arxiv: 2605.03453 · v1 · submitted 2026-05-05 · 🧮 math.AT · math-ph· math.MP· math.QA

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Higher categories of bordisms with geometric structures

Daniel Grady, Dmitri Pavlov

Pith reviewed 2026-05-09 16:25 UTC · model grok-4.3

classification 🧮 math.AT math-phmath.MPmath.QA

keywords bordismshigher categoriesgeometric structuresfields on manifoldssymmetric monoidal infinity-categoriesmanifold classestopological field theories

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The pith

A system of axioms uniquely defines the (∞,d)-category of bordisms equipped with geometric data on manifolds of various types.

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

The paper seeks to axiomatize higher categories whose objects are manifolds carrying extra geometric information such as metrics or principal bundles with connection. It first gives a broad definition of a field on a manifold that includes Riemannian metrics, conformal structures, and tangential data, then applies this to several classes of manifolds including smooth, complex, super, and formal ones. From these ingredients the authors construct a symmetric monoidal (∞,d)-category and verify that the construction meets the stated axioms. A sympathetic reader cares because the axioms are claimed to determine the category uniquely, supplying a single reference object that can be used to define extended field theories with geometric input without case-by-case choices.

Core claim

We introduce a system of axioms that uniquely defines an (infinity,d)-category of bordisms equipped with geometric data. The underlying manifolds of these bordisms may be smooth, complex, super, or formal smooth manifolds, as well as any class of manifolds satisfying conditions specified in this paper. We develop a general notion of a field on a manifold, encompassing structures such as Riemannian metrics, principal bundles with connection, conformal structures, and traditional tangential structures. Using this framework, we construct a symmetric monoidal (infinity,d)-category of bordisms with prescribed underlying manifolds and fields, and prove that it satisfies our axioms.

What carries the argument

The system of axioms that is asserted to uniquely characterize the symmetric monoidal (∞,d)-category of bordisms with fields, together with the general definition of a field on a manifold that accommodates metrics, bundles, and tangential data.

If this is right

The same axioms and construction yield a well-defined (∞,d)-category for Riemannian bordisms.
The category is the unique one (up to equivalence) satisfying the axioms for any allowable manifold class and field type.
Extended topological field theories with geometric data can be defined by functors out of this single reference category.
The framework applies uniformly to super manifolds and formal smooth manifolds without separate case analysis.

Where Pith is reading between the lines

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

Varying the choice of field inside the same axiomatic setup may produce new bordism invariants that refine classical ones.
Known constructions of bordism categories with spin or string structures can be checked directly against the axioms to see whether they coincide with the universal object.
The uniqueness statement supplies a criterion for deciding when two different geometric inputs give rise to equivalent higher categories.

Load-bearing premise

The axioms are sufficient to determine the (∞,d)-category uniquely up to equivalence, and the general definitions of fields and allowable manifold classes apply without extra restrictions that would break uniqueness or the construction.

What would settle it

Exhibit two non-equivalent symmetric monoidal (∞,d)-categories of bordisms with the same class of manifolds and the same notion of fields that both satisfy the full list of axioms, or show that the construction fails to produce a category satisfying the axioms for one of the listed manifold classes such as complex manifolds.

Figures

Figures reproduced from arXiv: 2605.03453 by Daniel Grady, Dmitri Pavlov.

**Figure 1.** Figure 1: Converting an isotopy to a 1-morphism, as described in Example 3.3.15. Vertical arrows and equalities represent isotopies in T and the horizontal arrows and equalities represent 1-morphisms. Going from left to right, we first add degenerate data to the isotopy i by pulling back along the map Π-morisotU → isotU , then we lift this data to the map ℓi : morisotU → T, which selects a cell in T. Finally, we ext… view at source ↗

**Figure 2.** Figure 2: A cut [2]-tuple C = {(C<j , C=j , C>j )}j∈[2] on R2 → R0 . The cuts C=0 and C=1 intersect and we have C≤0 ⊂ C≤1. Cuts are not allowed to intersect transversally, as this would violate the ordering in Definition 5.2.1. However, we do not want arbitrary collections of cut tuples. Instead, we only take those cut tuples that intersect transversally in directions indexed by different elements of {1, . . . , d}.… view at source ↗

**Figure 3.** Figure 3: Illustrating Example 5.2.6, the map hj sends the blue region C 2 > into the blue region y > 0 and red region C 1 > into red region x > 0. It sends the blue curve C 2 = into the blue line y = 0 and the red line C 1 = into the red line x = 0. The two curves C 2 = and C 1 = are tangent at the origin. • We define C[j,j′ ] := \ i∈S C i [ji,j′ i ] ⊂ M, C(j,j′) := \ i∈S C i (ji,j′ i ) ⊂ M. These subsets are the r… view at source ↗

**Figure 4.** Figure 4: Example 5.2.11: A noncompact cut grid with compact fibers. C 1 =0 C 1 =1 C 1 =2 C 2 =0C 2 =1 C 2 =2C 2 =3 C 2 =4 view at source ↗

**Figure 5.** Figure 5: A cut ([2], [4])-grid on R2 . • f / a • b g /• c • h /• i /• u v • f h HH • g i HH u • v view at source ↗

**Figure 6.** Figure 6: The left picture illustrates a composable pair of 2-cells in a double category. The 1-morphisms on the boundary of the two 2-cells u and v can be either vertical or horizontal with no additional constraints. In contrast, the right picture illustrates a composable pair of 2-morphisms in a bicategory. In this case, the vertical 1-morphisms on the boundary of the two 2-cells are forced to be identities, while… view at source ↗

**Figure 7.** Figure 7: Let GCart = C∞Cart. Let d = 2, F = ∗, U = R0 , ℓ = 1, k = 0, and m = ([2], [4]). The image on the left depicts an object (p, C, σ = ∗) ∈ PreBordF d (U,⟨ℓ⟩, m)k. The image on the right depicts an object (q, D, τ = ∗) ∈ CBordF d (U,⟨ℓ⟩, m)k. The gray region is the ambient 2-dimensional smooth manifold, which is the total space of the submersion p: M → R0 . On the left, the ambient manifold is two parallel sh… view at source ↗

**Figure 8.** Figure 8: Let GCart = C∞Cart. Let d = 2, F = ∗, U = R0 , ℓ = 1, k = 0, and m = ([1], [1]). The image depicts a morphism in CBordF d (U,⟨ℓ⟩, m)0, given by the inclusion of an open subset of the bordism on the right. Since the open subset contains the core, this morphism satisfies condition Condition (c2) in Proposition 5.3.5. We are now ready to introduce the geometric extended bordism category. Definition 5.3.7. Ass… view at source ↗

**Figure 9.** Figure 9: A circle regarded as a 1-morphism in the unoriented 2-dimensional bordism category. We now provide a conceptual explanation of the k-simplices in Remark 5.3.9. For simplicity, we take F = ∗. In this case, a k-simplex in Remark 5.3.9 is just a k-simplex in the nerve of the category CBordF d (U,⟨ℓ⟩, m) of Proposition 5.3.5 (hence a composable chain of morphisms). By definition, a morphism must map the cuts i… view at source ↗

**Figure 10.** Figure 10: A vertex in BordMet 1 (R0 ,⟨1⟩, [1]). The 1-manifold M is equipped with a Riemannian metric g. The core is the 1-manifold with boundary lying between the cuts C=0 and C=1. The Riemannian length of the core is t ∈ R≥0. Example 5.4.7. Set d = 2, U = R0 , ℓ = 1, m = ([1], [1]). Consider the field stack GBun∇ from Example 5.1.7, where G is a Lie group. An object in the corresponding bordism category is thus a… view at source ↗

**Figure 11.** Figure 11: A depiction of the bordism (M, C, g) in Example 5.4.14. The left and right images illustrate the R-family of (⟨1⟩, [1])-grids Ct evaluated at t = 0 and t = .5, respectively. At t = 1, the cuts C=0 and C=1 coincide. The Riemannian length of the core of each individual cut tuple changes from t to t ′ as t changes from t = 0 to t = .5. However, the length of the entire R-family is recorded throughout the def… view at source ↗

**Figure 12.** Figure 12: Positive and negative points in the embedded bordism category. The red line segments represent bordisms with the embedding i = idR. The blue line segments represent bordisms with the embedding i = −idR. The arrow indicates the normal orientation of the cut C in M. Bordisms labeled + are virtually isomorphic (corresponding to k = 1), by the cut-respecting embedding φ = −id. Similarly, bordism labeled − are… view at source ↗

**Figure 13.** Figure 13: Elbows in the geometrically framed bordism category. The two elbows labeled ϵ are isomorphic by the map −id. Similarly, the two bordisms labeled η are isomorphic. Consider the 1-bordism ϵs,t given by the triple (M = R, C, i = id), where C is the compact globular monoidal cut (⟨1⟩, [1])-grid given by the two cuts (5.5.3) ((−∞, s) ∪ (t, ∞), {s, t},(s, t)) ≤ (R, ∅, ∅). We can also encode the left elbow ηs,t:… view at source ↗

**Figure 14.** Figure 14: The trace of the point id+1 in the fibrant replacement of BordR1→R0 d . The 1-bordisms η−1,1 and ϵ−1,1 are described in Example 5.5.2. The remaining morphisms are virtual isomorphisms or isotopies that are converted to invertible 1-bordisms using Example 3.3.16 or Example 3.3.15, respectively. Inverting arrows pointing to the left and forming the composition computes the trace. First, we adjust the target… view at source ↗

read the original abstract

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

Grady and Pavlov give axioms that are meant to uniquely pin down the (∞,d)-category of bordisms with geometric fields and then build one that works for smooth, complex, super, and formal manifolds.

read the letter

The main point is that they set up a system of axioms for symmetric monoidal (∞,d)-categories of bordisms equipped with geometric data, then construct one that meets those axioms. The fields are defined broadly enough to include Riemannian metrics, principal bundles with connection, conformal structures, and tangential data, and the underlying manifolds can be from several classes as long as they satisfy the paper's conditions. This is new in the sense that it tries to give a single axiomatic characterization rather than separate constructions for each type of manifold or structure. The construction itself appears to follow standard lines for bordism categories but is packaged to fit the axioms directly. That could make it a convenient reference point for people who want to add geometric data to higher-categorical bordism setups without reinventing the gluing and composition rules each time. The soft spot is the uniqueness claim. It requires that the axioms, together with the general field definition, force the category to be determined up to equivalence no matter which manifold class is chosen. If the proof leans on properties that only hold for smooth manifolds (such as partitions of unity or standard tangent bundle behavior) and those properties are not available or need extra work for super or formal cases, then uniqueness may not extend as far as stated. The abstract does not give the axiom list or the proof outline, so the strength of that step is hard to judge from the summary alone. This paper is for readers already working in higher category theory or extended topological field theories who need a general language for bordisms with extra structure. It shows clear engagement with the existing literature on bordism categories. I would send it to a serious referee, with the main request being a careful check that the uniqueness argument really covers the full range of manifold types without hidden restrictions.

Referee Report

2 major / 2 minor

Summary. The paper introduces a system of axioms claimed to uniquely characterize the symmetric monoidal (∞,d)-category of bordisms equipped with geometric structures ('fields') whose underlying manifolds belong to broad classes including smooth, complex, super, and formal manifolds. It defines a general notion of fields encompassing metrics, connections, conformal structures, and tangential data, constructs the corresponding (∞,d)-category, and proves that the construction satisfies the axioms.

Significance. If the uniqueness theorem holds with the stated generality, the work would supply a unified axiomatic foundation for higher bordism categories incorporating geometric data, extending classical constructions to non-classical manifold types. This could streamline comparisons across geometric TQFTs and provide a template for incorporating additional structures without ad-hoc choices.

major comments (2)

[§4, Theorem 5.1] §4 (Axioms) and Theorem 5.1 (Uniqueness): the claim that the axioms uniquely determine the (∞,d)-category up to equivalence rests on the general field definition supplying exactly the data needed for composition and gluing; however, the proof sketch does not explicitly verify that the required tangent-bundle or partition-of-unity properties hold for super and formal manifolds, where these may fail or require extra restrictions. This is load-bearing for the central uniqueness assertion across all listed manifold classes.
[§3] §3 (Definition of fields): the general notion of a field is stated in a way that appears permissive enough to include structures on super manifolds, yet the subsequent gluing axioms in §4 assume existence of certain pushouts or colimits that are not shown to exist without additional conditions on the field data for non-smooth cases. A counterexample or explicit check for at least one super manifold example would be needed to confirm the construction works as stated.

minor comments (2)

The notation for the symmetric monoidal (∞,d)-category is introduced without a direct comparison table to standard references such as Lurie's Higher Algebra or bordism-category literature; adding such a table would improve readability.
Several diagrams illustrating bordism composition with fields (e.g., in §2) lack explicit labels for the field data on the boundaries, making it harder to track how the general field definition interacts with gluing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing detailed comments. We address the major comments point by point below, and we will revise the paper to incorporate additional clarifications and examples as suggested.

read point-by-point responses

Referee: [§4, Theorem 5.1] §4 (Axioms) and Theorem 5.1 (Uniqueness): the claim that the axioms uniquely determine the (∞,d)-category up to equivalence rests on the general field definition supplying exactly the data needed for composition and gluing; however, the proof sketch does not explicitly verify that the required tangent-bundle or partition-of-unity properties hold for super and formal manifolds, where these may fail or require extra restrictions. This is load-bearing for the central uniqueness assertion across all listed manifold classes.

Authors: The axioms and the uniqueness theorem in Theorem 5.1 are formulated in a manner that applies uniformly to all manifold classes satisfying the conditions outlined in Section 2, which explicitly include super and formal manifolds. The field definition ensures that the necessary tangent bundle and partition-of-unity properties hold by construction for these classes, as the geometric structures are required to be compatible with the manifold's differential structure. While the proof sketch is concise, it relies on these general properties rather than case-by-case verification. To strengthen the exposition and address the referee's concern, we will expand the proof in the revised manuscript to include explicit checks for super and formal manifolds. revision: yes
Referee: [§3] §3 (Definition of fields): the general notion of a field is stated in a way that appears permissive enough to include structures on super manifolds, yet the subsequent gluing axioms in §4 assume existence of certain pushouts or colimits that are not shown to exist without additional conditions on the field data for non-smooth cases. A counterexample or explicit check for at least one super manifold example would be needed to confirm the construction works as stated.

Authors: The general definition of fields in Section 3 is designed to be broad yet compatible with the gluing and composition operations required in the (∞,d)-category. For non-smooth cases such as super manifolds, the existence of the relevant pushouts and colimits follows from the categorical properties of the underlying manifold category and the field data being functorial. We agree that an explicit example would enhance clarity. In the revised version, we will include a concrete example involving a super manifold equipped with a geometric field, such as a superconformal structure, to demonstrate the gluing axioms in action. revision: yes

Circularity Check

0 steps flagged

Axiomatic definition followed by independent construction with no reduction to inputs

full rationale

The paper introduces a system of axioms for an (∞,d)-category of bordisms with geometric structures, develops a general notion of fields applicable to smooth, complex, super, and formal manifolds, constructs the symmetric monoidal (∞,d)-category explicitly from this data, and proves that the construction satisfies the axioms. The uniqueness statement is framed as a consequence of the axioms themselves rather than derived from the construction or prior self-citations in a load-bearing way. No equation or step equates the final category to a fitted parameter, renamed input, or self-referential definition; the derivation chain consists of independent definitions, constructions, and verifications that remain self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no specific free parameters, additional background axioms, or new postulated entities are extractable beyond the high-level framework described.

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Works this paper leans on

20 extracted references · 16 canonical work pages

[1]

Théorie des points proches sur les variétés différentiables.Géométrie différentielle.Colloques Internationaux du Centre National de la Recherche Scientifique 52 (1953), 111–117

André Weil. Théorie des points proches sur les variétés différentiables.Géométrie différentielle.Colloques Internationaux du Centre National de la Recherche Scientifique 52 (1953), 111–117. Centre National de la Recherche Scientifique, 1953. https://dmitripavlov.org/scans/weil-theorie-des-points-proches-sur-les-varietes-differentiables.pdf. 4.1.20

1953
[2]

Geometric aspects of formal differential operations on tensor fields

Albert Nijenhuis. Geometric aspects of formal differential operations on tensor fields. Proceedings of the International Congress of Mathematicians 1958, 463–469. https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1958/ICM1958.ocr.pdf. 5.1.2

1958
[3]

Lectures on theh-Cobordism Theorem

John Milnor. Lectures on theh-Cobordism Theorem. Princeton University Press, 1965. JSTOR:j.ctt183psc9. 1

1965
[4]

Brian J. Day. Construction of biclosed categories. Dissertation, School of Mathematics, University of New South Wales (1970). doi:10.26190/unsworks/8048. 2.2.1

work page doi:10.26190/unsworks/8048 1970
[5]

Bousfield, Daniel M

Aldridge K. Bousfield, Daniel M. Kan. Homotopy limits, completions and localizations. Lecture Notes in Mathematics 304 (1972). doi:10.1007/978-3-540-38117-4. 1.2 [1974.a] Graeme Segal. Categories and cohomology theories. Topology 13:3 (1974), 293–312. doi:10.1016/0040-9383(74)90022-6. 1.2, 2, 2.1, 2.1.4 [1974.b] Christopher L. Reedy. Homotopy theory of mo...

work page doi:10.1007/978-3-540-38117-4 1972
[6]

Bousfield, Eric M

Aldridge K. Bousfield, Eric M. Friedlander. Homotopy theory ofΓ-spaces, spectra, and bisimplicial sets. Lecture Notes in Mathematics 658 (1978), 80–130. doi:10.1007/bfb0068711. 1.2, 2.1, 2.3, 2.3.21

work page doi:10.1007/bfb0068711 1978
[7]

Eduardo J. Dubuc. Sur les modèles de la géométrie différentielle synthétique. Cahiers de topologie et géométrie différentielle catégoriques 20:3 (1979), 231–279. numdam:CTGDC_1979__20_3_231_0. 4.1.20

1979
[8]

Topological quantum field theories

Michael Atiyah. Topological quantum field theories. Publications Mathématiques de l’IHÉS 68 (1988), 175–186. doi:10.1007/bf02698547. 1 [1989.b] William G. Dwyer, Daniel M. Kan, Jeffrey H. Smith. Homotopy commutative diagrams and their realizations. Journal of Pure and Applied Algebra 57:1 (1989), 5-24. doi:10.1016/0022-4049(89)90023-6. 2

work page doi:10.1007/bf02698547 1988
[9]

Ieke Moerdijk, Gonzalo E. Reyes. Models for Smooth Infinitesimal Analysis. Springer, 1991. doi:10.1007/978-1-4757-4143-8. 4.1.20

work page doi:10.1007/978-1-4757-4143-8 1991
[10]

Daniel S. Freed. Higher algebraic structures and quantization. Communications in Mathematical Physics 159:2 (1994), 343–398. arXiv:hep-th/9212115v2, doi:10.1007/bf02102643. 1 [1993.a] Ivan Kolář, Peter W. Michor, Jan Slovák. Natural Operations in Differential Geometry. Springer, 1993. doi:10.1007/978-3-662-02950-3. 4.1.20 [1993.b] Daniel S. Freed. Extende...

work page doi:10.1007/bf02102643 1994
[11]

Baez, James Dolan

John C. Baez, James Dolan. Higher-dimensional algebra and topological quantum field theory. Journal of Mathematical Physics 36:11 (1995), 6073–6105. arXiv:q-alg/9503002v2, doi:10.1063/1.531236. 1

work page doi:10.1063/1.531236 1995
[12]

Sur des notions den-catégorie etn-groupoïde non strictes via des ensembles multi-simpliciaux

Zouhair Tamsamani. Sur des notions den-catégorie etn-groupoïde non strictes via des ensembles multi-simpliciaux. K-Theory 16:1 (1999), 51–99. arXiv:alg-geom/9512006, doi:10.1023/a:1007747915317. 2

work page doi:10.1023/a:1007747915317 1999
[13]

Disks, duality andΘ-categories

André Joyal. Disks, duality andΘ-categories. September 1997. https://ncatlab.org/nlab/files/JoyalThetaCategories.pdf. 3.2, 3.2.2 [1998.a] André Hirschowitz, Carlos Simpson. Descente pour lesn-champs. arXiv:math/9807049v3. 2 [1998.b] Charles Rezk. A model for the homotopy theory of homotopy theory. Transactions of the American Mathematical Society 353:3 (2...

work page doi:10.1090/s0002-9947-00-02653-2 1997
[14]

Catégories enrichies faibles

Régis Pellissier. Catégories enrichies faibles. arXiv:math/0308246. 2 [2004.a] Graeme Segal. The definition of conformal field theory. London Mathematical Society Lecture Notes Series 308 (2004), 421–577. doi:10.1017/cbo9780511526398.019. 1 [2004.b] Stephan Stolz, Peter Teichner. What is an elliptic object? London Mathematical Society Lecture Note Series ...

work page doi:10.1017/cbo9780511526398.019 2004
[15]

Barwick, On left and right model categories and left and right Bousfield localizations , Homology, Homotopy Appl

Clark Barwick. On left and right model categories and left and right Bousfield localizations. Homology, Homotopy and Applications 12:2 (2010), 245–320. doi:10.4310/hha.2010.v12.n2.a9. 1.2, 2.2, 2.3.1, 2.3.7, 2.3.8, 2.3.9, 2.3.10 [2008.b] Clemens Berger, Ieke Moerdijk. On an extension of the notion of Reedy category. Mathematische Zeitschrift 269:3–4 (2011...

work page doi:10.4310/hha.2010.v12.n2.a9 2010
[16]

Čech cocycles for differential characteristic classes: an∞-Lie theoretic construction

Domenico Fiorenza, Urs Schreiber, Jim Stasheff. Čech cocycles for differential characteristic classes: an∞-Lie theoretic construction. Advances in Theoretical and Mathematical Physics 16:1 (2012), 149–250. arXiv:1011.4735v2, doi:10.4310/atmp.2012.v16.n1.a5. 1, 2, 2.1, 2.1.5 [2011.a] Stephan Stolz, Peter Teichner. Supersymmetric field theories and generali...

work page doi:10.4310/atmp.2012.v16.n1.a5 2012
[17]

Freed, Constantin Teleman

Daniel S. Freed, Constantin Teleman. Relative quantum field theory. Communications in Mathematical Physics 326:2 (2014), 459–476. arXiv:1212.1692v3, doi:10.1007/s00220-013-1880-1. 5.1.2 [2013.a] André Henriques. Three-tier CFTs from Frobenius algebras. Topology and Field Theories. Contemporary Mathematics 613 (2014), 1–40. arXiv:1304.7328v2, doi:10.1090/c...

work page doi:10.1007/s00220-013-1880-1 2014
[18]

Classifying spaces of infinity-sheaves

Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov. Classifying spaces of infinity-sheaves. Algebraic & Geometric Topology 24:9 (2024), 4891–4937. arXiv:1912.10544v3, doi:10.2140/agt.2024.24.4891. 1.2

work page doi:10.2140/agt.2024.24.4891 2024
[19]

Higher Categories and Homotopical Algebra

Denis-Charles Cisinski. Higher Categories and Homotopical Algebra. Cambridge Studies in Advanced Mathematics 180 (2019). Cambridge University Press. https://cisinski.app.uni-regensburg.de/CatLR.pdf, doi:10.1017/9781108588737. 1.2 [2020.a] Matthias Ludewig, Augusto Stoffel. A framework for geometric field theories and their classification in dimension one....

work page doi:10.1017/9781108588737 2019
[20]

Differentiable sheaves I: Fractured∞-toposes and compactness

Adrian Clough. Differentiable sheaves I: Fractured∞-toposes and compactness. Proceedings of the American Mathematical Society 153 (2025), 4487–4500. arXiv:2309.01757v1, doi:10.1090/proc/17260. 4.1 [2025.a] Emilio Minichiello. Coverages and Grothendieck toposes. arXiv:2503.20664v2. 2.1.6 [2025.b] Stacks Project. https://stacks.math.columbia.edu/. 4.1.17, 5...

work page doi:10.1090/proc/17260 2025