# nLab model structure on simplicial algebras

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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 $T$ regarded as an (∞,1)-algebraic theory.

## Details

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:

### For algebras over ordinary Lawvere theories

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

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

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

Let $T$ be the syntactic category of a Lawvere theory, 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 finite products.

Then $T Alg^{\Delta^{op}}$ carries the structure of a model category $(T Alg^{\Delta^{op}})_{proj}$ 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 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

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

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 products up to weak homotopy equivalence.

This is due to (Badzioch 02, def. 5.2, prop. 5.4, cor. 5.7).

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

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

is an equivalence.

### For algebras over simplicial Lawvere theories

###### Definition

Let $\Gamma = (Skel(FinSet^{\ast/}))$ be Segal's category, the opposite category of a skeleton of finite pointed sets.

A simplicial Lawvere theory is a a pointed simplicial category $T$ equipped with a functor $i \;\colon\;\Gamma \to T$ such that

1. $T$ has the same set of objects as $\Gamma$;

2. $i$ is the identity on objects

3. $i$ preserves finite products

Given a simplicial theory $T$, then a simplicial $T$-algebra is a product preserving simplicial functor $X$ to the simplicial category of pointed simplicial sets. The simplicial set

$X(1_+) \in sSet$

(the value on the pointed 2-element set) is called the underlying simplicial set of the $T$-algebra.

A homomorphism of $T$-algebras is a simplicial natural transformation between such functors. Write

$T Alg \in sSet Cat$

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

$(T Alg)_{proj}$

for the category equipped with these classes of morphisms.

###### Remark

In the case that the simplicial Lawvere theory $T$ happens to be simplicially discrete, i.e. an ordinary category, then the category $(T Alg)_{proj}$ of simplicial $T$-algebras from def. with that from prop. .

###### Proposition

For $T$ a simplicial Lawvere theory (def. ) the category $(T Alg)_{poj}$ from def. is a simplicial model category.

This is due to (Reedy 74, theorem I), reviewed in (Schwede 01).

The analogous statement with the classical model structure on simplicial sets replaced by the classical model structure on topological spaces is due to (Schwänzl-Vogt 91)

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

### Other cases

###### Proposition

Let $\mathcal{C}$ be a category with enough projectives. Say that a morphism $f$ in the category $\mathcal{C}^{\Delta^{\mathrm{op}}}$ of simplicial objects in $\mathcal{C}$ is a weak equivalence or fibration if for every projective object $P \in \mathcal{C}$ then $Hom(P,f)$ 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,

• every object of $\mathcal{C}^{\Delta^{op}}$ is fibrant;

• $\mathcal{C}$ is closed under colimits and has a small set of projective generators;

then this makes $\mathcal{C}^{\Delta^{op}}$ a simplicial model category.

## 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 (Badzioch 02, theorem 1.3).

### 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 simplicial abelian 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. I believe this is the Badzioch paper cited above - JB

## Examples

### Simplicial associative $k$-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 (Toën-Vezzosi, section 2.2.1). According to (Schwede 97, Lemma 3.1.3), this model structure is proper.

### Simplicial commutative $k$-algebras

The following is a variant of the model structure on simplicial commutative algebras that is implied by the above general theorem.

###### Proposition

(second model structure)

Let $T$ be the Lawvere theory for (associative and) commutative algebras over a ring $k$. Then $(cAlg_k)^{\Delta^{op}}$ becomes a simplicial model category with

• weak equivalences the morphisms whose underlying morphism of simplicial sets are weak equivalences in the classical model structure on simplicial sets;

• 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 (Goerss-Schemmerhorn, theorem 4.17).

### Simplicial complete Hopf algebras

Complete rational Hopf algebras are not the models of a Lawvere theory, even though in some sense they are close (Quillen 67, bottom of p. 265 (61 of 92)).

But they do satisfy the assumptions of proposition (Quillen 69, appendix B, prop. 2.24). Hence simplicial rational complete Hopf algebras form a simplicial model category.

## References

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

with some further remarks in

• Dan Quillen, Section II.3 of: Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR, pdf)

The generalization to the case that the theory $T$ itself is allowed to be simplicial is due to

• Christopher Reedy, Homology of algebraic theories, Ph.D. Thesis, University of California, San Diego, 1974

and the topological version is due to

• Roland Schwänzl, Rainer Vogt, The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories, Archives of Mathematics 56 (1991) 405-411 (doi:10.1007/BF01198229)

More is in

• Stefan Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1-41 (pdf)

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

• Bernard Badzioch, Algebraic theories in homotopy theory, Annals of Mathematics, 155 (2002), 895-913 (JSTOR)

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

• Benoit Fresse, On the homotopy of simplicial algebras over an operad, Transactions of the AMS, volume 352, number 9 (jstor)

Last revised on July 22, 2021 at 08:27:59. See the history of this page for a list of all contributions to it.