and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
A minimal dg-module (Roig 92, Roig 94, section 1) is a minimal model in the context of the homotopy theory of dg-modules.
Hence over dgc-algebras in non-positve degree, minimal dg-modules are the analogue of minimal Sullivan model as one passes from dg-algebras to (just) dg-modules. Minimal KS-extensions hence play the role of formal duals of minimal fibration in some applications of rational homotopy theory.
(In Halperin 83 it has “Koszul-Sullivan extensions” for relative Sullivan algebras, and “KS” in “KS-models” refers to that usage.)
Throughout, let $A$ be a dg-algebra. Eventually this is thought of as being a Sullivan model for the rationalization of the quotient of a topological space by a circle action.
We take all differentials to have degree +1. For $V$ a vector space and $n$ a natural number, we write $V[n]$ for the chain complex concentrated on $V$ in degree $n$.
(Hirsch extension of dg-modules)
Let $N$ be a dg-module over $A$ and let $n \in \mathbb{N}$ be a natural number. Then a degree $n$ Hirsch extension of $N$ is a monomorphisms of dg-modules of the form
given by a choice of
a vector space $V$
a linear map $d_V \;\colon\; V \to N_{n+1}$
where the differential $d_{\phi}$ is $d_N$ on $N$, is $d_A$ on $A$, and is given on $V$ by $\phi$ followed by the action $\rho$ of $A$ on $V$:
It follows that for $N_1, N_2$ two $A$-dg-modules then homomorphisms $f$ out of a Hirsch extension of the former (def. )
is equivalently
a homomorphism of dg-modules $h \colon N_1 \to N_2$;
a linear map $g \colon V \to (N_2)_{n}$
such that
The following defines a kind of minimal cofibrations of dg-modules.
(minimal KS-extension)
For $N$ a dg-module over $A$, then a minimal KS-extension of $N$ is a certain transfinite composition of Hirsch extensions, namely a monomorphism
equipped with an exhaustive filtration $\{\hat N(n,q)\}_{n,q \in \mathbb{N}}$ such that:
$\hat N(0,0) \simeq N$;
the inclusions $\hat N(n,q) \hookrightarrow \hat N(n,q+1)$ are Hirsch extensions (def. ).
$N(n+1,0) = \underset{\longrightarrow}{\lim}_q \hat N(n,q)$ (colimit over the sequence of Hirsch extensions in the previous degree).
Accordingly, a minimal KS-factorization of a morphism $N_1 \to N_2$ of $A$-dg-modules is a factorization as a minimal KS-extension followed by a quasi-isomorphism
Finally a minimal KS-model is a dg-module $N$ such that $0 \hookrightarrow N$ is a minimal KS-fibration.
Let the ground field be of characteristic zero.
Let $f \colon N_1 \longrightarrow N_2$ be a morphism of $A$-dg-modules such that it induces a monomorphism in degree-0 cochain cohomology, $H^0(f) \colon H^0(N_1) \hookrightarrow H^0(N_2)$ then it admits a minimal KS-factorization (def. ).
In particular, every dg-module has a minimal KS-model (def. ).
Roig & Saralegi-Aranguren 00, theorem 1.3.1
Agustí Roig, Alguns punts d’àlgebra homotòpica, Barcelona (1992)
Agustí Roig, Minimal resolutions and other minimal models, Publicacions Matemàtiques (1993) Volume: 37, Issue: 2, page 285-303 (web)
Agustí Roig, Formalizability of dg modules and morphisms of cdg algebras, Volume 38, Issue 3 (1994), 434-451 (euclid)
Igor Kriz, Peter May, section IV.3 of Operads, Algebras, Modules and Motives, 1994 (pdf)
Agustí Roig, Martintxo Saralegi-Aranguren, Minimal Models for Non-Free Circle Actions, Illinois Journal of Mathematics, volume 44, number 4 (2000) (arXiv:math/0004141)
See also
Steve Halperin, Lectures on minimal models, Mem. Soc. Math. Franc. no 9/10 (1983) (web)
Flavio da Silveira, Rational homotopy theory of fibrations, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) (pdf)
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