The hunting of the Hopf ring

Stacey, Andrew; Whitehouse, Sarah

doi:10.4310/hha.2009.v11.n2.a6

Cited by 6 publications

(12 citation statements)

References 13 publications

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“…Evaluation on objects of D takes the form of a functor D → Alg M (Sets), encoding both the invariants assigned by these functors and the natural operations on them. These are examples of multi-sorted algebraic theory in the sense of Bergner [14], closely related to the work of [18,89]. We will return to the discussion of this structure in §4.4.…”

Section: Multicategoriesmentioning

confidence: 95%

E n -spectra and Dyer-Lashof operations

Lawson¹

2020

Handbook of Homotopy Theory

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Cohomology operations are absolutely essential in making cohomology an effective tool for studying spaces. In particular, the mod-p cohomology groups of a space X are enhanced with a binary cup product, a Bockstein derivation, and Steenrod's reduced power operations; these satisfy relations such as graded-commutativity, the Cartan formula, the Adem relations, and the instability relations [92]. The combined structure of these cohomology operations is very effective in homotopy theory because of three critical properties.These operations are natural. We can exclude the possibility of certain maps between spaces because they would not respect these operations.These operations are constrained. We can exclude the existence of certain spaces because the cup product and power operations would be incompatible with the relations that must hold. AcknowledgementsThe author would like to thank for discussions related to this material, and Haynes Miller for a careful reading of an earlier draft of this paper. The author also owes a significant long-term debt to Charles Rezk for what understanding he possesses.

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Section: Multicategoriesmentioning

confidence: 95%

E n -spectra and Dyer-Lashof operations

Lawson¹

2020

Handbook of Homotopy Theory

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“…We can now give a direct proof of a key result of Stacey and Whitehouse [12,Corollary 5.4]. The original proof is an application of a very abstract, but more general result.…”

Section: Topological Plethoriesmentioning

confidence: 97%

“…A priori, cohomology operations do not naturally fit into this framework due to considerations of grading and topologies on the algebraic structures. However, as shown by Stacey and Whitehouse [12], in sufficiently nice cases the operations admit the structure of a graded topologised plethory and this acts on completed cohomology algebras. A related approach by Bauer considers formal plethories, thus avoiding completion issues [3].…”

Section: Introductionmentioning

confidence: 93%

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The plethory of operations in complex topological K-theory

Mycroft,

Whitehouse

2020

Preprint

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We provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as a plethory, originally due to Tall and Wraith. Stacey and Whitehouse showed this structure is present on the cohomology operations for a suitable generalised cohomology theory. We compute an explicit expression for the plethory of operations for complex topological K-theory. This is formulated in terms of a plethory enhanced with structure corresponding to the looping of operations. In this context we show that the familiar λ-operations generate all the operations.

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“…In [SW09], we introduced the notion of a Tall-Wraith monoid as a way of describing the algebraic structure on the set of unstable operations of a suitable generalised cohomology theory. In brief, for a variety of algebras, V, a Tall-Wraith V-monoid is "that which can act on V-algebras".…”

Section: Introductionmentioning

confidence: 99%

Tall-Wraith Monoids

Stacey¹,

Whitehouse²

2011

Preprint

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Tall-Wraith monoids were introduced in [SW09] to describe the algebraic structure on the set of unstable operations of a suitable generalised cohomology theory. In this paper we begin the study of Tall-Wraith monoids in an algebraic and categorical setting. We show that for V a variety of algebras, applying the free V-algebra functor to a monoid in Set produces a Tall-Wraith monoid. We also study the example of the Tall-Wraith monoid defined by the self set-maps of a finite ring, an example closely related to the original motivation for Tall-Wraith monoids. * The authors acknowledge the support of the EPSRC, grant no.: GR/S76823 /01.

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The hunting of the Hopf ring

Cited by 6 publications

References 13 publications

E n -spectra and Dyer-Lashof operations

E n -spectra and Dyer-Lashof operations

The plethory of operations in complex topological K-theory

Tall-Wraith Monoids

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