$E$-theory of $X$-$C^{*}$-algebras and functor formalisms

Ulrich Bunke

arxiv: 2605.19863 · v2 · pith:MJWDM4UInew · submitted 2026-05-19 · 🧮 math.KT · math.AT· math.OA

E-theory of X-C^(*)-algebras and functor formalisms

Ulrich Bunke This is my paper

Pith reviewed 2026-05-22 09:35 UTC · model grok-4.3

classification 🧮 math.KT math.ATmath.OA

keywords E-theorysix-functor formalismC*-algebrassheavescosheavesK-theorylocally compact spaceslocales

0 comments

The pith

E-theory for locally compact Hausdorff spaces forms a six-functor formalism equivalent to that of E-valued sheaves.

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

The paper shows that E-theory, which assigns categories to locally compact Hausdorff spaces via associated C*-algebras, satisfies the full set of axioms for a six-functor formalism. This structure is proven equivalent to the six-functor formalism built directly from sheaves valued in E. A sympathetic reader cares because the result unifies an analytic K-theoretic invariant with a geometric sheaf-theoretic one, allowing each to be viewed through the lens of the other. The paper also establishes an equivalence between E-theory categories on certain locales and E-valued cosheaves.

Core claim

E-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of E-valued sheaves. Furthermore, the E-theory category for locales that can be written as unions of finite open sublocales is equivalent to the category of E-valued cosheaves.

What carries the argument

The six-functor formalism, a collection of six functors (direct and inverse images, proper direct images, and their adjoints) together with adjunctions, base-change isomorphisms, and projection formulas that govern behavior under continuous maps.

If this is right

E-theory computations on spaces can be carried out using the language and tools of sheaf theory.
Results about E-valued sheaves transfer directly to statements about E-theory categories.
For locales expressible as finite unions of open sublocales, E-theory categories coincide with cosheaf categories.
The six-functor structure supplies a uniform way to handle push-forwards, pull-backs, and base changes in both settings.

Where Pith is reading between the lines

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

The equivalence may allow importation of topos-theoretic techniques into the study of noncommutative C*-algebras.
Similar comparisons could be attempted for other variants of E-theory or for different base categories of spaces.
The cosheaf equivalence suggests that E-theory behaves like a homology theory with cosheaf coefficients on sufficiently simple locales.

Load-bearing premise

The six-functor formalism for E-valued sheaves is already defined independently on the relevant spaces so that equivalence with E-theory can be checked by direct comparison of the two structures.

What would settle it

A concrete locally compact Hausdorff space and a continuous map between such spaces where the E-theory functors violate proper base change or one of the other six-functor axioms would falsify the claimed equivalence.

read the original abstract

We show that $E$-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of $\mathrm{E}$-valued sheaves. We furthermore show that the $E$-theory category for locales that can be written as unions of finite open sublocales is equivalent to the category of $\mathrm{E}$-valued cosheaves.

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

Bunke shows E-theory on locally compact Hausdorff spaces gives a six-functor formalism equivalent to E-valued sheaves, plus a cosheaf version for finite-union locales.

read the letter

The key point is that this paper proves E-theory for locally compact Hausdorff spaces forms a six-functor formalism equivalent to that of E-valued sheaves. It also gives an equivalence for E-theory on locales that are finite unions of open sublocales with the category of E-valued cosheaves. Bunke does this by building the functors out of E-theory and showing they meet the same axioms and universal properties as the sheaf-theoretic versions. The comparison then yields the equivalence. This is a useful way to embed E-theory into the language of six-functor formalisms that many people in algebraic topology already use. The paper does a good job making these connections without adding extra assumptions. The constructions seem to handle the necessary properties like base change directly from the definitions of E-theory. One soft spot is that the results are tied to specific classes of spaces and locales. If someone wants a version for more general topological spaces, this does not provide it. Also, without seeing the full proofs in detail, it's hard to spot any subtle issues with the universal properties, but nothing in the outline suggests a problem. This work is for researchers who know E-theory and are interested in its functorial aspects or in comparing it to sheaf theory. A reader working on C*-algebras or K-theory with a background in categorical algebra would get the most out of it. It is not broad enough to interest people outside this niche. The thinking looks honest and engaged with the literature on these topics. The paper deserves a serious referee to check the details of the constructions and verifications. I recommend sending it to peer review. The equivalences are the sort of result that can be evaluated on its technical merits, and the field can use clear statements like these.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to construct E-theory for locally compact Hausdorff spaces as a six-functor formalism and to prove its equivalence with the six-functor formalism of E-valued sheaves. It further claims an equivalence between the E-theory category on locales that are finite unions of open sublocales and the category of E-valued cosheaves, proceeding via explicit functor constructions and verification that both structures satisfy the same axioms and universal properties.

Significance. If the equivalences are rigorously established, the result would connect E-theory arising from C*-algebras with classical sheaf and cosheaf theory, potentially enabling transfer of computational techniques and structural results between the two settings. The focus on verifying the full six-functor axioms (including base change and projection formulas) via universal properties is a methodological strength that could support broader applications in K-theory and noncommutative geometry.

minor comments (2)

[Abstract] The abstract states the main equivalences but does not indicate the key technical steps (e.g., how the E-theory functors are defined or which universal property is used for the comparison); a short sentence outlining the proof strategy would improve readability.
[Introduction] Notation for the six functors (f_!, f^*, etc.) and for the E-valued sheaf/cosheaf categories should be introduced with explicit references to the relevant axioms or prior literature in the first section where they appear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. The referee's summary accurately captures our main results on the six-functor formalism for E-theory of locally compact Hausdorff spaces and its equivalence to E-valued sheaves, as well as the equivalence for finite-open locales with E-valued cosheaves. We appreciate the positive assessment of the methodological approach via universal properties.

Circularity Check

0 steps flagged

No significant circularity; equivalence shown by independent comparison

full rationale

The paper constructs the six-functor formalism from E-theory for locally compact Hausdorff spaces and verifies it matches the independently defined six-functor formalism of E-valued sheaves, plus an equivalence for cosheaves on locales that are finite unions of open sublocales. The derivation proceeds by explicit functor construction and axiom verification rather than by defining one structure in terms of the other or fitting parameters to data. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to unverified inputs are present. The result is self-contained as a comparison against an external benchmark (the sheaf formalism) whose definition does not depend on the E-theory construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, new entities, or ad-hoc axioms are visible. The work relies on prior definitions of E-theory, six-functor formalisms, sheaves, and cosheaves from the literature.

axioms (1)

standard math Standard axioms of category theory and six-functor formalisms
Invoked implicitly when stating that E-theory constitutes such a formalism.

pith-pipeline@v0.9.0 · 5581 in / 1125 out tokens · 51542 ms · 2026-05-22T09:35:28.144595+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that E-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of E-valued sheaves.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

Bunke and B

U. Bunke and B. Duenzinger. E -theory is compactly assembled. https://arxiv.org/pdf/2402.18228.pdf https://arxiv.org/pdf/2402.18228.pdf, 2024

work page arXiv 2024
[2]

Bunke, A

U. Bunke, A. Engel, and M. Land. A stable -category for equivariant K\!K -theory. arxiv:2102.13372 https://arxiv.org/pdf/2102.13372.pdf

work page arXiv
[3]

Bentmann

R. Bentmann. Projective Dimension in Filtrated K-Theory , pages 41--62. Springer Berlin Heidelberg, 2013

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Bentmann

R. Bentmann. Homotopy-theoretic E -theory and $n$ -order. J. Homotopy Relat. Struct. , 9(2):455--463, 2014

work page 2014
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A. J. Blumberg, D. Gepner, and G. Tabuada. A universal characterization of higher algebraic K -theory. Geom. Topol. , 17(2):733--838, 2013

work page 2013
[6]

Blanchard

E. Blanchard. Tensor products of $C(X)$ -algebras over $C(X)$ . In Recent advances in operator algebras. Collection of talks given in the conference on operator algebras held in Orl\'eans, France in July 1992 , pages 81--92. Paris: Soci \'e t \'e Math \'e matique de France, 1995

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B. Blackadar. K -Theory for Operator Algebras . Cambridge University Press, 2nd edition, 1998

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N. P. Brown and N. Ozawa. $C^*$ -algebras and finite-dimensional approximations , volume 88 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2008

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[9]

U. Bunke. $KK$ - and $E$ -theory via homotopy theory. Orbita Math. , 1(2):103--210, 2024

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Bunke and M

U. Bunke and M. Volpe. A characterization of sheaves among six functor formalisms on LCH . In preparation

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Connes and N

A. Connes and N. Higson. Deformations, asymptotic morphisms and bivariant $K$ -theory. C. R. Acad. Sci., Paris, S \'e r. I , 311(2):101--106, 1990

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Dadarlat

M. Dadarlat. Fiberwise $KK$ -equivalence of continuous fields of $C^*$ -algebras. J. $K$-Theory , 3(2):205--219, 2009

work page 2009
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Drew and M

B. Drew and M. Gallauer. The universal six-functor formalism. Ann. $K$-Theory , 7(4):599--649, 2022

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J. Dixmier. $C^*$ -algebras. Transl . from the French by Francis Jellett , volume 15 of North-Holland Math. Libr. Elsevier (North-Holland), Amsterdam, 1983

work page 1983
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Dauser and J

A. Dauser and J. Kuijper. Uniqueness of six-functor formalisms. https://arxiv.org/abs/2412.15780 https://arxiv.org/abs/2412.15780, 2025

work page arXiv 2025
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E-theory C ^ * -algebras over topological spaces

Marius Dadarlat and Ralf Meyer. E-theory C ^ * -algebras over topological spaces. Journal of Functional Analysis , 263(1):216--247, July 2012

work page 2012
[17]

Dadarlat and P

M. Dadarlat and P. Vaidyanathan. $E$ -theory for $C[0, 1]$ -algebras with finitely many singular points. J. $K$-Theory , 13(2):249--274, 2014

work page 2014
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A. I. Efimov. K-theory and localizing invariants of large categories. https://arxiv.org/pdf/2405.12169.pdf https://arxiv.org/pdf/2405.12169.pdf, 05 2024

work page arXiv 2024
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Echterhoff and D

S. Echterhoff and D. P. Williams. Crossed products by $C_0(X)$ -actions. J. Funct. Anal. , 158(1):113--151, 1998

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Guentner, N

E. Guentner, N. Higson, and J. Trout. Equivariant E -theory for C ^ * -algebras. Memoirs of the American Mathematical Society , 148(703):0--0, 2000

work page 2000
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Gaitsgory and N

D. Gaitsgory and N. Rozenblyum. A study in derived algebraic geometry. Volume I : Correspondences and duality , volume 221 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2017

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N. Higson. Categories of fractions and excision in KK -theory. J. Pure Appl. Algebra , 65(2):119--138, 1990

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[23]

Heyer and L

C. Heyer and L. Mann. 6-functor Formalisms and Smooth Representations . https://arxiv.org/abs/2410.13038 https://arxiv.org/abs/2410.13038, 2024

work page arXiv 2024
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G. G. Kasparov. Equivariant K\!K -theory and the Novikov conjecture . Invent.\ Math. , 91(1):147--201, 1988

work page 1988
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u tzst \

A. Krause, Th. Nikolaus, and P. P \"u tzst \"u ck. Scheaves on manifolds. https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/sheaves-on-manifolds.pdf https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/sheaves-on-manifolds.pdf

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P.-Y. Le Gall. Equivariant Kasparov theory and groupoids. I . $K$-Theory , 16(4):361--390, 1999

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Land and T

M. Land and T. Nikolaus. On the relation between K- and L-theory of C^* -algebras . Math. Ann. , 371:517--563, 2018

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J. Lurie. Higher Algebra . www.math.harvard.edu/ lurie http://www.math.harvard.edu/ lurie/

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R. Meyer and R. Nest. The B aum-- C onnes conjecture via localisation of categories. Topology , 45(2):209--259, 2006

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[31]

Meyer and R

R. Meyer and R. Nest. C*-algebras over topological spaces: Filtrated K -theory. Canad. J. Math. 64 (2012), pp. 368-408 , 10 2008

work page 2012
[32]

Meyer and R

R. Meyer and R. Nest. $C^*$ -algebras over topological spaces: The bootstrap class. M \"u nster J. Math. , 2(1):215--252, 2009

work page 2009
[33]

Morel and V

F. Morel and V. Voevodsky. $ A ^1$ -homotopy theory of schemes. Publ. Math., Inst. Hautes \'E tud. Sci. , 90:45--143, 1999

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M. Nilsen. C ^* -bundles and C_0(X) -algebras. Indiana University Mathematics Journal , 45(2):0--0, 1996

work page 1996
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P. A. stv r. Homotopy theory of $C^*$ -algebras . Front. Math. Basel: Birkh \"a user, 2010

work page 2010
[36]

R. Popescu. Equivariant e-theory for groupoids acting on C^ * -algebras. Journal of Functional Analysis , 209(2):247--292, April 2004

work page 2004
[37]

Park and J

E. Park and J. Trout. Representable e-theory for C_ 0 (X) -algebras. Journal of Functional Analysis , 177(1):178--202, October 2000

work page 2000
[38]

P. Scholze. Six-functor Formalisms . https://arxiv.org/abs/2510.26269 https://arxiv.org/abs/2510.26269, 2025

work page arXiv 2025
[39]

M. Volpe. The six operations in topology. J. Topol. , 18(4):69, 2025

work page 2025
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Q. Zhu. Continuous six-functor formalism on locally compact Hausdorff spaces. https://arxiv.org/abs/2507.13537 https://arxiv.org/abs/2507.13537, 2025

work page arXiv 2025

[1] [1]

Bunke and B

U. Bunke and B. Duenzinger. E -theory is compactly assembled. https://arxiv.org/pdf/2402.18228.pdf https://arxiv.org/pdf/2402.18228.pdf, 2024

work page arXiv 2024

[2] [2]

Bunke, A

U. Bunke, A. Engel, and M. Land. A stable -category for equivariant K\!K -theory. arxiv:2102.13372 https://arxiv.org/pdf/2102.13372.pdf

work page arXiv

[3] [3]

Bentmann

R. Bentmann. Projective Dimension in Filtrated K-Theory , pages 41--62. Springer Berlin Heidelberg, 2013

work page 2013

[4] [4]

Bentmann

R. Bentmann. Homotopy-theoretic E -theory and \(n\) -order. J. Homotopy Relat. Struct. , 9(2):455--463, 2014

work page 2014

[5] [5]

A. J. Blumberg, D. Gepner, and G. Tabuada. A universal characterization of higher algebraic K -theory. Geom. Topol. , 17(2):733--838, 2013

work page 2013

[6] [6]

Blanchard

E. Blanchard. Tensor products of \(C(X)\) -algebras over \(C(X)\) . In Recent advances in operator algebras. Collection of talks given in the conference on operator algebras held in Orl\'eans, France in July 1992 , pages 81--92. Paris: Soci \'e t \'e Math \'e matique de France, 1995

work page 1992

[7] [7]

Blackadar

B. Blackadar. K -Theory for Operator Algebras . Cambridge University Press, 2nd edition, 1998

work page 1998

[8] [8]

N. P. Brown and N. Ozawa. \(C^*\) -algebras and finite-dimensional approximations , volume 88 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2008

work page 2008

[9] [9]

U. Bunke. \(KK\) - and \(E\) -theory via homotopy theory. Orbita Math. , 1(2):103--210, 2024

work page 2024

[10] [10]

Bunke and M

U. Bunke and M. Volpe. A characterization of sheaves among six functor formalisms on LCH . In preparation

work page

[11] [11]

Connes and N

A. Connes and N. Higson. Deformations, asymptotic morphisms and bivariant \(K\) -theory. C. R. Acad. Sci., Paris, S \'e r. I , 311(2):101--106, 1990

work page 1990

[12] [12]

Dadarlat

M. Dadarlat. Fiberwise \(KK\) -equivalence of continuous fields of \(C^*\) -algebras. J. \(K\)-Theory , 3(2):205--219, 2009

work page 2009

[13] [13]

Drew and M

B. Drew and M. Gallauer. The universal six-functor formalism. Ann. \(K\)-Theory , 7(4):599--649, 2022

work page 2022

[14] [14]

J. Dixmier. \(C^*\) -algebras. Transl . from the French by Francis Jellett , volume 15 of North-Holland Math. Libr. Elsevier (North-Holland), Amsterdam, 1983

work page 1983

[15] [15]

Dauser and J

A. Dauser and J. Kuijper. Uniqueness of six-functor formalisms. https://arxiv.org/abs/2412.15780 https://arxiv.org/abs/2412.15780, 2025

work page arXiv 2025

[16] [16]

E-theory C ^ * -algebras over topological spaces

Marius Dadarlat and Ralf Meyer. E-theory C ^ * -algebras over topological spaces. Journal of Functional Analysis , 263(1):216--247, July 2012

work page 2012

[17] [17]

Dadarlat and P

M. Dadarlat and P. Vaidyanathan. \(E\) -theory for \(C[0, 1]\) -algebras with finitely many singular points. J. \(K\)-Theory , 13(2):249--274, 2014

work page 2014

[18] [18]

A. I. Efimov. K-theory and localizing invariants of large categories. https://arxiv.org/pdf/2405.12169.pdf https://arxiv.org/pdf/2405.12169.pdf, 05 2024

work page arXiv 2024

[19] [19]

Echterhoff and D

S. Echterhoff and D. P. Williams. Crossed products by \(C_0(X)\) -actions. J. Funct. Anal. , 158(1):113--151, 1998

work page 1998

[20] [20]

Guentner, N

E. Guentner, N. Higson, and J. Trout. Equivariant E -theory for C ^ * -algebras. Memoirs of the American Mathematical Society , 148(703):0--0, 2000

work page 2000

[21] [21]

Gaitsgory and N

D. Gaitsgory and N. Rozenblyum. A study in derived algebraic geometry. Volume I : Correspondences and duality , volume 221 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2017

work page 2017

[22] [22]

N. Higson. Categories of fractions and excision in KK -theory. J. Pure Appl. Algebra , 65(2):119--138, 1990

work page 1990

[23] [23]

Heyer and L

C. Heyer and L. Mann. 6-functor Formalisms and Smooth Representations . https://arxiv.org/abs/2410.13038 https://arxiv.org/abs/2410.13038, 2024

work page arXiv 2024

[24] [24]

G. G. Kasparov. Equivariant K\!K -theory and the Novikov conjecture . Invent.\ Math. , 91(1):147--201, 1988

work page 1988

[25] [25]

u tzst \

A. Krause, Th. Nikolaus, and P. P \"u tzst \"u ck. Scheaves on manifolds. https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/sheaves-on-manifolds.pdf https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/sheaves-on-manifolds.pdf

work page

[26] [26]

G. Lehner. Algebraic k -theory of stably compact spaces. https://arxiv.org/pdf/2602.18245.pdf https://arxiv.org/pdf/2602.18245.pdf, 02 2026

work page arXiv 2026

[27] [27]

P.-Y. Le Gall. Equivariant Kasparov theory and groupoids. I . \(K\)-Theory , 16(4):361--390, 1999

work page 1999

[28] [28]

Land and T

M. Land and T. Nikolaus. On the relation between K- and L-theory of C^* -algebras . Math. Ann. , 371:517--563, 2018

work page 2018

[29] [29]

J. Lurie. Higher Algebra . www.math.harvard.edu/ lurie http://www.math.harvard.edu/ lurie/

work page

[30] [30]

Meyer and R

R. Meyer and R. Nest. The B aum-- C onnes conjecture via localisation of categories. Topology , 45(2):209--259, 2006

work page 2006

[31] [31]

Meyer and R

R. Meyer and R. Nest. C*-algebras over topological spaces: Filtrated K -theory. Canad. J. Math. 64 (2012), pp. 368-408 , 10 2008

work page 2012

[32] [32]

Meyer and R

R. Meyer and R. Nest. \(C^*\) -algebras over topological spaces: The bootstrap class. M \"u nster J. Math. , 2(1):215--252, 2009

work page 2009

[33] [33]

Morel and V

F. Morel and V. Voevodsky. \( A ^1\) -homotopy theory of schemes. Publ. Math., Inst. Hautes \'E tud. Sci. , 90:45--143, 1999

work page 1999

[34] [34]

M. Nilsen. C ^* -bundles and C_0(X) -algebras. Indiana University Mathematics Journal , 45(2):0--0, 1996

work page 1996

[35] [35]

P. A. stv r. Homotopy theory of \(C^*\) -algebras . Front. Math. Basel: Birkh \"a user, 2010

work page 2010

[36] [36]

R. Popescu. Equivariant e-theory for groupoids acting on C^ * -algebras. Journal of Functional Analysis , 209(2):247--292, April 2004

work page 2004

[37] [37]

Park and J

E. Park and J. Trout. Representable e-theory for C_ 0 (X) -algebras. Journal of Functional Analysis , 177(1):178--202, October 2000

work page 2000

[38] [38]

P. Scholze. Six-functor Formalisms . https://arxiv.org/abs/2510.26269 https://arxiv.org/abs/2510.26269, 2025

work page arXiv 2025

[39] [39]

M. Volpe. The six operations in topology. J. Topol. , 18(4):69, 2025

work page 2025

[40] [40]

Q. Zhu. Continuous six-functor formalism on locally compact Hausdorff spaces. https://arxiv.org/abs/2507.13537 https://arxiv.org/abs/2507.13537, 2025

work page arXiv 2025