An official website of the United States government
Abstract We give an operadic definition of a genuine symmetric monoidal $$G$$-category, and we prove that its classifying space is a genuine $$E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $$G$$-categories to genuine permutative $$G$$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $$G$$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $$G$$-categories as input to an equivariant infinite loop space machine that gives genuine $$\Omega $$-$$G$-spectra as output.
more »
« less
Models of G–spectra as presheaves of spectra
Let G be a finite group. We give Quillen equivalent models for the category of G–spectra as categories of spectrally enriched functors from explicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point–set-level categories of G–spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality.
more »
« less
- Award ID(s):
- 2003204
- PAR ID:
- 10552049
- Editor(s):
- Greenlees, John
- Publisher / Repository:
- Mathematical Sciences Publishers
- Date Published:
- Journal Name:
- Algebraic & Geometric Topology
- Volume:
- 24
- Issue:
- 3
- ISSN:
- 1472-2747
- Page Range / eLocation ID:
- 1225 to 1275
- Subject(s) / Keyword(s):
- equivariant stable homotopy theory, spectral Mackey functor, G–spectra, Atiyah duality
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We compute the RO(G)‐graded equivariant algebraic K‐groups of a finite field with an action by its Galois group G. Specifically, we show these K‐groups split as the sum of an explicitly computable term and the well‐studied RO(G)‐graded coefficient groups of the equivariant Eilenberg–MacLane spectrum HZ. Our comparison between the equivariant K‐theory spectrum and HZ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where G has prime order, we provide an explicit presentation of the equivariant K‐groups.more » « less
-
We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable -categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as -monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group , we give a symmetric monoidal stratification of genuine -spectra. In the case that is finite, this expresses genuine -spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory.more » « less
-
Tambara functors are an equivariant generalization of rings that appear as the homotopy groups of genuine equivariant commutative ring spectra. In recent work, Blumberg and Hill have studied the corresponding algebraic structures, called bi-incomplete Tambara functors, that arise from ring spectra indexed on incomplete G-universes. We answer a conjecture of Blumberg and Hill by proving a generalization of the Hoyer–Mazur theorem in the bi-incomplete setting. Bi-incomplete Tambara functors are characterized by indexing categories which parametrize incomplete systems of norms and transfers. In the course of our work, we develop several new tools for studying these indexing categories. In particular, we provide an easily checked, combinatorial characterization of when two indexing categories are compatible in the sense of Blumberg and Hill.more » « less
-
NA (Ed.)We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QK_{T}(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QK_{T}(G/B). In type A_{n-1}, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SL_{n}/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.2140/agt.2024.24.1225
