model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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, doi:10.1016/S0022-4049(00)00121-3
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
Last revised on March 22, 2023 at 11:59:41. See the history of this page for a list of all contributions to it.