J. Peter May is a homotopy theorist at the University of Chicago, inventor of operads as a technique for studying infinite loop spaces and spectra.
Peter May’s work makes extensive use of enriched- and model-category theory as power tools in algebraic topology/homotopy theory, notably in discussion of highly structured spectra in MMSS00‘s Model categories of diagram spectra (for exposition see Introduction to Stable homotopy theory – 1-2), or in the discussion of genuine equivariant spectra or K-theory of permutative categories, etc.. While he has co-edited a book collection on higher category theory (Baez-May 10) and eventually had high praise (May 16) for 2-category theory as a tool in algebraic topology/higher algebra, he has vocally warned against seeing abstract (∞,1)-category theory as a replacement for concrete realizations in model category-theory (P. May, MO comment Dec 2013).
Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (Euclid)
Infinite loop space theory revisited (pdf)
On higher algebra (brave new algebra) in stable homotopy theory, i.e. on ring spectra, module spectra etc.:
Rings, modules and algebras in stable homotopy theory, Mathematical Surveys and Monographs Volume 47, AMS 1997 (ISBN:978-0-8218-4303-1, pdf)
On module spectra:
On equivariant stable homotopy theory:
On equivariant complex oriented cohomology theory:
On tensor triangulated categories and traces:
On parametrized stable homotopy theory:
Specifically on 2-category theory as a tool in spectral algebraic geometry, equivariant homotopy theory and infinite loop space-theory:
equivariant homotopy theory (Bredon cohomology, equivariant stable homotopy theory, rational equivariant stable homotopy theory)
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