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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.
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Invertible topological field theories
A ‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal ‐category of ‐bordisms (embedded into and equipped with a tangential ‐structure) that lands in the Picard subcategory of the target symmetric monoidal ‐category. We classify these field theories in terms of the cohomology of the ‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the ‐category of bordisms with as an ‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math.202(2009), no. 2, 195–239) in the case , and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the ‐uple case. We also obtain results for the ‐category of ‐bordisms embedding into a fixed ambient manifold , generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN2011(2011), no. 3, 572–608) in the case . We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of ‐vector spaces (for ), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math.25(2013), no. 5, 1067–1106. arXiv:0912.4706).
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- Award ID(s):
- 2204297
- PAR ID:
- 10554625
- Publisher / Repository:
- Journal of Topology
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 17
- Issue:
- 2
- ISSN:
- 1753-8416
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/topo.12335
