Generalised cohomology theories are an important tool in algebraic topology. Part of the study of the theories themselves involves looking at their spaces of "operations" and "co-operations". In a paper in the Algebraic Handbook of Topology, Boardman, Johnson, and Wilson described the former as a comonad in a suitable category and the latter as an "enriched Hopf ring". The former of these is very elegant but not all that intuitive, the latter is less elegant and similarly not intuitive - at least, the "enriched" part. The missing part of these descriptions is a non-linear "tensor product", which was actually introduced by Tall and Wraith in 1970. I shall explain the various descriptions and show how both the unstable operations and co-operations have concise and straightforward descriptions as algebraic objects.

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I have currently given this talk three times: in the Sheffield Topology seminar, in the Glasgow Topology seminar, and at the Transpennine Topology Triangle. Although the mathematical content has remained the same, the style and method of delivery has changed considerably. Available here are the slides and notes from the Sheffield and Glasgow Topology seminars.

• Sheffield Topology Seminar:
  • Slides in pdf
  • Handout in pdf
  • Original LaTeX source
• Glasgow Topology Seminar:
  • Slides in pdf. The blank frame is intentional, it is to get the table of contents right.
  • Board notes in pdf
  • Article in pdf
  • Original LaTeX source