nLab
homotopy T-algebra

Idea

A homotopy $T$-algebra over a Lawvere theory $T$ is a model for an $\mathrm{\infty }$-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 $T\to \mathrm{Set}$ that preserves products, in that for all $n\in \mathbb{N}$ the canonical morphism

$$\prod _{i=1}^{n}A(p_i):A(x^n)\to (A(x){)}^n$$\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 $j:T^{\mathrm{op}}\to [T,\mathrm{sSet}]$. (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

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

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

Proof

The fibrant objects in $[T,\mathrm{sSet}{]}_{\mathrm{proj}}$ are precisely the Kan complex-valued co-presheaves. Because $F_T(n)$ is representable, it is cofibrant in $[T,\mathrm{sSet}{]}_{\mathrm{proj}}$ (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 $[T,\mathrm{sSet}]$ and so the degreewise fibrant $A$ being a local object means that all morphisms of sSet-hom-objects

$$[T,\mathrm{sSet}](F_T(n),A)\to [T,\mathrm{sSet}](\coprod _n{F}_T(1),A).$$[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

$$\cdots =\prod _n[T,\mathrm{sSet}](F_T(1),A).$$\cdots = \prod_n [T,sSet](F_T(1), A) \,.

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

$$A(x^n)\to A(x{)}^n.$$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 $[T,\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ of the projective model structure on simplicial presheaves $[T,\mathrm{sSet}{]}_{\mathrm{proj}}$ 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 $[T,\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ 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

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

has a left adjoint

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

Proof

The limits in $T\mathrm{Alg}$ 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:

• $T\mathrm{Alg}$ is complete;

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

• 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 $T{\mathrm{Alg}}_{\mathrm{proj}}^{{\Delta }^{\mathrm{op}}}$ 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

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

is a Quillen adjunction which is a Quillen equivalence

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

Related concepts

• algebra over a monad

  ∞-algebra over an (∞,1)-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

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

• David Spivak, Derived smooth manifolds (arXiv:0810.5174)