# nLab homotopy T-algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A homotopy $T$-algebra over a Lawvere theory $T$ is a model for an $\infty$-algebra over $T$, when the latter is regarded as an (∞,1)-algebraic theory.

As a model, homotopy $T$-algebras are equivalent to strict simplicial algebras.

## Definition

For $T$ (the syntactic category of) a Lawvere theory with generating object $x$ an ordinary algebra over a Lawvere theory functor $T \to Set$ that preserves products, in that for all $n \in \mathbb{N}$ the canonical morphism

$\prod_{i = 1}^n A(p_i) : A(x^n) \to (A(x))^n$

is an isomorphism.

###### Definition

A homotopy $T$-algebra is a functor $A : T \to$ sSet with values in Kan complexes such that for all $n \in \mathbb{N}$ this canonical morphism is a weak homotopy equivalence.

For $n \in \mathbb{N}$ write $F_T(n)$ for the free simplicial $T$-algebra on $n$-generators, which is the image of $x^n$ under the Yoneda embedding $j : T^{op} \to [T,sSet]$. (See Lawvere theory for more on this.)

###### Proposition

A homotopy $T$-algebra is precisely

• a fibrant object in the projective model structure on simplicial presheaves;

• which is a local object with respect to the canonical morphisms

$\coprod F_T(1) \to F_T(n)$

for all $n \in \mathbb{N}$.

###### Proof

The fibrant objects in $[T,sSet]_{proj}$ are precisely the Kan complex-valued co-presheaves. Because $F_T(n)$ is representable, it is cofibrant in $[T,sSet]_{proj}$ (as one easily checks). Therefore the derived hom-spaces between $F_T(\cdots)$ and a degreewise Kan complex-valued $A$ may be computed simply as the sSet-hom-objects of the simplicial model category $[T,sSet]$ and so the degreewise fibrant $A$ being a local object means that all morphisms of sSet-hom-objects

$[T,sSet](F_T(n),A) \to [T,sSet](\coprod_n F_T(1), A) \,.$

Due to the respect of the hom-functor for limits the expression on the right is

$\cdots = \prod_n [T,sSet](F_T(1), A) \,.$

Using the Yoneda lemma the morphism in question is indeed isomorphic to

$A(x^n) \to A(x)^n \,.$

This observation motivated the following definition.

###### Definition

The model category structure for homotopy $T$-algebras is the left Bousfield localization $[T,sSet]_{proj,loc}$ of the projective model structure on simplicial presheaves $[T,sSet]_{proj}$ at the set of morphisms $\{\coprod_n F_T(1) \to F_T(b)\}_{n \in \mathbb{N}}$.

## Properties

###### Proposition

The model structure for homotopy $T$-algebra $[T,sSet]_{proj,loc}$ is a left proper simplicial model category.

###### Proof

Because the model structure on simplicial presheaves is and left Bousfield localization of model categories preserves these properties.

###### Lemma

The inclusion

$i : T Alg^{\Delta^{op}} \hookrightarrow [T,sSet]$

$F : [T,sSet] \to T Alg^{\Delta^{op}}$
###### Proof

The limits in $T Alg$ are easily seen to be limits in the underlying sets. Hence $i$ preserves all limits. The statement then follows by observing that the assumptions of the special adjoint functor theorem are met:

• $T Alg$ is complete;

• it is a well powered category since $[T,Set]$ is and the subobject in $T Alg$ are special subobjects in $[T,Set]$;

• it has a small cogenerating set given by the representables.

###### Remark

An explicit description of $F$ is around HTT, lemma 5.5.9.5.

###### Theorem

Let $T Alg^{\Delta^{op}}_{proj}$ be the category of simplcial T-algebras equipped with the standard model structure on simplicial algebras (with weak equivalences and fibrations the degreewise weak equivalences and fibrations in simplicial sets).

The adjunction from the previous lemma

$T Alg^{\Delta^{op}} \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow} [T,sSet] = [T,Set]^{\Delta^{op}}$

is a Quillen adjunction which is a Quillen equivalence

$T Alg^{\Delta^{op}}_{proj} \simeq [T,sSet]_{proj,loc} \,.$

This is theorem 1.3 in (Badzioch)

## Examples

The model structure on homotopy $T$-algebras for $T =$ CartSp the Lawvere theory of smooth algebras is considered in (Spivak) in the study of derived smooth manifold. (There is also a bit of disucssion of the relation to the model structure on simplicial algebras there.)

## References

In

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

the model structure on homotopy $T$-algebras is discussed and its Quillen equivalence to simplcial $T$-algebras is proven.

A related discussion showing that simplicial $T$ algebras model all $\infty$-$T$-algebras is in

• Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.

The model structure on homotopy $T$-algebras for $T =$ CartSp the Lawvere theory of smooth algebras is considered in

Revised on November 25, 2010 00:26:13 by Urs Schreiber (87.212.203.135)