Algebras and modules
Model category presentations
Geometry on formal duals of algebras
In as far as an algebraic theory or Lawvere theory is nothing but a small category with finite products and an algebra for the theory a product-preserving functor to Set, the notion has an evident generalization to higher category theory and in particular to (∞,1)-category theory.
An -Lawvere theory is (given by a syntactic -category that is) an (∞,1)-category with finite (∞,1)-products. An -algebra for the theory is an (∞,1)-functor ∞Grpd that preserves these products.
The -category of ∞-algebras over an (∞,1)-algebraic theory is the full sub-(∞,1)-category
of the (∞,1)-category of (∞,1)-presheaves on on the product-preserving -functors
In a full -category theoretic context this appears as HTT, def. 18.104.22.168. A definition in terms of simplicially enriched categories and the model structure on sSet-categories to present -categories is in Ros. The introduction of that article lists further and older occurences of this definition.
Let be an (∞,1)-category with finite products. Then
is an accessible localization of the (∞,1)-category of (∞,1)-presheaves (on the opposite).
So in particular it is a locally presentable (∞,1)-category.
is a compactly generated (∞,1)-category.
The -Yoneda embedding factors through .
The full subcategory is stable under sifted colimits.
This is HTT, prop. 22.214.171.124.
There are various model category presentations of .
Recall that the (∞,1)-category of (∞,1)-presheaves itself is modeled by the model structure on simplicial presheaves
where we regard as a Kan complex-enriched category and have on the right the sSet-enriched functor category with the projective or injective model structure, and denoting the full enriched subcategory on fibrant-cofibrant objects.
This says in particular that every weak -functor is equivalent to a rectified on . And belongs to if preserves finite products weakly in that for a finite collection of objects, the canonical natural morphism
is a homotopy equivalence of Kan complexes.
If is an ordinary category with products, hence an ordinary Lawvere theory, then such a functor is called a homotopy T-algebra. There is a model category structure on these (see there).
We now look at model category structure on strictly product preserving functors , which gives an equivalent model for . See model structure on simplicial T-algebras.
This is due to (Quillen).
The inclusion into the projective model structure on simplicial presheaves evidently preserves fibrations and acylclic fibrations and gives a Quillen adjunction
This is due to (Bergner).
It follows that the natural -functor
is an equivalence.
A comprehensive statement of these facts is in HTT, section 5.5.9.
For (the syntactic category of) an ordinary algebraic theory (a Lawvere theory) let be the category of its ordinary algebras, the ordinary product-preserving functors .
We may regard as an -category and consider its -algebras. By the above discussion, these are modeled by product-presering functors . But this are equivalently simplicial objects in -algebras
There is a standard model structure on simplicial T-algebras and we find that simplicial -1-algebras model --algebras.
For an ordinary Lawvere theory, there is also a model category structure on ordinary functors that preserve the products only up to weak equivalence. Such functors are called homotopy T-algebras.
This model structure is equivalent to the model structure on simplicial T-algebras (see homotopy T-algebra for details) but has the advantage that it is a left proper model category.
There is a notion of simplicial algebraic theory that captures some class of -algebraic theories. For the moment see section 4 of (Rezk)
A pregeometry (for structured (∞,1)-toposes) is a (multi-sorted) -algebraic theory. A structure -sheaf on an (∞,1)-topos in the sense of structured (∞,1)-toposes is an -algebra over this theory
in the -topos – a special one satisfying extra conditions that make it indeed behave like a sheaf of function algebras .
Symmetric monoidal -Categories and -algebras
There is a -algebraic theory whose algebras in (∞,1)Cat are symmetric monoidal (∞,1)-categories. Hence monoids in these algebras are E-∞ algebras (see monoid in a monoidal (∞,1)-category).
This is in (Cranch). For more details see (2,1)-algebraic theory of E-infinity algebras.
The model structure presentation for the -category of -algebras goes back all the way to
- Dan Quillen, Homotopical Algebra Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)
A characterization of -categories of -algebras in terms of sifted colimits is given in
- J. Rosicky On homotopy varieties (pdf)
using the incarnation of -categories as simplicially enriched categories.
An -categorical perspective on these homotopy-algebraic theories is given in
- Andre Joyal, The theory of quasi-categories and its applications, lectures at CRM Barcelona February 2008, draft hc2.pdf_
from page 44 on.
A detailed account in the context of a general theory of (∞,1)-category of (∞,1)-presheaves is the context of section 5.5.8 of
The model category presentations of -algebras is studied in
where it is shown that every such model is Quillen equivalent to a left proper model category. The article uses a monadic definition of -algebras.
A discussion of homotopy T-algebras and their strictification is in
- Bernard Badzioch, Algebraic theories in homotopy theory Annals of Mathematics, 155 (2002), 895-913 (JSTOR)
and for multi-sorted theories in
- Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.
A discussion of E-∞ algebra-structures in terms of -algebraic theories is in