Homology Operations Revisited

1982 ◽  
Vol 34 (3) ◽  
pp. 700-717 ◽  
Author(s):  
Z. Fiedorowicz ◽  
J. P. May

There are two principal kinds of input data for infinite loop space theory, namelyE∞spaces à la Boardman-Vogt [3] and May [7] and Γ-spaces à la Segal [14]. May and Thomason [13] introduced a common generalization and used it to prove the equivalence of the output obtained from these two kinds of input.This suggests that any invariants of one kind of input should have analogs for the other. Homology operations are among the most basic invariants ofE∞spaces, and we here establish the analogous invariants for Γ-spaces. The definition is transparently obvious from the point of view of the common generalization but is at first sight rather surprising and unnatural from the point of view of Γ-spaces alone. Probably for this reason, there is no hint of the possibility of a direct definition of homology operations for Γ-spaces in the literature.

2019 ◽  
Vol 71 (1) ◽  
pp. 207-246
Author(s):  
Bertrand J Guillou ◽  
J Peter May ◽  
Mona Merling ◽  
Angélica M Osorno

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.


2014 ◽  
Vol 7 (4) ◽  
pp. 1077-1117 ◽  
Author(s):  
Matthew Ando ◽  
Andrew J. Blumberg ◽  
David Gepner ◽  
Michael J. Hopkins ◽  
Charles Rezk

2006 ◽  
Vol 205 (1) ◽  
pp. 163-228 ◽  
Author(s):  
A.D. Elmendorf ◽  
M.A. Mandell

Author(s):  
J. P. May

In this final sequel to (9), I shall prove a general consistency statement which seems to me to complete the foundations of infinite loop space theory. In particular, this result will specialize to yield the last step of the proof of the following theorem about the stable classifying spaces of geometric topology.


LingVaria ◽  
2019 ◽  
Vol 14 (28) ◽  
Author(s):  
Katarzyna Prorok

Polysemy and Synonymy in the Reconstruction of the Linguistic Image of Thistle in Polish Folklore In the first part of the article, the author attempts to determine which plant should be described in the entry OSET (‘thistle’) in Słownik stereotypów i symboli ludowych (‘A Dictionary of Folk Stereotypes and Symbols’), as the name oset is polysemic in Polish dialects and can refer not only to thistle (Carduus), but also to other similar plants with prickly leaves/stems and roundish red, purple or pink flowers, most often to cirsium (Cirsium), less often to cottonthistle (Onopordon) or milk thistle (Silybum). Villagers either identify these plants with each other, or distinguish them from each other, but nevertheless count them together in the common ‘category of thistles’. Hence the proposal that all the above mentioned ‘thistles’ should be included in the entry for OSET. The method of defining adopted in the Dictionary…, so-called cognitive definition, allows for it as its purpose is to reconstruct the colloquial, ‘stereotypical’ way of conceptualizing reality; it describes ‘mental objects’, ‘social notions of objects’ – not, as typically in dictionary definitions, words or the real-world objects that correspond to them. In the second part of the article, the author analyses synonyms and hyponyms of all of the ‘thistles’ that have been selected for description. Although from a scientific point of view some of them are only quasi-synonyms, they form a semantically consistent group, and are worth being included in the definition of OSET, because they strengthen or profile the basic image of OSET that has been reconstructed on the basis of other sources (proverbs, songs, tales, legends, descriptions of beliefs and practices, etc.). For example, bodziak (lit. ‘one that gores’), drapacz (lit. ‘one that scratches’) talk about thistle as a sharp, prickly plant; kolkowe ziele (lit. ‘colic herb’) – a plant believed to heal stabbing pains; diabelski oset (lit. ‘devil’s thistle’), diabelskie nasienie (lit. ‘devil’s seed’) – a plant perceived as devilish; czartopłoch (lit. ‘one that scares the devil away’) – an apotropaic plant, etc.


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