geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
Let $G$ be a finite group.
There is a model category-structure on the category
of connective $G$-equivariant cochain complexes (i.e. with differential of degree +1) over the rational numbers, whose
$\mathrm{W}$ – weak equivalences are the quasi-isomorphisms over each $G/H \in G Orbits$;
$Cof$ – cofibrations are the positive-degree wise injections over each $G/H \in G Orbits$;
$Fib$ – fibrations are the morphisms which over each $G/H \in G Orbits$ are degree-wise surjections whose degreewise kernels are injective objects (in the category of vector G-spaces).
For $G = 1$ the trivial group, this reduces to the injective model structure on connective cochain complexes.