nLab
model structure on simplicial algebras

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Homotopy theory

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

Contents

Idea

For TT a Lawvere theory and TAlgT Alg the category of algebra over a Lawvere theory, there is a model category structure on the category TAlg Δ opT Alg^{\Delta^{op}} of simplicial TT-algebras which models the \infty-algebras for TT rregarded as an (∞,1)-algebraic theory.

Details

Recall that the (∞,1)-category of (∞,1)-presheaves PSh (,1)(C op)PSh_{(\infty,1)}(C^{op}) itself is modeled by the model structure on simplicial presheaves

PSh (,1)(C op)[C,sSet] , PSh_{(\infty,1)}(C^{op}) \simeq [C, sSet]^\circ \,,

where we regard CC 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 (,1)(\infty,1)-functor f:CGrpf : C \to \infty \mathrm{Grp} is equivalent to a rectified one F:CKanCplxF : C \to KanCplx. And fPSh (,1)(C op)f \in PSh_{(\infty,1)}(C^{op}) belongs to Alg (,1)(C)Alg_{(\infty,1)}(C) if FF preserves finite products weakly in that for {c iC}\{c_i \in C\} a finite collection of objects, the canonical natural morphism

F(c 1×,c n)F(c 1)××F(c n) F(c_1 \times \cdots, \c_n) \to F(c_1) \times \cdots \times F(c_n)

is a homotopy equivalence of Kan complexes.

We now look at model category structure on strictly product preserving functors CsSetC \to sSet, which gives an equivalent model for Alg (,1)(C)Alg_{(\infty,1)}(C). See model structure on homotopy T-algebras.

Proposition

(first model structure)

Let TT be a category with finite products, and let TAlg Δ opFunc(T,sSet)T Alg^{\Delta^{op}} \subset Func(T,sSet) be the full subcategory of the functor category from TT to sSet on those functors that preserve these products.

Then TAlg Δ opT Alg^{\Delta^{op}} carries the structure of a model category sAlg(C) projsAlg(C)_{proj} where the weak equivalences and the fibrations are objectwise those in the standard model structure on simplicial sets.

This is due to (Quillen).

The inclusion i:sAlg(C)sPSh(C op) proji : 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) projisPSh(C op). sAlg(C)_{proj} \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}} sPSh(C^{op}) \,.
Proposition

The total right derived functor

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

is a full and faithful functor and an object FsPSh(C op)F \in sPSh(C^{op}) belongs to the essential image of i\mathbb{R}i precisely if it preserves product up to weak homotopy equivalence.

This is due to (Bergner).

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

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

is an equivalence.

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

Proposition

(second model structure)

Let TT be the Lawvere theory for commutative associative algebras over a ring kk. Then CAlg kCAlg_k becomes a simplicial model category with

  • weak equivalences the morphisms whose underlying morphusms of simplicial sets are weak equivalences;

  • fibrations the morphisms XYX \to Y such that Xπ 0X× π 0YYX \to \pi_0 X \times_{\pi_0 Y} Y is a degreewise surjection.

This appears as (GoerssSchemmerhorn, theorem 4.17).

Properties

Relation to homotopy TT-algebras

Theorem

There is a Quillen equivalence between the model structure on simplicial TT-algebras and the model structure for homotopy T-algebras. (See there).

This is theorem 1.3 in (Badzioch).

Homotopy groups

Let TT be an abelian Lawvere theory, a theory that contains the theory of abelian group, AbTAb \to T. Then every simplicial TT-algebra has an underlying abelian simplicial group and is necessarily a Kan complex.

Observation

The homotopy groups π *\pi_* of a simplicial abelian TT-agebra form an \mathbb{N}-graded TT-algebra π *(A)\pi_*(A)

Observation

The inclusion of the full subcategory i:TAlgTAlg Δ opi : T Alg \hookrightarrow T Alg^{\Delta^{op}} of ordinary TT-algebra as the simplicially constant ones constitutes a Quillen adjunction

(π 0i):TAlgiπ 0TAlg Δ op (\pi_0 \dashv i) : T Alg \stackrel{\overset{\pi_0}{\leftarrow}}{\underset{i}{\hookrightarrow}} T Alg^{\Delta^{op}}

from the trivial model structure on TAlgT Alg.

The derived functor i:TAlgHo(TAlg Δ op)\mathbb{R} i : T Alg \to Ho(T Alg^{\Delta^{op}}) is a full and faithful functor.

This allows us to think of ordinary TT-algebras a sitting inside \infty-TT-algebras.

all this is certainly true for ordinary kk-algebras. Need to spell out general proof.

Examples

Simplicial ordinary kk-algebras

A simplicial rings is a simplicial TT-algebras for TT the Lawvere theory of rings.

Let kk be an ordinary commutative ring and TT the theory of commutative associative algebras over kk. We write TAlgT Alg as sCAlg ksCAlg_k or CAlg k opCAlg_k^{op}.

Such simplicial kk-algebras are discussed for instance in (ToënVezzosi, section 2.2.1).

Simplicial Lie algebras

See at model structure on simplicial Lie algebras.

References

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

  • Dan Quillen, Homotopical Algebra Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)

The simplicial model structure on ordinary simplicial algebras is in

In

it is discussed that every model category of simplicial TT-algebras is Quillen equivalent to a left proper model category.

The fact that the model structure on simplicial TT-algebras serves to model \infty-algebras is in

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

The Quillen equivalence to the model structure on homotopy TT-algebras is in

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

Discussion of simplicial commutative associative algbras over a ring in the context of derived geometry is in

Revised on September 18, 2014 09:41:21 by Urs Schreiber (185.26.182.29)