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**The hunting of the Hopf ring.**
*(English)*
Zbl 1194.55019

The unstable cohomology operations for a generalized cohomology theory form a set with a rich algebraic structure. Previous descriptions of this structure (for suitable nice theories) have depicted it as a monad on a certain category, or as the dual of an enriched Hopf ring. The most amenable description, however, has been to depict this set as the dual of a Hopf ring, discarding its enrichment, and this is the description that has been most studied.

Since it is the enrichment which captures the key operation of composition, this omission is unfortunate and the paper under review aims to rectify this, offering a new algebraic description with composition at its heart.

The paper describes the set of unstable operations as forming a graded, completed Tall-Wraith monoid. D. O. Tall and G. C. Wraith in [Proc. Lond. Math. Soc., III. Ser. 20, 619–643 (1970; Zbl 0226.13007)] introduced these monoids to describe the algebraic structure necessary to act on a certain category; for example, a ring is a Tall-Wraith monoid for the category of abelian groups. The bulk of the paper is concerned with adapting this notion to cope with the grading (which is relatively straightforward) and the topology.

Since it is the enrichment which captures the key operation of composition, this omission is unfortunate and the paper under review aims to rectify this, offering a new algebraic description with composition at its heart.

The paper describes the set of unstable operations as forming a graded, completed Tall-Wraith monoid. D. O. Tall and G. C. Wraith in [Proc. Lond. Math. Soc., III. Ser. 20, 619–643 (1970; Zbl 0226.13007)] introduced these monoids to describe the algebraic structure necessary to act on a certain category; for example, a ring is a Tall-Wraith monoid for the category of abelian groups. The bulk of the paper is concerned with adapting this notion to cope with the grading (which is relatively straightforward) and the topology.

Reviewer: Martin D. Crossley (Swansea)

### MSC:

55S25 | \(K\)-theory operations and generalized cohomology operations in algebraic topology |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

16W99 | Associative rings and algebras with additional structure |