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GavinWraith comments on "A topos-theoretic look at Steenrod operations" (62659)
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https://nforum.ncatlab.org/discussion/7762/?Focus=62659#Comment_62659Sun, 14 May 2017 18:06:31 -0400GavinWraith
I recently found some notes written in 1992 after receiving a preprint by N.J.Kuhn, Generic Representations of the Finite General Linear Groups and the Steenrod Algebra, I & II, which I am sorry to say disappeared years ago. I realized that some of its results could be recast in a form that would be suggestive for topos-theorists. According to some handwritten notes, Two Theorems of J.Lannes , Frank Adams had known these results. I mention it because it might tickle a reader to verify it for themself. The mod p Steenrod algebra can be described, I believe, as the endo-Hopf-algebra of the graded symmetric algebra of the generic differential graded F_p-vector space. Recall that commutative (dg, ... whatever) algebras are enriched, tensored and cotensored over the Cartesian closed category of commutative (dg ... whatever) coalgebras. This holds in particular for (dg ... whatever) algebras in a topos. For mod 2 forget the differential bit (which gives the Bockstein). The key is the Milnor dual basis description of the Steenrod algebra and the fact that the only extra natural linear operations we have available are got from the Frobenius p-th power.
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