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:
A model category structure on the category of dg-coalgebras.
Let $k$ be a field of characteristic 0.
There is a pair of adjoint functors
between the category of dg-Lie algebrasand that of dg cocommutative coalgebras, where the right adjoint sends a dg-Lie algebra $(\mathfrak{g}_\bullet, [-,-])$ to its Chevalley-Eilenberg coalgebra, whose underlying coalgebra is the free graded co-commutative coalgebra on $\mathfrak{g}[1]$ and whose differential is given on the tensor product of two generators by the Lie bracket $[-,-]$.
For (pointers to) the details, see at model structure on dg-Lie algebras – Relation to dg-coalgebras.
There exists a model category structure on $dgCoCAlg_k$ for which
the cofibrations are the (degreewise) injections;
the weak equivalences are those morphisms that become quasi-isomorphisms under the functor $\mathcal{L}$ from prop. .
Moreover, this is naturally a simplicial model category structure.
This is (Hinich98, theorem, 3.1). More details on this are in the relevant sections at model structure for L-infinity algebras.
Throughout, let $k$ be of characteristic zero.
(Chevalley-Eilenberg dg-coalgebra)
Write
for the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to the dg-coalgebra
where on the right the extension of $\partial$ and $[-,-]$ to graded derivations is understood.
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, prop 6.2). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.2).
For $(X,D) \in dgCoCAlg_k$ write
where
$\overline{X} \coloneqq ker(\epsilon)$ is the kernel of the counit, regarded as a chain complex;
$F$ is the free Lie algebra functor (as graded Lie algebras);
on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra derivation
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, prop 6.1). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.1).
The functors from def. and def. are adjoint to each other:
Moreover, for $X \in dgCoCAlg_k$ and $\mathfrak{g} \in dgLieAlg_k$ then the adjoint hom sets are naturally isomorphic
to the Maurer-Cartan elements in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, somewhere). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.5).
The adjunction $(\mathcal{L} \dashv CE)$ from prop. is a Quillen adjunction between then projective model structure on dg-Lie algebras as the model structure on dg-coalgebras
(Hinich 98, lemma 5.2.2, lemma 5.2.3)
Moreover:
In non-negatively graded dg-coalgebras, both Quillen functors $(\mathcal{L} \dashv CE)$ from prop. preserve all quasi-isomorphisms, and both the adjunction unit and the adjunction counit are quasi-isomorphisms.
For dg-algebras in degrees $\geq n \geq 1$ this is (Quillen 76, theorem 7.5). In unbounded degrees this is (Hinich 98, prop. 3.3.2)
The Quillen adjunctin from prop. is a Quillen equivalence:
(Hinich 98, theorem 3.2) using (Quillen 76 II 1.4)
In characteristic zero and in positive degrees the model structure is due to
in non-negative degrees in
and in unbounded degrees in
Vladimir Hinich, DG coalgebras as formal stacks, Journal of Pure and Applied Algebra Volume 162, Issues 2–3, 24 August 2001, Pages 209–250 (arXiv:math/9812034)
Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)
See also
Review with discussion of homotopy limits and homotopy colimits is in