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). According to (Schwede 97, Lemma 3.1.3), this model structure is proper.
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
Some more details on this model structure are in section 3.1 of
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