Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Paths and cylinders
For a Lawvere theory and the category of algebra over a Lawvere theory, there is a model category structure on the category of simplicial -algebras which models the -algebras for rregarded as an (∞,1)-algebraic theory.
Recall that the (∞,1)-category of (∞,1)-presheaves itself is modeled by the model structure on simplicial presheaves
where we regard as a Kan complex-enriched category and have on the right the sSet-enriched functor category with the projective or injective model structure, and denoting the full enriched subcategory on fibrant-cofibrant objects.
This says in particular that every weak -functor is equivalent to a rectified one . And belongs to if preserves finite products weakly in that for a finite collection of objects, the canonical natural morphism
is a homotopy equivalence of Kan complexes.
We now look at model category structure on strictly product preserving functors , which gives an equivalent model for . See model structure on homotopy T-algebras.
This is due to (Quillen).
The inclusion into the projective model structure on simplicial presheaves evidently preserves fibrations and acyclic fibrations and gives a Quillen adjunction
This is due to (Bergner).
It follows that the natural -functor
is an equivalence.
A comprehensive statement of these facts is in HTT, section 5.5.9.
(second model structure)
Let be the Lawvere theory for commutative associative algebras over a ring . Then becomes a simplicial model category with
weak equivalences the morphisms whose underlying morphusms of simplicial sets are weak equivalences;
fibrations the morphisms such that is a degreewise surjection.
This appears as (GoerssSchemmerhorn, theorem 4.17).
Relation to homotopy -algebras
This is theorem 1.3 in (Badzioch).
Let be an abelian Lawvere theory, a theory that contains the theory of abelian group, . Then every simplicial -algebra has an underlying abelian simplicial group and is necessarily a Kan complex.
The homotopy groups of a simplicial abelian -agebra form an -graded -algebra
The inclusion of the full subcategory of ordinary -algebra as the simplicially constant ones constitutes a Quillen adjunction
from the trivial model structure on .
The derived functor is a full and faithful functor.
This allows us to think of ordinary -algebras a sitting inside --algebras.
all this is certainly true for ordinary -algebras. Need to spell out general proof.
Simplicial ordinary -algebras
A simplicial rings is a simplicial -algebras for the Lawvere theory of rings.
Let be an ordinary commutative ring and the theory of commutative associative algebras over . We write as or .
Such simplicial -algebras are discussed for instance in (ToënVezzosi, section 2.2.1). According to (Schwede 97, Lemma 3.1.3), this model structure is proper.
Simplicial Lie algebras
See at model structure on simplicial Lie algebras.
The classical reference for the transferred model structure on simplicial -algebras is
- Dan Quillen, Homotopical Algebra Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)
The simplicial model structure on ordinary simplicial algebras is in
Some more details on this model structure are in section 3.1 of
- Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997), pp. 77-104, pdf.
it is discussed that every model category of simplicial -algebras is Quillen equivalent to a left proper model category.
The fact that the model structure on simplicial -algebras serves to model -algebras is in
- Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007. .
The Quillen equivalence to the model structure on homotopy -algebras is in
- Bernard Badzioch, Algebraic theories in homotopy theory Annals of Mathematics, 155 (2002), 895-913 (JSTOR)
Discussion of simplicial commutative associative algbras over a ring in the context of derived geometry is in