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Journal articles on the topic 'Equivariant localization'

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Published: 14 March 2023

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1

Alekseev, A., E. Meinrenken, and C. Woodward. "Group-valued equivariant localization." Inventiones Mathematicae 140, no. 2 (May 1, 2000): 327–50. http://dx.doi.org/10.1007/s002220000056.

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2

Katz, Gabriel. "Localization in equivariant bordisms." Mathematische Zeitschrift 213, no. 1 (May 1993): 617–45. http://dx.doi.org/10.1007/bf03025741.

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3

Tirkkonen, Olav. "Equivariant BRST and localization." Theoretical and Mathematical Physics 98, no. 3 (March 1994): 344–49. http://dx.doi.org/10.1007/bf01102211.

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4

Kusakabe, Yuta. "An implicit function theorem for sprays and applications to Oka theory." International Journal of Mathematics 31, no. 09 (July 17, 2020): 2050071. http://dx.doi.org/10.1142/s0129167x20500718.

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We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition [Formula: see text], and the equivariant localization principle is also given.
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5

Bruzzo, Ugo, and Vladimir Rubtsov. "On localization in holomorphic equivariant cohomology." Central European Journal of Mathematics 10, no. 4 (April 19, 2012): 1442–54. http://dx.doi.org/10.2478/s11533-012-0054-2.

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6

Smithies, Laura. "Equivariant analytic localization of group representations." Memoirs of the American Mathematical Society 153, no. 728 (2001): 0. http://dx.doi.org/10.1090/memo/0728.

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7

Dykstra, Hans M., Joseph D. Lykken, and Eric J. Raiten. "Exact path integrals by equivariant localization." Physics Letters B 302, no. 2-3 (March 1993): 223–29. http://dx.doi.org/10.1016/0370-2693(93)90388-x.

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8

Karpenko, Nikita, and Alexander Merkurjev. "Equivariant connective 𝐾-theory." Journal of Algebraic Geometry 31, no. 1 (October 28, 2021): 181–204. http://dx.doi.org/10.1090/jag/773.

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For separated schemes of finite type over a field with an action of an affine group scheme of finite type, we construct the bi-graded equivariant connective K K -theory mapping to the equivariant K K -homology of Guillot and the equivariant algebraic K K -theory of Thomason. It has all the standard basic properties as the homotopy invariance and localization. We also get the equivariant version of the Brown-Gersten-Quillen spectral sequence and study its convergence.
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9

Duflot, J. "Erratum to "Localization of Equivariant Cohomology Rings"." Transactions of the American Mathematical Society 290, no. 2 (August 1985): 857. http://dx.doi.org/10.2307/2000321.

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10

Duflot, J. "Erratum to: ‘‘Localization of equivariant cohomology rings”." Transactions of the American Mathematical Society 290, no. 2 (February 1, 1985): 857. http://dx.doi.org/10.1090/s0002-9947-1985-0792834-5.

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11

Pedroza, Andrés, and Loring W. Tu. "On the localization formula in equivariant cohomology." Topology and its Applications 154, no. 7 (April 2007): 1493–501. http://dx.doi.org/10.1016/j.topol.2005.10.013.

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12

Harada, Megumi, and Yael Karshon. "Localization for equivariant cohomology with varying polarization." Communications in Analysis and Geometry 20, no. 5 (2012): 869–947. http://dx.doi.org/10.4310/cag.2012.v20.n5.a1.

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13

Fink, Alex, and David E. Speyer. "$K$ -classes for matroids and equivariant localization." Duke Mathematical Journal 161, no. 14 (November 2012): 2699–723. http://dx.doi.org/10.1215/00127094-1813296.

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14

Takeda, Yuichiro. "Localization theorem in equivariant algebraic K-theory." Journal of Pure and Applied Algebra 96, no. 1 (September 1994): 73–80. http://dx.doi.org/10.1016/0022-4049(94)90088-4.

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15

Bruzzo, U., L. Cirio, P. Rossi, and V. Rubtsov. "Equivariant cohomology and localization for Lie algebroids." Functional Analysis and Its Applications 43, no. 1 (March 2009): 18–29. http://dx.doi.org/10.1007/s10688-009-0003-4.

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16

VAJIAC, ADRIAN. "EQUIVARIANT LOCALIZATION TECHNIQUES IN TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 07, no. 02 (March 2010): 247–66. http://dx.doi.org/10.1142/s0219887810004038.

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This paper proposes a different approach to derive Witten's formula relating Donaldson and Seiberg–Witten invariants for four-manifolds, via equivariant localization techniques. Our approach proposes a direct study on the Donaldson–Witten and Seiberg–Witten configuration spaces, not making use the theory of non-abelian monopoles.
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17

Goresky, Mark, Robert Kottwitz, and Robert MacPherson. "Equivariant cohomology, Koszul duality, and the localization theorem." Inventiones Mathematicae 131, no. 1 (December 17, 1997): 25–83. http://dx.doi.org/10.1007/s002220050197.

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18

Oda, Nobuyuki, and Yoshimi Shitanda. "Localization, completion and detecting equivariant maps on skeletons." Manuscripta Mathematica 65, no. 1 (March 1989): 1–18. http://dx.doi.org/10.1007/bf01168363.

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19

Barannikov, Serguei. "EA-Matrix Integrals of Associative Algebras and Equivariant Localization." Arnold Mathematical Journal 5, no. 1 (March 2019): 97–104. http://dx.doi.org/10.1007/s40598-019-00111-0.

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20

Anderson, Dave, Richard Gonzales, and Sam Payne. "Equivariant Grothendieck–Riemann–Roch and localization in operational K-theory." Algebra & Number Theory 15, no. 2 (April 7, 2021): 341–85. http://dx.doi.org/10.2140/ant.2021.15.341.

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21

Edidin, Dan, and William Graham. "Localization in equivariant intersection theory and the Bott residue formula." American Journal of Mathematics 120, no. 3 (1998): 619–36. http://dx.doi.org/10.1353/ajm.1998.0020.

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22

Blau, Matthias, and George Thompson. "Equivariant Kähler geometry and localization in the G/G model." Nuclear Physics B 439, no. 1-2 (April 1995): 367–94. http://dx.doi.org/10.1016/0550-3213(95)00058-z.

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23

Bunk, Severin, and Richard J. Szabo. "Topological insulators and the Kane–Mele invariant: Obstruction and localization theory." Reviews in Mathematical Physics 32, no. 06 (December 9, 2019): 2050017. http://dx.doi.org/10.1142/s0129055x20500178.

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We present homotopy theoretic and geometric interpretations of the Kane–Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to non-equivariant homotopy. We prove a Mayer–Vietoris Theorem for manifolds with [Formula: see text]-actions which intertwines Real and [Formula: see text]-equivariant de Rham cohomology groups, and apply it to derive a new localization formula for the Kane–Mele invariant. This provides a unified cohomological explanation for the equivalence between the discrete Pfaffian formula and the known local geometric computations of the index for periodic lattice systems. We build on the relation between the Kane–Mele invariant and the theory of bundle gerbes with [Formula: see text]-actions to obtain geometric refinements of this obstruction and localization technique. In the preliminary part we review the Freed–Moore theory of band insulators on Galilean spacetimes with emphasis on geometric constructions, and present a bottom-up approach to time-reversal symmetric topological phases.
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24

Chidester, Benjamin, Tianming Zhou, Minh N. Do, and Jian Ma. "Rotation equivariant and invariant neural networks for microscopy image analysis." Bioinformatics 35, no. 14 (July 2019): i530—i537. http://dx.doi.org/10.1093/bioinformatics/btz353.

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Abstract Motivation Neural networks have been widely used to analyze high-throughput microscopy images. However, the performance of neural networks can be significantly improved by encoding known invariance for particular tasks. Highly relevant to the goal of automated cell phenotyping from microscopy image data is rotation invariance. Here we consider the application of two schemes for encoding rotation equivariance and invariance in a convolutional neural network, namely, the group-equivariant CNN (G-CNN), and a new architecture with simple, efficient conic convolution, for classifying microscopy images. We additionally integrate the 2D-discrete-Fourier transform (2D-DFT) as an effective means for encoding global rotational invariance. We call our new method the Conic Convolution and DFT Network (CFNet). Results We evaluated the efficacy of CFNet and G-CNN as compared to a standard CNN for several different image classification tasks, including simulated and real microscopy images of subcellular protein localization, and demonstrated improved performance. We believe CFNet has the potential to improve many high-throughput microscopy image analysis applications. Availability and implementation Source code of CFNet is available at: https://github.com/bchidest/CFNet. Supplementary information Supplementary data are available at Bioinformatics online.
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25

Chorny, Boris. "Localization with respect to a class of maps I — Equivariant localization of diagrams of spaces." Israel Journal of Mathematics 147, no. 1 (December 2005): 93–139. http://dx.doi.org/10.1007/bf02785361.

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26

Legendre, Eveline. "Localizing the Donaldson–Futaki invariant." International Journal of Mathematics 32, no. 08 (June 22, 2021): 2150055. http://dx.doi.org/10.1142/s0129167x21500555.

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We use the equivariant localization formula to prove that the Donaldson–Futaki invariant of a compact smooth (Kähler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone.
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27

SEMENOFF, GORDON W., and RICHARD J. SZABO. "EQUIVARIANT LOCALIZATION, SPIN SYSTEMS AND TOPOLOGICAL QUANTUM THEORY ON RIEMANN SURFACES." Modern Physics Letters A 09, no. 29 (September 21, 1994): 2705–18. http://dx.doi.org/10.1142/s0217732394002550.

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We study equivariant localization formulas for phase space path-integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the non-trivial homology classes of the phase space. We explicitly construct the coherent states in the canonical quantum theory and show that the Hilbert space is finite-dimensional with the wave functions carrying a projective representation of the discrete homology group of the phase space. The corresponding coherent state path-integral then describes the quantum dynamics of a novel spin system given by the quantization of a nonsymmetric coadjoint Lie group orbit. We also briefly discuss the geometric structure of these quantum systems.
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28

Bytsenko, A. A., M. Libine, and F. L. Williams. "Localization of Equivariant Cohomology for Compact and Non-Compact Group Actions." Journal of Dynamical Systems and Geometric Theories 3, no. 2 (January 2005): 171–95. http://dx.doi.org/10.1080/1726037x.2005.10698497.

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29

Chen, Harrison. "Equivariant localization and completion in cyclic homology and derived loop spaces." Advances in Mathematics 364 (April 2020): 107005. http://dx.doi.org/10.1016/j.aim.2020.107005.

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30

Edidin, Dan, and William Graham. "Nonabelian localization in equivariant K-theory and Riemann–Roch for quotients." Advances in Mathematics 198, no. 2 (December 2005): 547–82. http://dx.doi.org/10.1016/j.aim.2005.06.010.

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31

Gonzales, Richard P. "Localization in equivariant operational K-theory and the Chang–Skjelbred property." manuscripta mathematica 153, no. 3-4 (October 10, 2016): 623–44. http://dx.doi.org/10.1007/s00229-016-0890-7.

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32

Perrot, Denis. "The equivariant index theorem in entire cyclic cohomology." Journal of K-Theory 3, no. 2 (May 28, 2008): 261–307. http://dx.doi.org/10.1017/is008004027jkt047.

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AbstractLet G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.
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33

Szabo, Richard J., and Gordon W. Semenoff. "Phase space isometries and equivariant localization of path integrals in two dimensions." Nuclear Physics B 421, no. 2 (June 1994): 391–412. http://dx.doi.org/10.1016/0550-3213(94)90333-6.

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34

Backelin, Erik, and Kobi Kremnizer. "Quantum flag varieties, equivariant quantum D-modules, and localization of quantum groups." Advances in Mathematics 203, no. 2 (July 2006): 408–29. http://dx.doi.org/10.1016/j.aim.2005.04.012.

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35

Morin, Baptiste. "Utilisation d’une cohomologie étale équivariante en topologie arithmétique." Compositio Mathematica 144, no. 1 (January 2008): 32–60. http://dx.doi.org/10.1112/s0010437x07003168.

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AbstractSikora has given results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora’s formulas which leads us to clarify and to extend the dictionary of arithmetic topology.
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36

Kaptanoglu, Semra Öztürk. "Betti numbers of fixed point sets and multiplicities of indecomposable summands." Journal of the Australian Mathematical Society 74, no. 2 (April 2003): 165–72. http://dx.doi.org/10.1017/s1446788700003220.

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AbstractLet G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set XCn and the multiplicities of indecomposable summands of M considered as a kCn-module are related via a localization theorem in equivariant cohomology, where Cn is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.
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37

Carrell, Jim, and Kiumars Kaveh. "Springer’s Weyl Group Representation via Localization." Canadian Mathematical Bulletin 60, no. 3 (September 1, 2017): 478–83. http://dx.doi.org/10.4153/cmb-2017-016-9.

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AbstractLet G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra 𝔤. The Springer variety Bx is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of 𝔤 containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of Bx even though W rarely acts on Bx itself. Well-known constructions of this action due to Springer and others use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H*S (Bx), where S is a certain algebraic subtorus of G. This action descends to H*(Bx) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H*S (Bx) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the ûxed point set of a torus action on a space lifts to the full cohomology algebra of the space.
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38

NIEMI, ANTTI J., and KAUPO PALO. "ON QUANTUM INTEGRABILITY AND THE LEFSCHETZ NUMBER." Modern Physics Letters A 08, no. 24 (August 10, 1993): 2311–21. http://dx.doi.org/10.1142/s0217732393003615.

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Certain phase space path integrals can be evaluated exactly using equivalent cohomology and localization in the canonical loop space. Here we extend this to a general class of models. We consider Hamiltonians which are a priori arbitrary functions of the Cartan subalgebra generators of a Lie group which is defined on the phase space. We evaluate the corresponding path integral and find that it is closely related to the infinitesimal Lefschetz number of a Dirac operator on the phase space. Our results indicate that equivariant characteristic classes could provide a natural geometric framework for understanding quantum integrability.
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39

Chang, Huai-Liang, and Jun Li. "A Vanishing Associated With Irregular MSP Fields." International Mathematics Research Notices 2020, no. 20 (April 17, 2020): 7347–96. http://dx.doi.org/10.1093/imrn/rnaa049.

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Abstract In [ 7] and [ 8], the notion of Mixed-Spin-P (MSP) fields is introduced and their ${\mathbb{C}}^\ast $-equivariant moduli space ${{\mathcal{W}}}_{g,\gamma ,{\textbf d}}$ is constructed. In this paper, we prove a vanishing of a class of localization terms in $[(\mathcal{W}_{g,\gamma ,\mathbf{d}})^{\mathbb{C}^*}]^{\textrm{vir}}$, which implies the only quintic FJRW invariants that contribute to the relations derived from the theory of MSP fields are those with pure insertions $2/5$. It is critical in a proof of BCOV Feynman sum formula for quintic Calabi–Yau three-folds.
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40

Knutson, Allen. "A Compactly Supported Formula for Equivariant Localization and Simplicial Complexes of Bialynicki-Birula Decompositions." Pure and Applied Mathematics Quarterly 6, no. 2 (2010): 501–44. http://dx.doi.org/10.4310/pamq.2010.v6.n2.a9.

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41

Chorny, Boris. "Localization with respect to a class of maps II — Equivariant cellularization and its application." Israel Journal of Mathematics 147, no. 1 (December 2005): 141–55. http://dx.doi.org/10.1007/bf02785362.

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42

Paradan, Paul-Emile, and Michéle Vergne. "Witten Non Abelian Localization for Equivariant K-Theory, and the [𝑄,𝑅]=0 Theorem." Memoirs of the American Mathematical Society 261, no. 1257 (September 2019): 0. http://dx.doi.org/10.1090/memo/1257.

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43

Su, Yukun, Guosheng Lin, Yun Hao, Yiwen Cao, Wenjun Wang, and Qingyao Wu. "Self-Supervised Object Localization with Joint Graph Partition." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 2 (June 28, 2022): 2289–97. http://dx.doi.org/10.1609/aaai.v36i2.20127.

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Object localization aims to generate a tight bounding box for the target object, which is a challenging problem that has been deeply studied in recent years. Since collecting bounding-box labels is time-consuming and laborious, many researchers focus on weakly supervised object localization (WSOL). As the recent appealing self-supervised learning technique shows its powerful function in visual tasks, in this paper, we take the early attempt to explore unsupervised object localization by self-supervision. Specifically, we adopt different geometric transformations to image and utilize their parameters as pseudo labels for self-supervised learning. Then, the class-agnostic activation map (CAAM) is used to highlight the target object potential regions. However, such attention maps merely focus on the most discriminative part of the objects, which will affect the quality of the predicted bounding box. Based on the motivation that the activation maps of different transformations of the same image should be equivariant, we further design a siamese network that encodes the paired images and propose a joint graph cluster partition mechanism in an unsupervised manner to enhance the object co-occurrent regions. To validate the effectiveness of the proposed method, extensive experiments are conducted on CUB-200-2011, Stanford Cars and FGVC-Aircraft datasets. Experimental results show that our method outperforms state-of-the-art methods using the same level of supervision, even outperforms some weakly-supervised methods.
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44

Kaneda, Masaharu, and Jiachen Ye. "Equivariant localization of D¯-modules on the flag variety of the symplectic group of degree 4." Journal of Algebra 309, no. 1 (March 2007): 236–81. http://dx.doi.org/10.1016/j.jalgebra.2006.07.023.

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45

Kimura, Takashi, and Ross Sweet. "Adams operations on the virtual K-theory of ℙ(1,n)." Journal of Algebra and Its Applications 16, no. 08 (August 9, 2016): 1750149. http://dx.doi.org/10.1142/s0219498817501493.

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We analyze the structure of the virtual (orbifold) [Formula: see text]-theory ring of the complex orbifold [Formula: see text] and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin–Graham [D. Edidin and W. Graham, Nonabelian localization in equivariant [Formula: see text]-theory and Riemann–Roch for quotients, Adv. Math. 198(2) (2005) 547–582]. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual [Formula: see text]-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual [Formula: see text]-theory ring of [Formula: see text] to the ordinary [Formula: see text]-theory ring of a crepant resolution of the cotangent bundle of [Formula: see text] which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of [Formula: see text] which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin–Jarvis–Kimura [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial [Formula: see text]-theory, Ann. K-Theory (2016)], who proved the latter for [Formula: see text].
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46

Leslie, Spencer, and Gus Lonergan. "Parity sheaves and Smith theory." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 777 (May 8, 2021): 49–87. http://dx.doi.org/10.1515/crelle-2021-0018.

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Abstract Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory of parity complexes in a certain 2-periodic localization of the equivariant constructible derived category D ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} . Under certain assumptions, we use this to define a functor from the category of parity sheaves on X to the category of parity sheaves on the fixed-point locus X ℤ / p ⁢ ℤ {X^{\mathbb{Z}/p\mathbb{Z}}} . This may be thought of as a categorification of Smith theory. When X is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.
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47

PELAYO, ALVARO, and SUSAN TOLMAN. "Fixed points of symplectic periodic flows." Ergodic Theory and Dynamical Systems 31, no. 4 (June 16, 2010): 1237–47. http://dx.doi.org/10.1017/s0143385710000295.

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AbstractThe study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least $\frac {1}{2}\,{\dim M}+1$ fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah–Bott–Berline–Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective—the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.
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48

Hill, Michael A. "Equivariant chromatic localizations and commutativity." Journal of Homotopy and Related Structures 14, no. 3 (November 27, 2018): 647–62. http://dx.doi.org/10.1007/s40062-018-0226-2.

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49

Henn, Hans-Werner, Jean Lannes, and Lionel Schwartz. "Localizations of unstableA-modules and equivariant modp cohomology." Mathematische Annalen 301, no. 1 (January 1995): 23–68. http://dx.doi.org/10.1007/bf01446619.

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50

Škoda, Zoran. "Some Equivariant Constructions in Noncommutative Algebraic Geometry." gmj 16, no. 1 (March 2009): 183–202. http://dx.doi.org/10.1515/gmj.2009.183.

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Abstract We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
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