related by the Dold-Kan correspondence
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 regarded as an (∞,1)-algebraic theory.
First we consider the case of simplicial objects in algebras over an ordinary Lawvere theory:
Then we generalize to the case that the Lawvere theory itself is simplicial:
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
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.
Then carries the structure of a model category where the weak equivalences and the fibrations are those maps underlying which are weak equivalences or fibrations, respectively, in the classical model structure on simplicial sets.
This is due to (Quillen 67, II.4 theorem 4).
The total right derived functor
This is due to (Badzioch 02, def. 5.2, prop. 5.4, cor. 5.7).
It follows that the natural -functor
is an equivalence.
(the value on the pointed 2-element set) is called the underlying simplicial set of the -algebra.
for the resulting simplicial category.
A homomorphism is called a weak equivalence or a fibration if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets. Write
for the category equipped with these classes of morphisms.
A comprehensive statement of these facts is in HTT, section 5.5.9.
Let be a category with enough projectives. Say that a morphism in the category of simplicial objects in is a weak equivalence or fibration if for every projective object then is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets (i.e. a Kan fibration or weak homotopy equivalence, respectively). If now at least one of the following conditions is satisfied,
then this makes a simplicial model category.
This is (Badzioch 02, theorem 1.3).
The homotopy groups of a simplicial abelian -agebra form an -graded -algebra
from the trivial model structure on .
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.
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 .
The following is a variant of the model structure on simplicial commutative algebras that is implied by the above general theorem.
(second model structure)
weak equivalences the morphisms whose underlying morphism of simplicial sets are weak equivalences in the classical model structure on simplicial sets;
fibrations the morphisms such that is a degreewise surjection.
This appears as (Goerss-Schemmerhorn, theorem 4.17).
The classical reference for the transferred model structure on simplicial -algebras is
with some extra remarks in
The generalization to the case that the theory itself is allowed to be simplicial is due to
and the topological version is due to
More is in
The fact that the model structure on simplicial -algebras serves to model -algebras is in
and the multi-sorted version is in
The simplicial model structure on ordinary simplicial algebras (i.e. simplicial associative algebras) is discussed in
Discussion of simplicial commutative associative algbras over a ring in the context of derived geometry is in
Discussion of divided power operations on simplicial algebras is in