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I finally learned about the general abstract story behind the notion of orientation in $R$-cohomology, for $R$ an $E_\infty$-ring, in terms of trivialization of $GL_1(R)$-associated $\infty$-bundles – from this lecture by Mike Hopkins
I added some remark about that to orientation in generalized cohomology. Needs more polishing and expansion, but I have to interrupt for the moment.
Another discussion of orientation is also in Stong’s notes :)
I have added to orientation in generalized cohomology a section with the traditional definition (here). It includes statement and proof (here) that traditonally defined universal multiplicative $E$-orientation for vector bundles with $G$-structure is equivalently the class of a ring spectrum homomorphism $M G \longrightarrow E$.
Added a rather unusual reference
with Latin section titles.
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