There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at the nlab page on cohomology

I would like to understand if there's any non-contrived way that Floer-theoretic invariants fit into this picture.

I'm far from an expert. I'm aware of a number of works elucidating homotopy-theoretic aspects of Floer theory: for example the construction of Cohen-Jones-Segal, and the infinity category of Lagrangian cobordisms defined by Nadler-Tanaka. Do any of these constructions provide an answer to the question above?

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    $\begingroup$ I know nothing about Floer homology, but the observation of Urs Schreiber is just a complicated way of stating that we like when our cohomology theories are (co)representable (be it by spectra, motives or whatever object you find suitable). $\endgroup$ – Denis Nardin Jul 12 '18 at 7:47
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    $\begingroup$ Wikipedia mentions some recent work of Manolescu extending Cohen-Jones-Segal, seemingly in the direction of your question $\endgroup$ – მამუკა ჯიბლაძე Jul 12 '18 at 11:14
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    $\begingroup$ You can also do worse than study the works of Thomas Kragh $\endgroup$ – Thomas Rot Jul 12 '18 at 11:49
  • $\begingroup$ Thanks! I was aware of the Manolescu and Abouzaid-Kragh lines of work, but didn't want to turn the question into a glossary of vaguely relates things none of which I understand very well :( $\endgroup$ – zzz Jul 12 '18 at 13:20
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    $\begingroup$ Cohomology \emph{theory} generally has a precise meaning i.e. there are certain axioms your cohomology groups satisfy(e.g. excision). Floer (co)homologies don't usually satisfy those literally. Though it is of course interesting to think about situations where Mayer-Vietoris, gluing formulas etc hold. $\endgroup$ – algebrachallenged Jul 12 '18 at 14:16

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