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
The category of simplicial abelian groups carries the structure of a model category $sAbGrp_{proj}$ whose weak equivalences and fibrations are those of the underlying morphisms in the classical model structure on simplicial sets, hence whose
weak equivalences are the simplicial weak homotopy equivalences,
fibrations are the Kan fibrations
of underlying simplicial sets.
Hence the free/underlying-adjoint functors (where the free functor produces free simplicial abelian groups $\mathbb{Z}[-]$) is a Quillen adjunction with the classical model structure on simplicial sets $sSet_{Qu}$:
With respect to the degreewise tensor product of abelian groups this is a monoidal model category (Schwede & Shipley 2003, p. 312 (26 of 48)).
The Dold-Kan correspondence yields (see there for more) a Quillen equivalence to the projective model structure on connective chain complexes.
(Quillen 67, Section II.4 item 5, see also Schwede-Shipley 03, section 4.1, p.17, Jardine 03, Lemma 1.5).
In fact, much more is true: all five classes of maps in a model category (weak equivalences, (acyclic) cofibrations, and (acyclic) fibrations) are preserved and reflected by both of these equivalences. That is to say, each model structure is obtained from the other one by transferring it along the corresponding equivalence of categories.
Due to:
Further discussion:
Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories , Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv:math.AT/0209342, euclid:euclid.agt/1513882376)
J. F. Jardine, Lemma 1.5 in: Presheaves of chain complexes, K-theory 30.4 (2003): 365-420 (pdf, pdf)
Last revised on July 14, 2021 at 10:50:43. See the history of this page for a list of all contributions to it.