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Previously asked on math.stackexchange, but perhaps this is more appropriate for MathOverflow:

Let $X$ be a spectrum. I've heard that the action of the Steenrod algebra on $H^*(X; \mathbb{F}_p)$ and the action of the Dyer-Lashof algebra on the homology of the associated infinite loop space, $H_*(\Omega^{\infty}X; \mathbb{F}_p)$, are "dual" in some sense.

To my knowledge, this has something to do with Koszul duality, but I've never seen this spelled out in detail. Anyone willing to enlighten me on this? Or point me toward a helpful reference?

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    $\begingroup$ Have you seen math.uchicago.edu/~may/PAPERS/11.pdf? The first section describes this duality in detail. $\endgroup$ – Daniel Grady Jun 16 '17 at 13:38
  • $\begingroup$ Daniel- I don't see the discussion you're referring to. Just some mention of Nishida relations and a little about the dual of the Dyer-Lashof algebra. The point here, I think, is that the homology of the infinite looping is very different from the homology of the spectrum... $\endgroup$ – Dylan Wilson Jun 16 '17 at 23:07
  • $\begingroup$ ...(contd): On the other hand, the cohomology together with the action of the Steenrod algebra is kind of an approximation to the spectrum itself (via the Adams spectral sequence). Likewise, the homology of the infinite looping is kind of an approximation to the spectrum itself (via the Miller spectral sequence). that's maybe why one would guess you could move between these two pieces of data by some procedure (Koszul duality here) $\endgroup$ – Dylan Wilson Jun 16 '17 at 23:09
  • $\begingroup$ But I don't know enough to figure out how that goes... or even if it's Koszul duality at play or something else. To me it seems closer to like.. describing spectra (or p-complete spectra) in two different ways as roughly algebras for some monad or comonad and then recording the descent data in each case? $\endgroup$ – Dylan Wilson Jun 16 '17 at 23:14
  • $\begingroup$ @DylanWilson hmm...I may have been too quick to point to that reference. I recalled seeing there that the dual of the Dyer-Lashof algebra was related to a quotient of the cohomology of $K(\mathbb{Z}_p,n))$, and so the Steenrod operations make an appearance...but this seems to indeed be a far stretch from OP's question. $\endgroup$ – Daniel Grady Jun 16 '17 at 23:48
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The original paper on Koszul algebras, [Stewart Priddy, Koszul resolutions. Trans. Amer. Math. Soc. 152 (1970) 39–60], was, in essence, written to explain this example. Well almost: he was considering the Steenrod algebra and the Lambda algebra. The Dyer Lashof algebra is pretty much the Lambda algebra with some unstable side conditions.

Haynes Miller makes the connection very explicitly in [Miller, Haynes, A spectral sequence for the homology of an infinite delooping. Pacific J. Math. 79 (1978), no. 1, 139–155]. This influenced other papers of his, and also my work on the Whitehead conjecture.

An alternative delooping spectral sequence was defined in my paper [The McCord model for the tensor product of a space and a commutative ring spectrum. Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), 213–236, Progr. Math., 215, Birkhäuser, Basel, 2004.] The geometric objects involve the Lie operad, and the associated cohomology operations lead to Steenrod operations.

Dually, a looping spectral sequence was studied here: [Kuhn, Nicholas; McCarty, Jason, The mod 2 homology of infinite loopspaces. Algebr. Geom. Topol. 13 (2013), no. 2, 687–745.] Now the geometric objects involve the commutative operad, and the associated homology operations lead to Dyer-Lashof operations.

The geometric underpinnings of this sort of situation have been studied by Arone and Ching.

So yeah, you have heard right.

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This is not the duality that was asked for in the question, but it does show a way in which the Steenrod algebra structure is related to the Dyer-Lashof algebra structure. The Dyer-Lashof algebra can act both on the homology of infinite loop spaces and $E_{\infty}$ ring spectra. If I consider the latter and look at $H_*(F(X,S^0);\mathbb{F}_p)$ then this has an action of the Dyer-Lashof algebra. It also happens to be isomorphic to the cohomology of $X$ as an algebra. In fact, the action of the Dyer-Lashof algebra "is" the action of the Steenrod algebra (there is an op that needs to be thrown in).

I learned this from chapter 3 of http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf.

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