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I am collecting some basics on rational models for possibly non-free circle actions, due to
I thought I’d write the main result into circle action (?) but I am now splitting off ingredients. First of all:
Does KS stand for anything?
It’s for “Koszul-Sullivan” in Halperin 83 for dg-algebras. But Roig uses “KS-extension” specifically for the variant for dg-modules, so I thought I’d follow that. But maybe one should merge the entry with the one on Sullivan models, not sure.
I have added the statement of the minimal KS-model for $S^4 // S^1$ to 4-sphere – circle action.
I have changed the title to “minimal dg-module”.
The concept actually goes back to Roig’s thesis, from 1992. I have added further references.
So how to think of the minimal dg-modules in Roig-Saleri 02, page 2: they should be rational models of parameterized spectra, I suppose.
Let’s see. Let $X$ be a simply connected space and $A(X)$ the minimal Sullivan model for its rationalization. Then I suspect the homotopy category of $A(X)$-dg-modules is equivalent to rational parameterized $H \mathbb{Q}$-module spectra over $X$. (Is this right?)
Then consider the canonical map $S^4/S^1 \to S^3$ (from the homotopy quotient to the naive quotient of the canonical $S^1$-action on $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$). Consider the fiberwise stabilization and rationalize everything. The result, under the above equivalence, should equivalently be the minimal $A(S^3)$-dg-module (here) produced by Roig-Salieri 00, p. 2.
Does that sound plausible?
I was wondering about the ’general abstract’ account of rational homotopy, which I guess is still to appear rational homotopy theory in an (infinity,1)-topos. The pointer there goes to function algebras on infinity-stacks.
There’s something fracture theorem-like going on in rationalization, but that’s only for spectra?
The entry on rational homotopy theory is badly in need of improvement.
The hints on the abstract picture there refer to the perspective given by Toën in his “Champs affine”, which is discussed in some detail at “function algebras on $\infty$-stacks”: Localizing at the line gives a $\mathcal{O} \dashv Spec$-adjunction, and composing this with the inclusion $const$ of locally constant $\infty$-stacks gives the Sullivan functor $\mathcal{O} \circ const \;\colon\; \inftyGrpd \simeq L_{whe} Top \longrightarrow dgAlg^{op}$.
There are however other abstract perspectives on rational homotopy theory, which seem different in nature.
For instance forming supension spectra $\Sigma^\infty$ is an oplax monoidal $\infty$-functor, so it sends co-monoids in spaces to co-monoids in spectra, hence to $\infty$-co-algebras. Now every space is canonically a co-monoid, namely via the diagonal map. Hence every suspension spectrum is canonically an $\infty$-co-algebra in spectra. The rationalization of this is a rational $\infty$-co-algebra, and these are equivalently just the $L_\infty$-algebras of rational homotopy theory. In fact this may be lifted to “integral homotopy theory”.
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