Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
A model category structure on the category of dg-coalgebras.
Let be a field of characteristic 0.
For (pointers to) the details, see at model structure on dg-Lie algebras – Relation to dg-coalgebras.
This is (Hinich98, theorem, 3.1). More details on this are in the relevant sections at model structure for L-infinity algebras.
Relation to dg-Lie algebras
Throughout, let be of characteristic zero.
for the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra to the dg-coalgebra
where on the right the extension of and to graded derivations is understood.
For dg-Lie algebras concentrated in degrees this is due to (Quillen 69, appendix B, prop 6.2). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.2).
is the kernel of the counit, regarded as a chain complex;
is the free Lie algebra functor (as graded Lie algebras);
on the right we are extending as a Lie algebra derivation
For dg-Lie algebras concentrated in degrees 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. 1 and def. 2 are adjoint to each other:
Moreover, for and then the adjoint hom sets are naturally isomorphic
to the Maurer-Cartan elements in the Hom-dgLie algebra from to .
For dg-Lie algebras concentrated in degrees this is due to (Quillen 69, appendix B, somewhere). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.5).
(Hinich 98, lemma 5.2.2, lemma 5.2.3)
For dg-algebras in degrees this is (Quillen 76, theorem 7.5). In unbounded degrees this is (Hinich 98, prop. 3.3.2)
The Quillen adjunctin from prop. 3 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
- Dan Quillen, section II.5 and appendix B of Rational homotopy theory, Annals of Math., 90(1969), 205–295 (JSTOR, pdf)
in non-negative degrees in
and in unbounded degrees in
Review with discussion of homotopy limits and homotopy colimits is in