The paper ["A history of duality in algebraic topology" by Becker and Gottlieb] (math.purdue.edu/~gottlieb/Bibliography/53.pdf) is a very nice read. Several concepts of duality are discussed, along with their interactions.

From what I've read, your "standard fact" was first proven by Kadeishvili and can be seen in some sources cited as "Kadeishvili's theorem".

Another reference for the $\mathbb{F}_p$-homology of the $K(\mathbb{F}_p, n)$ (following Cartan's method) is in the thesis of Alain Prouté: logique.jussieu.fr/~alp/these_A_Proute-TAC.pdf

There's a theorem of Adams that tells you that the only fiber bundles which have spheres as base space, total space and fiber, are the four Hopf bundles.

I think the paper "Classification theorem for fibre spaces" by Stasheff (1963) should be mentioned. He proves that for a finite CW complex $Y$, if $F$ is the topological monoid of self-homotopy equivalences of $Y$, then $[X,BF]$ is in bijection with "fiber homotopy-equivalence classes of Hurewicz fibrations with fiber homotopy equivalent to $Y$".

@JairoBochi I think I learned the word "porism" by reading Weibel's Introduction to Homological Algebra, so there's at least one modern source where it's used. It would sound fancy IMHO, but it seems perfectly correct. You will have to think what you prefer, to be very precise about your choice of words or to possibly save your reader a little visit to a dictionary :)

Isn't "porism" better?

Snaith, 1979, Algebraic cobordism and K-theory, section 9: I have written this section in terms of spaces (infinite loopspaces) rather than spectra in order to emphasis the familiar space, BU, rather than the more metaphysical spectrum, BU.

For a discussion of why "replace all maps with cofibrations" is not always the way to obtain a cofibrant diagram, see Strom, Modern Classical Homotopy Theory, section 6.4.2, paragraph "A common misconception". He defines what it means for a small category to be "tree-like", and proves that in this case it is true that a diagram is cofibrant iff all the arrows are cofibrant.

I just stumbled on a reference for this fact and I recalled this question. You can find an argument at the beginning of section 1 of John Rognes' "The product on topological Hochschild homology of the integers with mod 4 coefficients", and he gives a reference to a paper of Barratt: "Track groups (II)", 1951.

Almost seven years after you posted this answer, I have a question, if you don't mind. I'm not sure I really understand their construction: the map $G\times G^n \to G^n$, $(g,g_1,\dots,g_n)\mapsto (gg_1,\dots,gg_n)$ does not commute with the bar construction differentials: e.g. taking $n=2$, $gg_1g_2 \neq gg_1gg_2$. What am I missing? Thank you.

Thanks. One more question: when you say "Similarly, a strong symmetric monoidal morphism..." you mean a strong symmetric morphism of symmetric pseudomonoids, right? (I believe a couple of "symmetric" are still missing in that last part, by the way!). Sorry for the silly questions, I haven't dealt with these concepts before.

Thanks a lot for your reply. I'm a bit confused by your last paragraph. When you say "a lax symmetric monoidal functor can be identified with a pseudomonoid", do you mean a symmetric pseudomonoid? What do you mean by "oplax transformation" in this context? (I'm only aware of (op)lax transformations between (op)lax monoidal functors between monoidal categories)

I'm sorry, but again there is something eluding me. At the beginning you define a colax structure on $|-|$. However, you're still assuming (and using) that $F$ is lax, so I don't see what monoidal structure the functors $F|-|$ and $|F-|$ could have (in particular, how to make sense of a monoidal transformation between them?)

It's clearer now, thanks! $$$$ You meant "lax structure of F" rather than "colax", and in the end, you meant "depends on the structure of $\Delta^n \times \Delta^m$" (which is not $\Delta^{n+m})$). As for your last comment, I believe what's key is that $|-|:sSet\to Top$ is strong symmetric monoidal: that's where the non-trivial topology comes in. From this and a couple more properties of $|-|$, I think one can deduce formally that $|-|:sTop\to Top$ is strong symmetric monoidal. I agree with you that we cannot expect the result to hold in general without getting some control on the $|-|$'s.

Thanks for the clarification. However, I don't see why you end up in a coend over $\Delta\times \Delta$. Shouldn't the arrow point to $\int^n X_n \otimes Y_n \otimes \Delta^n \otimes \Delta^n$?