associativity coherence of the new composition is only sketched in Appendix B, and well-definedness of right whiskering is explicitly deferred in Appendix C.3. Those are not cosmetic gaps; they sit under the main construction. If either fails, B(W^{-1}) as defined is not a bicategory. I don't think they do fail—the sketches are plausible and the authors clearly know the machinery—but as written, the paper asks the reader to take a lot on faith in exactly the places where the risk concentrates.

The strengths are real. The weakened axioms WB1–WB5 are carefully chosen, and the quotient/closure equivalence with the classical Pronk conditions (Theorem 4.9) is a solid contribution. The weakly initial subclass idea is simple and useful for small hom-categories; Corollary 4.5 genuinely strengthens Roberts's local essential-smallness result. Section 5's co-ﬀ theorem (uniqueness of representatives with a given left-hand 2-cell, and 2-full/2-faithfulness of JW) is elegant and directly addresses the known awkwardness of equivalence classes of 2-cells. The pseudo-pullback simplifications in Section 6 are also well motivated, assuming the hypotheses hold; the orbispace application in Section 7 is honestly labeled as a sketch with proofs deferred, so I would not count it against the paper beyond noting it is not yet evidence.

The reader's conditional verdict is the right one. I agree with the stress-test note: the self-flagged gaps are real. Appendix B.4 divides the pentagon into eight regions and says each commutes by one of three propositions but leaves the details; Appendix C.3 explicitly says “interested reader is welcome to contact the authors.” A referee should press hard on these two points. That said, the surrounding arguments in Appendices A, C.1, and C.2 are detailed, and the main strategy—using Proposition 2.6 to mediate between choices—is coherent enough that I'd expect the gaps to be fillable rather than fatal.

Who should read this: anyone doing localizations of bicategories in stacks, orbifolds, or internal category theory. It earns a serious referee. My recommendation: send it out, but insist that the associativity pentagon and right-whiskering well-definedness be either completed or explicitly stated as assumptions, perhaps in a companion note.

Referee Report

3 major / 5 minor

Summary. The paper revisits the construction of bicategories of fractions. It introduces a weaker set of axioms [WB1]–[WB5] replacing Pronk's BF1–BF5, with [WB2] replacing closure of W under composition by a weaker condition that still allows composition of spans after precomposing with a suitable arrow. The authors construct a bicategory B(W^{-1}) whose 1-cells are single spans, define vertical and horizontal composition of equivalence classes of 2-cell diagrams, and state a universal property (Theorem 3.6). They also introduce weakly initial subclasses to obtain locally small hom-categories, prove an equivalence between the new construction and the classical bicategory of fractions for the closure of W under composition and invertible 2-cells, and study canonical representatives for 2-cells under full/co-full and fully faithful/co-fully faithful conditions. A final section sketches applications to orbispaces, with details deferred to a companion paper in preparation.

Significance. If fully correct, the paper makes a substantive contribution: it weakens the known axioms for a bicalculus of fractions, gives a pathway to locally small hom-categories via weakly initial subclasses, and provides canonical 2-cell representatives that simplify horizontal composition. The constructions and statements are original and do not reduce to cited results; the proofs are self-contained in style, and the paper explicitly builds on rather than assumes the main universal property. However, the central existence theorem 3.6 depends on associativity coherence (Appendix B.4) and on well-definedness of right whiskering (Appendix C.3), both of which are only sketched, with details explicitly left to the reader. These are load-bearing gaps, so the paper in its current form is not yet fully verified. The potential value of the results, especially for orbispaces, is high if the missing proofs are supplied.

major comments (3)

[Appendix B.4, Proposition B.4] The proof of the associativity coherence pentagon is only a sketch. The text states 'We will only sketch the proof for the associativity pentagon' and then divides the pentagon into eight regions, asserting that each commutes by one of Propositions A.1, B.1, and B.2, 'leaving the details for the reader.' A bicategory requires the full pentagon coherence for its associativity 2-cells, and Theorem 3.6 asserts that B(W^{-1}) is a bicategory. The hypotheses of Propositions B.1 and B.2 include specific membership conditions such as w1w2w3si in W, and the paper does not verify these for each region. The unit coherence laws are also dismissed as 'straight forward' without proof. As written, this is not a proof of the associativity coherence condition, and Theorem 3.6 is therefore not established.
[Appendix C.3, Proposition C.3] Right whiskering is an essential part of horizontal composition of 2-cells, and its well-definedness on equivalence classes is required for the composition operation in B(W^{-1}) to be defined. Proposition C.3 begins: 'We will sketch the proof of this result as the details get rather involved and don't necessarily illuminate the idea behind the proof. Any interested reader is welcome to contact the authors for further details.' The subsequent argument is a lengthy diagram chase but explicitly leaves out details. Since Theorem 3.6 depends on right whiskering being well-defined, this is a load-bearing gap. The paper's own text flags this as missing support, and the reader cannot verify the claim from the manuscript.
[Section 7, Theorems 7.1 and 7.3] The theorems about orbispaces state that the class of essential equivalences satisfies the necessary properties and that the resulting bicategory of fractions has small hom-categories, with 2-fully faithful universal maps and canonical pseudo-pullback representatives. The proofs are deferred to [6], which is 'in preparation'. While these results are not needed for the categorical core of the paper, they are billed as applications of the main results. If the authors intend to claim them, they should either include the proofs or explicitly mark the statements as conditional on the forthcoming paper.

minor comments (5)

[Abstract and Introduction] There is a typo in the abstract: 'adresses' should be 'addresses'.
[Theorem 4.3] The statement says 'a class W⊆ V which is initial in V', but Definition 4.1 defines 'weakly initial'. This should be corrected for consistency.
[Figures and diagrams] Several diagrams, especially in Appendix C (e.g. diagrams (62) and (63)), are extremely dense and difficult to read. Splitting the pasting into smaller steps or adding labels to the arrows would improve verifiability.
[Notation 3.2, [C1]] The notation wu,v is introduced for the arrow chosen via [WB2], but the superscript/subscript convention becomes hard to track in later sections; a summary table of the chosen arrows and their W-membership properties would help.
[Section 7] The statement that the class C of essential covering maps is 'locally small' would benefit from a precise definition of what smallness means here (e.g., small over each object), since this is a main selling point of the orbifold application.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is self-contained, and the sketch-level proofs in the appendices are completeness gaps, not circular imports.

full rationale

The paper's central theorem constructs B(W^{-1}) explicitly from spans and equivalence classes of 2-cell diagrams, and then verifies the bicategory axioms. The new conditions [WB1]-[WB5] are not defined in terms of the conclusion; [WB2] is a deliberate weakening of the earlier composition-closure axiom BF2, and the construction is adapted to this weakening rather than assuming the old condition. There are no fitted parameters, so nothing is fitted and then renamed as a prediction. Where prior work is cited, it is used as foundation or comparison, not as the source of the new derivation: [5] supplies the original bicalculus framework and the standard universal-property argument, [12] supplies choice-independence of 2-cell representatives, and [9] supplies the WISC/local-smallness context. These are not ways of making the present theorem true by definition. Section 4's comparison between B(W^{-1}) and B(hat W^{-1}) is an additional equivalence theorem proved from the universal property, not an input to the construction. The genuine weakness is proof completeness: Appendix B.4 says 'We will only sketch the proof for the associativity pentagon' and Appendix C.3 says 'We will sketch the proof of this result as the details get rather involved and don't necessarily illuminate the idea.' Those are correctness risks that could invalidate Theorem 3.6 if the omitted coherence verifications fail, but they are not circularity: the construction is not secretly assumed in its own hypotheses, and no conclusion is imported through a self-citation chain. Accordingly the circularity score is 0.

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

The central framework is a pure existence/construction proof; it introduces no fitted parameters and no invented entities. The main external dependencies are standard bicategory theory and the prior construction of bicategories of fractions by Pronk [5], plus unproved orbispace properties deferred to [6].

assumptions (4)

standard math Bicategorical foundations: all results take place in the standard theory of bicategories, homomorphisms, and transformations.
The paper presupposes the usual axioms of bicategories and the calculus of mates; no new foundational system is introduced.
domain assumption Choice for the constructions [C1]-[C7] in Notation 3.2.
The definition of B(W^{-1}) requires choosing an arrow for each [WB2] pair, a square for each [WB3] cospan, and a lifting for each [WB4] cell. The paper assumes these choices can be made, which is a standard use of choice in category theory.
domain assumption The universal property of the original bicategory of fractions [5, Theorem 21] transfers to the new construction.
In Theorem 3.6 the proof says 'the remainder of the proof goes as in [5]' and 'the proof of [5, Theorem 21] does not depend on any specific properties of the choices made.' The paper relies on this cited theorem rather than re-proving it.
domain assumption Properties of essential equivalences and essential covering maps for etale groupoids, cited from [2,4] and deferred to [6].
Section 7 asserts that essential equivalences are fully faithful and co-fully-faithful, pullback closed, and that essential covering maps satisfy WB1-WB5 and are weakly initial. Some of these facts are cited to prior work; several are stated without proof with details in [6] in preparation.

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Pith. "Pith review of Bicategories of fractions revisited: towards small homs and canonical 2-cells." pith.science (2026). https://pith.science/paper/DAN3HUNJ

@misc{pith2026190801215,
  author       = {Pith},
  title        = {Pith review of: Bicategories of fractions revisited: towards small homs and canonical 2-cells},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAN3HUNJ}},
  note         = {Machine review of arXiv:1908.01215}
}

read the original abstract

This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is weaker than the one given in Pronk, Etendues and stacks as bicategories of fractions, but still allows a bicalculus of fractions. These conditions allow us to invert a smaller collection of arrows so that in some cases we may obtain a bicategory of fractions with small hom-categories. We adapt the construction of the bicategory of fractions to work with the weaker conditions. The second issue is the difficulty in dealing with 2-cells, which are defined by equivalence classes. We discuss conditions under which there are canonical representatives for 2-cells, and how pasting of 2-cells can be simplified in the presence of certain pseudo pullbacks. We also discuss how both of these improvements apply in the category of orbispaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[6]

Pronk, L

D.A. Pronk, L. Scull, Mapping groupoids for orbifolds, in preparation

work page
[1]

Abbad, E

O. Abbad, E. Vitale, Faithful calculus of fractions. Cah. Top. G´ eom. Diﬀ. Cat´ eg.54 (2013), pp. 22–239

work page 2013
[2]

Vesta Coufal, Dorette Pronk, Carmen Rovi, Laura Scull, Courtney Thatcher, Orbispaces via Groupoids: A Categorical Approach, in Contemporary Mathematics 641 (2015), Women in Topology: Collaborations in Homotopy Theory , editors: Maria Basterra, Kristine Bauer, Kathryn Hess, Brenda Johnson, pp. 135–166

work page 2015
[3]

Gabriel, M

P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory , Springer Verlag, New York, 1967

work page 1967
[4]

Moerdijk, D.A

I. Moerdijk, D.A. Pronk, Orbifolds, groupoids and sheaves, K-theory 12 (1997), p. 3–21

work page 1997
[5]

Pronk, Etendues and stacks as bicategories of fractions, Compositio Math., 102 (1996), pp

D.A. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math., 102 (1996), pp. 243–303

work page 1996
[7]

David Michael Roberts, The elementary construction of formal anafunctors, arXiv preprint, 1808.04552

work page Pith review arXiv
[8]

David Michael Roberts, On certain 2-categories admitting localisation by bicategories of fractions, Applied Categorical Structures 24 (2016), pp. 373–384

work page 2016

Show all 13 references

[9]

David Michael Roberts, Internal categories, anafunctors and localisations, Theory and Ap- plications of Categories, 26 (2012), pp. 788–829

2012
[10]

Satake, On a generalization of the notion of manifold, Proc

I. Satake, On a generalization of the notion of manifold, Proc. of the Nat. Acad. of Sc. U.S.A., 42 (1956), pp. 359–363

1956
[11]

Matteo Tommasini, private communication
[12]

Matteo Tommasini, Some insights on bicategories of fractions: representations and composi- tions of 2-morphisms, Theory and Applications of Categories , 31 (2016), pp. 257–329

2016
[13]

" t2 "" t1 // vβ′ ⇐ vw1

Matteo Tommasini, Right saturations and induced pseudofunctors between bicategories of fractions, Journal of Pure and Applied Algebra , 222 (2018), pp. 241–290. BICATEGORIES OF FRACTIONS REVISITED 49 Appendix Appendix A Associativity Part I: Associativity 2-cells The goal of t...

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Pronk, L

D.A. Pronk, L. Scull, Mapping groupoids for orbifolds, in preparation

work page

[2] [1]

Abbad, E

O. Abbad, E. Vitale, Faithful calculus of fractions. Cah. Top. G´ eom. Diﬀ. Cat´ eg.54 (2013), pp. 22–239

work page 2013

[3] [2]

Vesta Coufal, Dorette Pronk, Carmen Rovi, Laura Scull, Courtney Thatcher, Orbispaces via Groupoids: A Categorical Approach, in Contemporary Mathematics 641 (2015), Women in Topology: Collaborations in Homotopy Theory , editors: Maria Basterra, Kristine Bauer, Kathryn Hess, Brenda Johnson, pp. 135–166

work page 2015

[4] [3]

Gabriel, M

P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory , Springer Verlag, New York, 1967

work page 1967

[5] [4]

Moerdijk, D.A

I. Moerdijk, D.A. Pronk, Orbifolds, groupoids and sheaves, K-theory 12 (1997), p. 3–21

work page 1997

[6] [5]

Pronk, Etendues and stacks as bicategories of fractions, Compositio Math., 102 (1996), pp

D.A. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math., 102 (1996), pp. 243–303

work page 1996

[7] [7]

David Michael Roberts, The elementary construction of formal anafunctors, arXiv preprint, 1808.04552

work page Pith review arXiv

[8] [8]

David Michael Roberts, On certain 2-categories admitting localisation by bicategories of fractions, Applied Categorical Structures 24 (2016), pp. 373–384

work page 2016