model structure on dg-coalgebras
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.
There is a pair of adjoint functors
between the category of dg-Lie algebras (on unbounded chain complexes) and that of dg cocommutative coalgebras, where the right adjoint sends a dg-Lie algebra to its “Chevalley-Eilenberg coalgebra”, whose underlying coalgebra is the free graded co-commutative coalgebra on and whose differential is given on the tensor product of two generators by the Lie bracket .
This is (Hinich98, theorem, 3.1). More details on this are in the relevant sections at model structure for L-infinity algebras.
Relation to the model structure on dg-Lie algebras
(Hinich98, lemma 5.3.2)
Hence is a Quillen adjunction to the model structure on dg-algebras.
This is (Hinich98, theore, 3.2).
Dan Quillen, Rational homotopy theory , Annals of Math., 90(1969), 205–295. (see Appendix B)
Ezra Getzler, Paul Goerss, A model category structure for differential graded coalgebras (ps, pdf)
Vladimir Hinich, Homological algebra of homotopy algebras , Comm. in algebra, 25(10)(1997), 3291–3323.
Vladimir Hinich, DG coalgebras as formal stacks (arXiv:math/9812034)
Review with discussion of homotopy limits and homotopy colimits is in
Revised on January 16, 2016 02:39:28
by Urs Schreiber