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model structure on simplicial algebras

Idea

For $T$ a Lawvere theory and $T\mathrm{Alg}$ the category of algebra over a Lawvere theory, there is a model category structure on the category $T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}$ of simplicial $T$-algebras which models the $\mathrm{\infty }$-algebras for $T$ rregarded as an (∞,1)-algebraic theory.

Details

Recall that the (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{(\mathrm{\infty },1)}(C^{\mathrm{op}})$ itself is modeled by the model structure on simplicial presheaves

$${\mathrm{PSh}}_{(\mathrm{\infty },1)}(C^{\mathrm{op}})\simeq [C,\mathrm{sSet}{]}^{\circ },$$PSh_{(\infty,1)}(C^{op}) \simeq [C, sSet]^\circ \,,

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 $(\mathrm{\infty },1)$-functor $f:C\to \mathrm{\infty }\mathrm{Grp}$ is equivalent to a rectified one $F:C\to \mathrm{KanCplx}$. And $f\in {\mathrm{PSh}}_{(\mathrm{\infty },1)}(C^{\mathrm{op}})$ belongs to ${\mathrm{Alg}}_{(\mathrm{\infty },1)}(C)$ if $F$ preserves finite products weakly in that for $\{c_i\in C\}$ a finite collection of objects, the canonical natural morphism

$$F(c_1\times \cdots ,c_n)\to F(c_1)\times \cdots \times F(c_n)$$F(c_1 \times \cdots, \c_n) \to F(c_1) \times \cdots \times F(c_n)

is a homotopy equivalence of Kan complexes.

We now look at model category structure on strictly product preserving functors $C\to \mathrm{sSet}$, which gives an equivalent model for ${\mathrm{Alg}}_{(\mathrm{\infty },1)}(C)$. See model structure on homotopy T-algebras.

Proposition

(first model structure)

Let $T$ be a category with finite products, and let $T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}\subset \mathrm{Func}(T,\mathrm{sSet})$ be the full subcategory of the functor category from $T$ to sSet on those functors that preserve these products.

Then $T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}$ carries the structure of a model category $\mathrm{sAlg}(C{)}_{\mathrm{proj}}$ 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 $i:\mathrm{sAlg}(C)\hookrightarrow \mathrm{sPSh}(C^{\mathrm{op}}{)}_{\mathrm{proj}}$ into the projective model structure on simplicial presheaves evidently preserves fibrations and acyclic fibrations and gives a Quillen adjunction

$$\mathrm{sAlg}(C{)}_{\mathrm{proj}}\overleftarrow{\underset{i}{\hookrightarrow }}\mathrm{sPSh}(C^{\mathrm{op}}).$$sAlg(C)_{proj} \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}} sPSh(C^{op}) \,.

Proposition

The total right derived functor

$$ℝi:\mathrm{Ho}(\mathrm{sAlg}(C{)}_{\mathrm{proj}})\to \mathrm{Ho}(\mathrm{sPSh}(C^{\mathrm{op}}{)}_{\mathrm{proj}})$$\mathbb{R}i : Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj})

is a full and faithful functor and an object $F\in \mathrm{sPSh}(C^{\mathrm{op}})$ belongs to the essential image of $ℝi$ precisely if it preserves product up to weak homotopy equivalence.

This is due to (Bergner).

It follows that the natural $(\mathrm{\infty },1)$-functor

$$(\mathrm{sAlg}(C{)}_{\mathrm{proj}}{)}^{\circ }\stackrel{}{\to }{\mathrm{PSh}}_{(\mathrm{\infty },1)}(C^{\mathrm{op}})$$(sAlg(C)_{proj})^\circ \stackrel{}{\to} PSh_{(\infty,1)}(C^{op})

is an equivalence.

A comprehensive statement of these facts is in HTT, section 5.5.9.

Proposition

(second model structure)

Let $T$ be the Lawvere theory for commutative associative algebras over a ring $k$. Then ${\mathrm{CAlg}}_k$ 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).

Properties

Relation to homotopy $T$-algebras

Theorem

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).

Homotopy groups

Let $T$ be an abelian Lawvere theory, a theory that contains the theory of abelian group, $\mathrm{Ab}\to T$. Then every simplicial $T$-algebra has an underlying abelian simplicial group and is necessarily a Kan complex.

Observation

The homotopy groups ${\pi }_{*}$ of a simplicial abelian $T$-agebra form an $\mathbb{N}$-graded $T$-algebra ${\pi }_{*}(A)$

Observation

The inclusion of the full subcategory $i:T\mathrm{Alg}\hookrightarrow T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}$ of ordinary $T$-algebra as the simplicially constant ones constitutes a Quillen adjunction

$$({\pi }_0⊣i):T\mathrm{Alg}\stackrel{\stackrel{{\pi }_0}{\leftarrow }}{\underset{i}{\hookrightarrow }}T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}$$(\pi_0 \dashv i) : T Alg \stackrel{\overset{\pi_0}{\leftarrow}}{\underset{i}{\hookrightarrow}} T Alg^{\Delta^{op}}

from the trivial model structure on $T\mathrm{Alg}$.

The derived functor $ℝi:T\mathrm{Alg}\to \mathrm{Ho}(T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}})$ is a full and faithful functor.

This allows us to think of ordinary $T$-algebras a sitting inside $\mathrm{\infty }$-$T$-algebras.

all this is certainly true for ordinary $k$-algebras. Need to spell out general proof.

Examples

Simplicial ordinary $k$-algebras

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 $T\mathrm{Alg}$ as ${\mathrm{sCAlg}}_k$ or ${\mathrm{CAlg}}_k^{\mathrm{op}}$.

Such simplicial $k$-algebras are discussed for instance in (ToënVezzosi, section 2.2.1).

Simplicial Lie algebras

See at model structure on simplicial Lie algebras.

Related concepts

• model structure on monoids

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

  • model structure on algebras over a monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

  • model structure on algebras over an operad

References

The classical reference for the transferred model structure on simplicial $T$-algebras is

• Dan Quillen, Homotopical Algebra Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)

The simplicial model structure on ordinary simplicial algebras is in

• Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods (pdf)

In

• Charles Rezk, Every homotopy theory of simplicial algebras admits a proper model (math/0003065) ,

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 $\mathrm{\infty }$-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 $T$-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

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv) .