# nLab model structure on simplicial algebras

model category

for ∞-groupoids

## Theorems

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

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.

## Details

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

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

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 $(\infty,1)$-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

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

is a homotopy equivalence of Kan complexes.

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

###### Proposition

(first model structure)

Let $T$ be a category with finite products, and let $T Alg^{\Delta^{op}} \subset Func(T,sSet)$ 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

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

The total right derived functor

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

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

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

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 $CAlg_k$ 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, $Ab \to T$. 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 $i : T Alg \hookrightarrow T Alg^{\Delta^{op}}$ of ordinary $T$-algebra as the simplicially constant ones constitutes a Quillen adjunction

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

from the trivial model structure on $T Alg$.

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.

## 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 $T Alg$ as $sCAlg_k$ or $CAlg_k^{op}$.

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

## 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

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

• 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

Revised on September 18, 2014 09:41:21 by Urs Schreiber (185.26.182.29)