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A homotopy -algebra over a Lawvere theory is a model for an -algebra over , when the latter is regarded as an (∞,1)-algebraic theory.
As a model, homotopy -algebras are equivalent to strict simplicial algebras.
For (the syntactic category of) a Lawvere theory with generating object an ordinary algebra over a Lawvere theory functor that preserves products, in that for all the canonical morphism
is an isomorphism.
For write for the free simplicial -algebra on -generators, which is the image of under the Yoneda embedding . (See Lawvere theory for more on this.)
A homotopy -algebra is precisely
The fibrant objects in are precisely the Kan complex-valued co-presheaves. Because is representable, it is cofibrant in (as one easily checks). Therefore the derived hom-spaces between and a degreewise Kan complex-valued may be computed simply as the sSet-hom-objects of the simplicial model category and so the degreewise fibrant being a local object means that all morphisms of sSet-hom-objects
Due to the respect of the hom-functor for limits the expression on the right is
Using the Yoneda lemma the morphism in question is indeed isomorphic to
This observation motivated the following definition.
has a left adjoint
The limits in are easily seen to be limits in the underlying sets. Hence preserves all limits. The statement then follows by observing that the assumptions of the special adjoint functor theorem are met:
it is a well powered category since is and the subobject in are special subobjects in ;
it has a small cogenerating set given by the representables.
Let 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
is a Quillen adjunction which is a Quillen equivalence
This is theorem 1.3 in (Badzioch)
The model structure on homotopy -algebras for 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.)
- Bernard Badzioch, Algebraic theories in homotopy theory Annals of Mathematics, 155 (2002), 895-913 (JSTOR)
the model structure on homotopy -algebras is discussed and its Quillen equivalence to simplcial -algebras is proven.
A related discussion showing that simplicial algebras model all --algebras is in
- Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.
The model structure on homotopy -algebras for CartSp the Lawvere theory of smooth algebras is considered in