Existence of $B^k_{\alpha,\beta}$-Structures on $C^k$-Manifolds

Rodney Josu\'e Biezuner; Yuri Ximenes Martins

REVIEW 3 major objections 3 minor 22 references

Existence of $B^k_{\alpha,\beta}$-Structures on $C^k$-Manifolds

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper defines a broad class of structured manifolds and proves that, under algebraic conditions on the structure presheaf, every $C^k$-manifold automatically carries such a structure.

desk verdict Novel categorical framework with an unproved central existence theorem; Theorem B needs an explicit inversion-closure condition and Proposition 2.1 is false. read the letter →

arxiv 1908.04442 v3 pith:L73YGLW7 submitted 2019-08-13 math.DG

Yuri Ximenes Martins , Rodney Josué Biezuner This is my paper · ORCID

classification math.DG MSC 58A0553C1018F2046E35

keywords structuredmanifoldsG-structuresboundedgeometrypresheavesofFrechetspacesadjointfunctorsSobolevtransitionfunctions

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 introduces $B^k_{\alpha,\beta}$-manifolds as a common generalization of smooth manifolds with $G$-structure and manifolds with $k$-bounded geometry: they are $C^k$-manifolds whose transition functions $\phi_{ji}$ satisfy $\partial^\mu \phi_{ji} \in B_{\alpha(r)} \cap C^{k-\beta(r)}$ for every multi-index $\mu$ with $|\mu|=r$, where $B=(B_r)_{r\in\Gamma}$ is a sequence of presheaves of Fréchet spaces and $\alpha,\beta$ are parameter functions. The paper's central claim is an existence theorem: if $B$ is ordered, fully left- or right-absorbing, and has retractible $(B,k,\alpha,\beta)$-diffeomorphisms, then the forgetful functor from $B^k_{\alpha,\beta}$-manifolds to $C^k$-manifolds has an adjoint that does not depend on the retraction, and as a consequence every $C^k$-manifold admits a $(B^k_{\alpha,\beta},X)$-structure. A sympathetic reader should care because the framework unifies two previously separate sources of extra regularity on manifolds — Lie-group reductions such as almost-complex or semi-Riemannian structures, and analytic bounds such as Sobolev or bounded-geometry atlases — and reduces the existence of more-regular geometric objects to the existence of such structured atlases. The paper additionally proves embedding theorems for the corresponding categories of structural presheaves, describing how these structures behave when the smoothness order, the ambient dimension, or the parameter function $\beta$ changes.

What carries the argument

The load-bearing object is the structural presheaf $B=(B_r)_{r\in\Gamma}$: a family of presheaves of nuclear Fréchet spaces on $\mathbb{R}^n$ equipped with multiplicative, additive, distributive, and intersection structures, required to be compatible with the classical presheaf $C^{k-\beta}$ in an intersection presheaf $X$ — this compatibility is what makes $B$ a $\mathcal{C}^{k,\alpha}_{n,\beta}$-presheaf. Three named mechanisms carry the argument. The ordering property ensures that the composite of two $(B,k,\alpha,\beta)$-functions is again one, via Faà di Bruno's formula applied block-wise to higher derivatives (Lemma 5.1). The absorption properties ensure that composing a $(B,k,\alpha,\beta)$-diffeomorphism with an ordinary $C^k$-diffeomorphism on either side stays within the class. The retraction presheaf $r$ is a splitting of the inclusion $B^k_{\alpha,\beta}(U;X)\hookrightarrow C^k(U;\mathbb{R}^n)$ that is required to send $C^k$-diffeomorphisms to $(B,k,\alpha,\beta)$-diffeomorphisms; it is the mechanism that turns a plain atlas into a refined one chart by chart, and the adjointness theorem turns that refinement into a functor with a universal property.

What would settle it

A concrete test: take $B$ to be the $L^p$ Sobolev presheaf with $1\le p<\infty$ and a single diffeomorphism of $\mathbb{R}^n$ such as a translation or dilation, and check whether some splitting $r$ of the inclusion $B^k_{\alpha,\beta}(U;X)\hookrightarrow C^k(U;\mathbb{R}^n)$ sends it and its inverse into $\operatorname{Diff}^{k}_{\alpha,\beta}$. If no such splitting exists, retractible $(B,k,\alpha,\beta)$-diffeomorphisms fail for this $B$, and the charts built in Theorem 6.1 need not have $B$-regular transitions. Alternatively, for the $G$-structure presheaf with $G=GL(n,\mathbb{C})$, the hypotheses of Theorem B would force every $C^k$-manifold to admit an almost-complex structure; since $S^4$ admits none, testing which hypothesis fails for this presheaf would falsify or precisely delimit the theorem.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of theorems about the categories of structured manifolds. Theorem A states that the category $\mathcal{C}^{k,\alpha}_{n,\beta}$ of structural presheaves embeds fully into larger categories when the smoothness order is lowered (when $l\le k$), when the model space $\mathbb{R}^n$ is carried into $\mathbb{R}^r$ by a continuous injective map, and when the parameter function $\beta$ is lowered. Theorem B states that if $B$ is ordered, fully left-absorbing (respectively fully right-absorbing), and has retractible $(B,k,\alpha,\beta)$-diffeomorphisms in one intersection presheaf $X$, then any choice of retraction $r$ induces a left-adjoint (respectively right-adjoint) for the forgetful functor $F: \operatorname{Diff}^{B,k}_{\alpha,\beta}(X)\to\operatorname{Diff}^k$, independent of $r$; if $B$ is fully absorbing, $F$ is ambidextrous. The decisive intermediate step, Theorem 6.1, constructs the structured atlas explicitly: from any $C^k$-atlas with charts $\phi_i$ and transition functions $\phi_{ji}=\phi_j\circ\phi_i^{-1}$, the retraction produces new charts $\phi_{j;i}=r(\phi_{ji})\circ(\phi_i|_{U_{ij}})$, whose transition functions $\phi_{k;l}\circ\phi_{j;i}^{-1}=r(\phi_{kl})\circ(\phi_{li}|_{U_{ijkl}})\circ r(\phi_{ji})^{-1}$ lie in the $(B,k,\alpha,\beta)$-class precisely because of the absorbing, ordering, and retraction hypotheses. Restricted to the core category (diffeomorphisms only), the same construction yields a functor, and the fully absorbing version extends the adjunction from the core to all structured morphisms, so that the category of structured manifolds inherits limits and colimits from the category of $C^k$-manifolds.

Load-bearing premise

The argument rests on being able to choose the retraction $r$ so that it sends ordinary diffeomorphisms of Euclidean open sets to diffeomorphisms that stay inside the structured class, including their inverses, so that $r(\phi_{ji})^{-1}$ is again a $(B,k,\alpha,\beta)$-function; the paper assumes such a retraction exists and uses it without proof in Theorem 6.1.

Editorial extensions

If this is right

Under the hypotheses, every $C^k$-manifold admits a $(B^k_{\alpha,\beta},X)$-structure: the refinement $\kappa_r(A')=r(A')$ produces the structured subatlas explicitly (Theorem 6.1).
The category $\operatorname{Diff}^{B,k}_{\alpha,\beta}(X)$ inherits all small limits and colimits that exist in $\operatorname{Diff}^k$, and in particular has finite products and coproducts when $B$ is fully absorbing (Corollary 6.2).
Left-adjointness (respectively right-adjointness) of the forgetful functor means that structured manifolds can be freely generated from, or cofreely coarsened from, plain $C^k$-manifolds, and the resulting structure does not depend on which retraction was chosen.
The classical motivating classes — $G$-structures and Sobolev or bounded-geometry structures — are instances of this framework, so the existence theorem applies to them whenever their structure presheaves satisfy the absorption and retraction hypotheses.
For ordered presheaves, composition of structured morphisms is well-defined, so the category $\operatorname{Diff}^{B,k}_{\alpha,\beta}(X)$ genuinely exists and the adjunction is between honest categories (Proposition 5.2).

Reading between the lines

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

The retraction hypothesis is the least algebraically anchored part of the theorem: the paper obtains $r$ as a vector-space splitting, but nothing forces a splitting to respect the diffeomorphism group, so the effective content of the hypothesis is a geometric existence claim. If it fails for a given $B$, the charts constructed in Theorem 6.1 need not have $B$-regular transitions.
For the $G$-structure presheaf with $G=GL(n,\mathbb{C})$, the hypotheses would force every $C^k$-manifold to admit an almost-complex structure; since $S^4$ does not, one of the hypotheses must fail for that presheaf, and locating the failure would mark the exact boundary between this theorem and classical obstruction theory.
For the $L^p$ presheaf, the theorem implies every $C^k$-manifold should carry an atlas whose transition functions and derivatives up to order $k$ lie in $L^p\cap C^{k-\beta}$; this is a concrete analytic assertion that could be checked directly on simple diffeomorphisms before relying on the full adjunction.
Read categorically, the adjunction suggests viewing '$B^k_{\alpha,\beta}$-structurization' as a completion functor on manifolds, which may be the categorical shadow of analytic regularization results such as the existence of bounded-geometry metrics in conformal classes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague — here is the honest take on Ximenes Martins–Biezuner. The paper is a genuine piece of formalism: it defines a common generalization of G-structures and Sobolev/bounded-geometry manifolds via presheaves B^k_{α,β}, and the embedding theorem (Theorem A) appears to be a real result, modulo strong but clearly stated hypotheses. That part deserves credit.

The problem is the main existence/adjunction theorem. Theorem B rests on Theorem 6.1, and the proof needs the inverse of a (B,k,α,β)-diffeomorphism to again be one. That is not in the definition and is not generally true for the intended examples (L^p, Sobolev). The paper simply asserts it. Without it, the constructed charts do not have B-regular transitions. The stress-test note is right: the right-absorbing half cannot be rescued by regrouping, because the absorption direction does not match; a left-absorbing half might be repaired by two absorption steps, but that is not what the paper does. You would need to add an explicit inversion-closure axiom or change the construction.

Second, Proposition 2.1 is false. The pullback of two bilinear maps does not generally contain a copy of the four factors; for maps to a one-dimensional space it can be a single line. That proposition is used to show the main examples are C^{k,α}_{n,β}-presheaves, so those examples are not established as written. The framework might still work with a different intersection structure, but the paper's own examples do not currently justify it.

Third, the hypotheses for Theorem B are very strong and no nontrivial example is shown to satisfy them. The retractibility and absorption conditions look hard to verify; the paper gives none. So even if the proof gap were fixed, the theorem might be vacuous.

There are also minor typos and index inconsistencies in the main statements, but those are not the issue. The core issue is that the central existence theorem is unsupported.

Who is this for? A reader interested in categorical approaches to regularity structures might get ideas from the formalism and from Theorem A. But no one should rely on Theorem B as stated.

Recommendation: if this lands on my desk, I would send it to a referee, because the framework is novel and the flaw is concrete and possibly repairable. A report that points to the inversion-closure gap and Proposition 2.1 could lead to a substantially better revision. As it stands, the main theorem should not be accepted.

Referee Report

3 major / 3 minor

Summary. The paper introduces a general framework of B^k_{\alpha,\beta}-manifolds, defined as C^k-manifolds whose transition functions and their derivatives belong to prescribed intersection presheaves B_{\alpha(r)} \cap C^{k-\beta(r)}. It develops a presheaf-categorical machinery of (\Gamma,\epsilon)-spaces, intersection structures, and C^{k,\alpha}_{n,\beta}-presheaves, and proves two main results: Theorem A, giving full embeddings between categories of such presheaves under changes of k, n, and \beta; and Theorem B, asserting that, under hypotheses of being ordered, fully left- or right-absorbing, and having retractible (B,k,\alpha,\beta)-diffeomorphisms, the forgetful functor from B^k_{\alpha,\beta}-manifolds to C^k-manifolds has adjoints, so that every C^k-manifold admits a B^k_{\alpha,\beta}-structure. The paper is written as the first in a planned series, with applications to G-structures, Sobolev manifolds, and bounded geometry as motivation.

Significance. The proposed framework is ambitious and, if fully established, would provide a unified categorical treatment of several established notions of geometric regularity, with Theorem B giving a clean adjunction statement and consequences for limits and colimits. The abstract embedding results of Theorem A are a genuine structural contribution, and the paper is honest about some of its own limitations, e.g., in Remark 4.1. However, the central existence theorem is not established as stated: the proof of Theorem 6.1 silently assumes that inverses of (B,k,\alpha,\beta)-diffeomorphisms are again (B,k,\alpha,\beta)-functions, an assumption that is not part of the definitions and is not derived from them. In addition, Proposition 2.1, used to certify the main examples of C^k_{n,\beta}-presheaves, is false. Because these issues affect the load-bearing construction rather than only its presentation, the paper's central claim currently rests on unproved and, in one case, false assertions.

major comments (3)

[Section 6, proof of Theorem 6.1] The proof of Theorem 6.1 uses without proof the assertion that r(\varphi_{ji})^{-1} is a (B,k,\alpha,\beta)-function. By definition, Diﬀ^k_{\alpha,\beta}(U,V;X) is only the largest subset of B^k_{\alpha,\beta}(U,V;X) for which a dotted arrow to Diﬀ^k(U,V) exists: a C^k inverse is required, but the inverse itself is not required to belong to B^k_{\alpha,\beta}. Since the new transition functions are written as r(\varphi_{kl}) \circ \varphi_{li} \circ r(\varphi_{ji})^{-1}, both the left-absorbing and the right-absorbing cases need r(\varphi_{ji})^{-1} \in B^k_{\alpha,\beta}; the ordering hypothesis only controls composition of two B-functions, not inversion. The same unproved step is used in equations (9) and (11) of Theorem 6.2 and is inherited by Theorem B. A repair would require either an explicit axiom that (B,k,\alpha,\beta)-diffeomorphisms are closed under inverses or a different construction that never requires r(\varphi_{ji})^{-1} to lie in B.
[Section 2, Proposition 2.1] Proposition 2.1 is false as stated, and its proof is incorrect. Take V=W=V'=W'=\mathbb{R} and Z=\mathbb{R}^2, let T,T':\mathbb{R}\otimes\mathbb{R}\to\mathbb{R}^2 be the linear maps sending 1\otimes 1 to e_1 and e_2, respectively. Both multiplicative structures are nontrivial, but the pullback of T and T' is { (a,b) : a e_1 = b e_2 } = 0. Thus nontrivial multiplicative structures need not have nontrivial intersection in a vectorial intersection structure. This proposition is invoked in Examples 3.3 and 3.4 to conclude that the C^{k-} and L presheaves are C^k_{\alpha,\beta}-presheaves, so the intended instantiations of the hypotheses of Theorem B are not justified as they stand. The proposition should either be corrected with additional hypotheses or replaced by a direct verification in the examples.
[Section 6, absorbing definitions and Proposition 6.1] The terminology 'left-absorbing' and 'right-absorbing' is inconsistent with the ideal terminology used in Proposition 6.1. In the lower square, a C^k map is composed on the right of a (B,k,\alpha,\beta)-diffeomorphism, i.e., the B-map is the outer left factor; this is a right-ideal property in the standard convention used in the preceding magma discussion, yet it is called left-absorbing. The upper square has the opposite behavior yet is called right-absorbing. Consequently the asserted equivalence in Proposition 6.1 is not correct under the stated definitions, and the proof of Theorem 6.2 uses an absorption direction that appears to be opposite to the one named in the text. The labels and equivalences should be fixed, and the adjunction directions in Theorem B should be rechecked after relabeling.

minor comments (3)

[Throughout] The manuscript contains many typos and misspellings, including 'indenpendently' in Proposition 2.1, 'condiser' in Section 3, 'Simlarly' in Example 2.9, 'aborving' in the fully absorbing paragraph, 'presehaf' in Remark 5.2, and 'Straighforward' in several places. A careful proofread is needed.
[Section 6, after diagram (8)] The definition of a retraction presheaf is hard to parse: the diagrams are said to be 'not necessarily making the first diagram commutative', but the quantifier over r_{U,V} and the precise retraction identities are not stated explicitly. Please spell out the exact equalities required of r_{U,V}.
[Theorem 4.1(2)] The proof of part (2) asserts that f^{-1} becomes an embedding when f is injective. This is used for presheaves of Fréchet spaces, not only presheaves of sets; the verification that hom-sets are mapped bijectively should be written out, especially the action on morphisms between (\Gamma,\epsilon)-spaces.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem B is derived from explicit absorption/retraction axioms, with no self-citation or fitted input; the terse inversion step in Theorem 6.1 is a proof-detail issue, not a circular reduction.

full rationale

The claimed derivation is self-contained in the sense relevant to the circularity pass. Theorem B is not obtained by fitting parameters, by renaming a known result, or by importing an author-specific uniqueness theorem; the reference list contains no overlapping self-citation, and no load-bearing premise is justified only by the authors' prior work. The central construction in Theorem 6.1 starts from the explicitly assumed retraction presheaf r and the ordered/absorbing axioms: the new charts are defined as φ_{j;i} = r(φ_{ji}) ∘ φ_i|Uij, and the transition functions are then checked against the stated hypotheses. This is a verification using the assumptions, not an identity forced by a definition. The retractibility assumption is strong — it locally supplies B-diffeomorphisms for every Euclidean C^k-diffeomorphism — but it is not equivalent to the theorem's conclusion: the conclusion asserts existence of B-structures and adjoints for arbitrary C^k-manifolds, while the hypothesis is a section over the small category of Euclidean open sets; the paper then proves the functorial extension K_r and the adjunction bijection (10). The only passage worth flagging is the terse assertion in the proof of Theorem 6.1 that 'r(φ_{ji})^{-1} is also a (B,k,α,β)-function in X'; the text does not spell out the derivation. Whether or not that omitted argument is fully valid, this is a proof-detail or correctness question about a step inside the proof, not a case where the conclusion is equivalent to an input by construction. Accordingly, no formal circularity is found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central theorems rest on new, strong conditions (ordered, absorbing, retractible diffeomorphisms) that are not demonstrated for any nontrivial example, on a false lemma about pullbacks, and on an unproven inverse-closure step. Standard categorical background is leveraged correctly, but the paper's own definitions carry most of the load.

assumptions (6)

ad hoc to paper B is ordered, fully left-absorbing (resp. right-absorbing), and has retractible (B,k,α,β)-diffeomorphisms in X (Theorem 6.1/Theorem B hypotheses).
These conditions are introduced in this paper and are not shown to hold for any nontrivial example; they are precisely what makes the proof go through, including the unproven inverse-closure of B-diffeomorphisms.
ad hoc to paper Vector subspaces of C^k(U;R^n) admit algebraic complements, giving linear retractions r_U; these retractions can be chosen to map C^k-diffeomorphisms to B-diffeomorphisms.
The vector-space part is standard (choice), but the diffeomorphism-preserving part is assumed as 'retractible (B,k,α,β)-diffeomorphisms' in Section 6 and is load-bearing.
ad hoc to paper For linear maps T:V⊗W→Z and T':V'⊗W'→Z, the pullback contains a copy of V⊕W⊕V'⊕V' (Proposition 2.1).
This is false; e.g., T=T'=id on R gives a diagonal line. Used to show nontrivial multiplicative structures always intersect and to justify examples of C^{k,α}_{n,β}-presheaves.
standard math Composition of (B,k,α,β)-functions is controlled by Fàa di Bruno and yields an ordered structure (Lemma 5.1).
Relies on Hörmander [13] for Fàa di Bruno and on the multiplicative/additive compatibility of B; used to define the category.
standard math Standard categorical machinery: cores, adjoints, Kan extensions, limits/colimits preservation by adjoints (Mac Lane [16], Borceux [7]).
Used throughout Sections 4 and 6.
standard math C^k-manifolds have finite limits and colimits (Moerdijk-Reyes [18], Baez-Hoffnung [4]).
Used in Corollary 6.2.

invented entities (1)

B^k_{\alpha,\beta}-manifold framework
purpose: Unify G-structures, Sobolev manifolds, and bounded geometry under a categorical presheaf formalism; enables statement of existence and adjunction theorems.
The framework is defined in this paper; no independent evidence outside the paper exists, and no known nontrivial B is shown to satisfy all Theorem B hypotheses.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Existence of $B^k_{\alpha,\beta}$-Structures on $C^k$-Manifolds." pith.science (2026). https://pith.science/paper/L73YGLW7

@misc{pith2026190804442,
  author       = {Pith},
  title        = {Pith review of: Existence of $B^k_\alpha,\beta$-Structures on $C^k$-Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L73YGLW7}},
  note         = {Machine review of arXiv:1908.04442}
}

abstract

In this paper we introduce $B_{\alpha,\beta}^{k}$-manifolds as generalizations of the notion of smooth manifolds with $G$-structure or with $k$-bounded geometry. These are $C^{k}$-manifolds whose transition functions $\varphi_{ji}=\varphi_{j}\circ\varphi_{i}^{-1}$ are such that $\partial^{\mu}\varphi_{ji}\in B_{\alpha(r)}\cap C^{k-\beta(r)}$ for every $\vert\mu\vert=r$, where $B=(B_{r})_{r\in\Gamma}$ is some sequence of presheaves of Fr\'echet spaces endowed with further structures, $\Gamma\subset\mathbb{Z}_{\geq0}$ is some parameter set and $\alpha,\beta$ are functions. We present embedding theorems for the presheaf category of those structural presheaves $B$. The existence problem of $B_{\alpha,\beta}^{k}$-structures on $C^{k}$-manifolds is studied and it is proved that under certain conditions on $B$, $\alpha$ and $\beta$, the forgetful functor from $C^{k}$-manifolds to $B_{\alpha,\beta}^{k}$-manifolds has adjoints.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Adachi, M., Embeddings and Immersions , AMS, 2012

work page 2012
[2]

Aguiar, M., Mahajan, S., Monoidal Functors, Species and Hopf Algebras , AMS 2010

work page 2010
[3]

Atiyah, M., Complex analytic connections in ﬁbre bundles , Trans. Amer. Math. Soc. 85 (1957), 181–207

work page 1957
[4]

C., Hoﬀnung, A

Baez, J. C., Hoﬀnung, A. E., Convenient Categories of Smooth Spaces , Trans. Amer. Math. Soc. Vol. 363, No. 11 (2011), pp. 5789-5825

work page 2011
[5]

Berline, N., Getzler, E., Vergne, M., Heat Kernels and Dirac Operators , Springer, 2004

work page 2004
[6]

Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J., Schwa chhofer, L., Symplectic connections, arXiv:math/0511194

work page Pith review arXiv
[7]

Borceux, F., Handbook of Categorical Algebra I , Cambridge University Press, 2008

work page 2008
[8]

Costello, K., Renormalization and Eﬀective Field Theory , AMS 2011

work page 2011

Show all 22 references

[9]

Eichhorn, J., The Boundedness of Connection Coeﬃcients and their Derivative s, Math. Nachr. 152 (1991) 145-158

1991
[10]

Fedosov, B., A simple geometrical construction of deformation quantiza tion, Journal of Diﬀer- ential Geometry, 40(2):213–238, 1994

1994
[11]

I., Manin, Y

Gelfand, S. I., Manin, Y. I., Methods of Homological Algebra , Springer, 2003

2003
[12]

E., Complete metrics of bounded curvature on noncompact manifol ds, Arch

Greene, R. E., Complete metrics of bounded curvature on noncompact manifol ds, Arch. Math (1978) 31: 89

1978
[13]

H¨ ormander, L.,The analysis of linear partial diﬀerential operators I , Springer, 1983

1983
[14]

Kobayashi, S., Diﬀerential Geometry of Complex Vector Bundle , Princeton University Press, 2014

2014
[15]

Kobayashi, S., Nomizu, K., Foundations of Diﬀerential Geometry, Vol.1 , Wiley-Interscience, 1996

1996
[16]

MacLane, S., Categories for the Working Mathematician , 2nd edition, Springer, 1978

1978
[17]

MacLane, S., Moerdijk, I., Sheaves in Geometry and Logic , Springer, 1994

1994
[18]

E., Models for Smooth Inﬁnitesimal Analysis , Springer, 1991

Moerdijk, I., Reyes, G. E., Models for Smooth Inﬁnitesimal Analysis , Springer, 1991

1991
[19]

M¨ uller, O., Nardmann, M., Every conformal class contains a metric of bounded geometry , Mathematische Annalen 363 (2015), 143-174. 26

2015
[20]

Sternberg, S., Lectures on Diﬀerential Geometry , AMS, 1999

1999
[21]

Treves, F., Topological Vector Spaces, Distributions and Kernels , Academic Press, 1967

1967
[22]

K., Connections with Lp bounds on curvature , Comm

Uhlenbeck, K. K., Connections with Lp bounds on curvature , Comm. Math. Phys. Volume 83, Number 1 (1982), 31-42. 27

1982

Pith tools

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

[1] [1]

Adachi, M., Embeddings and Immersions , AMS, 2012

work page 2012

[2] [2]

Aguiar, M., Mahajan, S., Monoidal Functors, Species and Hopf Algebras , AMS 2010

work page 2010

[3] [3]

Atiyah, M., Complex analytic connections in ﬁbre bundles , Trans. Amer. Math. Soc. 85 (1957), 181–207

work page 1957

[4] [4]

C., Hoﬀnung, A

Baez, J. C., Hoﬀnung, A. E., Convenient Categories of Smooth Spaces , Trans. Amer. Math. Soc. Vol. 363, No. 11 (2011), pp. 5789-5825

work page 2011

[5] [5]

Berline, N., Getzler, E., Vergne, M., Heat Kernels and Dirac Operators , Springer, 2004

work page 2004

[6] [6]

Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J., Schwa chhofer, L., Symplectic connections, arXiv:math/0511194

work page Pith review arXiv

[7] [7]

Borceux, F., Handbook of Categorical Algebra I , Cambridge University Press, 2008

work page 2008

[8] [8]

Costello, K., Renormalization and Eﬀective Field Theory , AMS 2011

work page 2011