on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
equivalences in/of $(\infty,1)$-categories
For $T$ a Lawvere theory and $T Alg$ the category of algebra over a Lawvere theory, there is a model category structure on the category $T Alg^{\Delta^{op}}$ of simplicial $T$-algebras which models the $\infty$-algebras for $T$ rregarded as an (∞,1)-algebraic theory.
Recall that the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C^{op})$ itself is modeled by the model structure on simplicial presheaves
where we regard $C$ as a Kan complex-enriched category and have on the right the sSet-enriched functor category with the projective or injective model structure, and $(-)^\circ$ denoting the full enriched subcategory on fibrant-cofibrant objects.
This says in particular that every weak $(\infty,1)$-functor $f : C \to \infty \mathrm{Grp}$ is equivalent to a rectified one $F : C \to KanCplx$. And $f \in PSh_{(\infty,1)}(C^{op})$ belongs to $Alg_{(\infty,1)}(C)$ if $F$ preserves finite products weakly in that for $\{c_i \in C\}$ 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 $C \to sSet$, which gives an equivalent model for $Alg_{(\infty,1)}(C)$. See model structure on homotopy T-algebras.
(first model structure)
Let $T$ be a category with finite products, and let $T Alg^{\Delta^{op}} \subset Func(T,sSet)$ be the full subcategory of the functor category from $T$ to sSet on those functors that preserve these products.
Then $T Alg^{\Delta^{op}}$ carries the structure of a model category $sAlg(C)_{proj}$ where the weak equivalences and the fibrations are objectwise those in the standard model structure on simplicial sets.
This is due to (Quillen).
The inclusion $i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}$ into the projective model structure on simplicial presheaves evidently preserves fibrations and acyclic fibrations and gives a Quillen adjunction
The total right derived functor
is a full and faithful functor and an object $F \in sPSh(C^{op})$ belongs to the essential image of $\mathbb{R}i$ precisely if it preserves product up to weak homotopy equivalence.
This is due to (Bergner).
It follows that the natural $(\infty,1)$-functor
is an equivalence.
A comprehensive statement of these facts is in HTT, section 5.5.9.
(second model structure)
Let $T$ be the Lawvere theory for commutative associative algebras over a ring $k$. Then $CAlg_k$ becomes a simplicial model category with
weak equivalences the morphisms whose underlying morphusms of simplicial sets are weak equivalences;
fibrations the morphisms $X \to Y$ such that $X \to \pi_0 X \times_{\pi_0 Y} Y$ is a degreewise surjection.
This appears as (GoerssSchemmerhorn, theorem 4.17).
There is a Quillen equivalence between the model structure on simplicial $T$-algebras and the model structure for homotopy T-algebras. (See there).
This is theorem 1.3 in (Badzioch).
Let $T$ be an abelian Lawvere theory, a theory that contains the theory of abelian group, $Ab \to T$. Then every simplicial $T$-algebra has an underlying abelian simplicial group and is necessarily a Kan complex.
The homotopy groups $\pi_*$ of a simplicial abelian $T$-agebra form an $\mathbb{N}$-graded $T$-algebra $\pi_*(A)$
The inclusion of the full subcategory $i : T Alg \hookrightarrow T Alg^{\Delta^{op}}$ of ordinary $T$-algebra as the simplicially constant ones constitutes a Quillen adjunction
from the trivial model structure on $T Alg$.
The derived functor $\mathbb{R} i : T Alg \to Ho(T Alg^{\Delta^{op}})$ is a full and faithful functor.
This allows us to think of ordinary $T$-algebras a sitting inside $\infty$-$T$-algebras.
all this is certainly true for ordinary $k$-algebras. Need to spell out general proof.
A simplicial rings is a simplicial $T$-algebras for $T$ the Lawvere theory of rings.
Let $k$ be an ordinary commutative ring and $T$ the theory of commutative associative algebras over $k$. We write $T Alg$ as $sCAlg_k$ or $CAlg_k^{op}$.
Such simplicial $k$-algebras are discussed for instance in (ToënVezzosi, section 2.2.1).
See at model structure on simplicial Lie algebras.
algebra over an algebraic theory
∞-algebra over an (∞,1)-algebraic theory
The classical reference for the transferred model structure on simplicial $T$-algebras is
The simplicial model structure on ordinary simplicial algebras is in
In
it is discussed that every model category of simplicial $T$-algebras is Quillen equivalent to a left proper model category.
The fact that the model structure on simplicial $T$-algebras serves to model $\infty$-algebras is in
The Quillen equivalence to the model structure on homotopy $T$-algebras is in
Discussion of simplicial commutative associative algbras over a ring in the context of derived geometry is in