and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(also nonabelian homological algebra)
In analogy to rational homotopy theory, the idea of real homotopy theory is to study those aspects of homotopy types that are visible when the ground ring is the real numbers, such as their real cohomology-groups and the tensor product of their homotopy groups with the additive abelian group of real numbers. This is of central relevance in relation to differential cohomology theory, which needs real instead of rational coefficients in order to connect to de Rham complexes of differential forms on smooth manifolds.
But a technical issue with generalizing the fundamental theorem of dg-algebraic rational homotopy theory to the case of real homotopy theory is that the PL de Rham-Quillen adjunction between simplicial sets and connective dgc-algebras – which does exist over any ground field $k$ of characteristic zero and relates to the one over the rational numbers by derived extension of scalars (all reviewed in FSS 2020, Sec. 3.2) – models $k$-localization only for $k = \mathbb{Q}$ the rational numbers – then: rationalization.
From the point of algebra, the issue is (see this Remark) that other $k$ are not solid rings, so that the iterations of the tensor products $(-) \otimes_{\mathbb{Z}} k$ on homotopy groups are not, in general, isomorphic to each other, for $k \neq \mathbb{Q}$.
On the other hand, from the point of topology the issue is that the real numbers $k = \mathbb{R}$ are naturally a topological ring, whose topology, however, is ignored in the standard Quillen adjunction between simplicial sets and connective dgc-algebras over the real numbers.
Therefore, the real homotopy theory of Brown & Szczarba 1995 replaces, in the fundamental theorem of dg-algebraic rational homotopy theory, the plain real cohomology-groups by real continuous cohomology and plain dgc-algebras with topological algebras.
The authors of (Brown-Szczarba 95) show how the Bousfield-Gugenheim approach to rational homotopy theory may be extended to the framework of continuous cohomology. In the usual theory, the extension from the rationals to the reals has always been effected by a formal tensoring with the reals. While this allows a connection to de Rham theory, it has always been preferable to have a direct construction of real homotopy theory akin to the Postnikov tower-like construction of rational homotopy theory. Unfortunately, in the usual situation, such a construction is impossible due to the incompatibility of the algebra of $\mathbb{R}$ (as an abelian group) with its natural topology. The authors of (Brown-Szczarba 95) remedy this problem by creating a theory of minimal models which incorporates the usual topology of the reals and a generalization of van Est's theorem saying that $H^*(K(\mathbb{R},n); \mathbb{R})$ is a freely generated algebra on one generator (a result which is not true in the discrete topology case). By using the natural topology of $\mathbb{R}$, there is an immediate connection to continuous cohomology and applications such as characteristic classes of foliations.
(from the Zentralblatt review of (Brown-Szczarba 95))
If one thinks of simplicial topological spaces as simplicial presheaves on (some small version of) Top, and hence as presentations for topological $\infty$-stacks, then localization of these at the topological real line coefficient is the topological analog of passing to schematic homotopy types, in the sense of Toën 2006, which generalizes to D-topological $\infty$-groupoids (and further to smooth $\infty$-groupoids etc.) along the lines of Stel 2010.
On real homotopy theory (rational homotopy theory over the real numbers) using continuous real cohomology and topological dgc-algebras:
Edgar Brown, Robert H. Szczarba, Real and Rational Homotopy Theory, Chapter 17 in: Handbook of Algebraic Topology, Elsevier, 1995 869-915 (doi:10.1016/B978-044481779-2/50018-3, ZB)
(also introducing Borel-equivariant rational homotopy theory)
with emphasis on globally Kan fibrant simplicial topological spaces (such as simplicial topological groups):
with arbitrary fundamental group:
Edgar Brown, Robert H. Szczarba, Continuous cohomology and Real homotopy type II Asterisque 191, Societe Mathematique De France (1990) (numdam:AST_1990__191__45_0)
Edgar Brown, Robert H. Szczarba, Real and rational homotopy theory for spaces with arbitrary fundamental group, Duke Mathematical Journal 71. (1993): 229-316 (doi:10.1215/S0012-7094-93-07111-6)
Discussion of the plain Quillen adjunction between simplicial sets and connective dgc-algebras over the real numbers, and its relation to rationalization followed by derived extension of scalars: