Parametrized homotopy theory

By Jon P. May, J. Sigurdsson. 2004

278 citations recorded by the source corpus.

Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The $qf$-model structure on $\mathcal{K}_B$ Equivariant $qf$-type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho$G\mathcal{K}_B$ and $hG\mathcal{W}_B$ Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and $G$-categories The category of orthogonal $G$-spectra over $B$ Model structures for parametrized $G$-spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/47deb68acd3ef195628e16227fd7111a38e8eb98

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Citations in the corpus

What precisely are E∞ring spaces and E∞ring spectra? (2009, background): “We are following [35].”
HOMOTOPY LIMITS AND COLIMITS AND ENRICHED HOMOTOPY THEORY (2006, methodology): “In this section, we use the term “cofibration” without prejudice as to meaning, adopting from [EKMM97] and [MS06] the terminology q-cofibration for the model structure cofibrations.”
A simple universal property of Thom ring spectra (2014, unknown): “From a more conceptual point of view, work of Lewis and May [LMSM86], Sullivan [Sul05], and May and Sigurdsson [MS06] places the theory of Thom spectra in the context of parametrized spectra and thus reveals the underlying geometry of the construction.”
Principal ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-bundles: general theory (2012, background): “Whereas ordinary cohomology is given by a derived hom-∞-groupoid, twisted cohomology is given by the ∞-groupoid of sections of a local coefficient bundle in an ∞-topos, which in turn is an associated ∞-bundle induced via a representation of an ∞-group G from a G-principal ∞-bundle (this is a geometr”
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture (2011, background): “The fact that we will mostly use these notions for compact Hausdorff spaces makes many things easier, however for a general treatment we refer to [14].”
The Mukai pairing, I: a categorical approach (2007, background): “objects are dual in a monoidal category if and only if the corresponding 1-morphisms are adjoint in the corresponding 2-category-with-one-object. Indeed, taking this point of view, May and Sigurdsson [12] refer to what is here called adjunction as duality. It is at this point that the utility of the”
Topological cyclic homology (2019, background): “In classical language, this is a parametrized spectrum over S in the sense of May–Sigurdsson, [MS06], and it has been discussed in this form by Klein, [Kle07, Section 5].”
Twists of K-theory and TMF (2010, methodology): “Although this category has a model structure induced by the standard model structure on T /X , one of the key insights of May and Sigurdsson is that for the purposes of parametrized homotopy theory it is essential to work with a variant they call the qf -model structure [MS06, 6.2.6].”
Traces in symmetric monoidal categories (2011, background): “In [31], May and Sigurdsson construct a category SpB of parametrized spectra over B, which is symmetric monoidal, has a symmetric monoidal homotopy category Ho(SpB), and admits a suspension functor ΣB,+ from Top/B that is similar to Σ ∞ + .”
Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory (2014, unknown): “For an associative S -algebra R , one can carry out a similar analysis after passing to a coﬁbrant–ﬁbrant replacement of R as an S - algebra, and the pullback description of GL 1 R in fact yields a genuine topological monoid [ 29 , 22.2.3].”
Twisted Differential String and Fivebrane Structures (2009, background): “To stay within the spectrum point of view, May and Sigurdsson suggested [60] that twisted abelian cohomology should instead be formalized in terms of parameterized homotopy theory, where one thinks of P ×G A as a parameterized family of spectra.”