nLab model structure on simplicial algebras



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

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

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


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



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 regarded as an (∞,1)-algebraic theory.


First we consider the case of simplicial objects in algebras over an ordinary Lawvere theory:

Then we generalize to the case that the Lawvere theory itself is simplicial:

For algebras over ordinary Lawvere theories

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.


Let TT be the syntactic category of a Lawvere theory, 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 finite products.

Then TAlg Δ opT Alg^{\Delta^{op}} carries the structure of a model category (TAlg Δ op) proj(T Alg^{\Delta^{op}})_{proj} where the weak equivalences and the fibrations are those maps underlying which are weak equivalences or fibrations, respectively, in the classical model structure on simplicial sets.

This is due to (Quillen 67, II.4 theorem 4).

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}) \,.

The total right derived functor

i:Ho(sAlg(C) proj)Ho(sPSh(C op) proj) \mathbb{R}i \;\colon\; 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 products up to weak homotopy equivalence.

This is due to (Badzioch 02, def. 5.2, prop. 5.4, cor. 5.7).

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

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

is an equivalence.

For algebras over simplicial Lawvere theories


Let Γ=(Skel(FinSet */))\Gamma = (Skel(FinSet^{\ast/})) be Segal's category, the opposite category of a skeleton of finite pointed sets.

A simplicial Lawvere theory is a a pointed simplicial category TT equipped with a functor i:ΓTi \;\colon\;\Gamma \to T such that

  1. TT has the same set of objects as Γ\Gamma;

  2. ii is the identity on objects

  3. ii preserves finite products

Given a simplicial theory TT, then a simplicial TT-algebra is a product preserving simplicial functor XX to the simplicial category of pointed simplicial sets. The simplicial set

X(1 +)sSet X(1_+) \in sSet

(the value on the pointed 2-element set) is called the underlying simplicial set of the TT-algebra.

A homomorphism of TT-algebras is a simplicial natural transformation between such functors. Write

TAlgsSetCat T Alg \in sSet Cat

for the resulting simplicial category.

A homomorphism is called a weak equivalence or a fibration if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets. Write

(TAlg) proj (T Alg)_{proj}

for the category equipped with these classes of morphisms.

(Schwede 01, def. 2.1 and def. 2.2 and beginning of section 3)


In the case that the simplicial Lawvere theory TT happens to be simplicially discrete, i.e. an ordinary category, then the category (TAlg) proj(T Alg)_{proj} of simplicial TT-algebras from def. with that from prop. .


For TT a simplicial Lawvere theory (def. ) the category (TAlg) proj(T Alg)_{proj} from def. is a simplicial model category.

This is due to (Reedy 74, theorem I), reviewed in (Schwede 01).

The analogous statement with the classical model structure on simplicial sets replaced by the classical model structure on topological spaces is due to (Schwänzl-Vogt 91)

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

Other cases


Let 𝒞\mathcal{C} be a category with enough projectives. Say that a morphism ff in the category 𝒞 Δ op\mathcal{C}^{\Delta^{\mathrm{op}}} of simplicial objects in 𝒞\mathcal{C} is a weak equivalence or fibration if for every projective object P𝒞P \in \mathcal{C} then Hom(P,f)Hom(P,f) is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets (i.e. a Kan fibration or weak homotopy equivalence, respectively). If now at least one of the following conditions is satisfied,

then this makes 𝒞 Δ op\mathcal{C}^{\Delta^{op}} a simplicial model category.

(Quillen 67, II.4, theorem 4)


Relation to homotopy TT-algebras


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 (Badzioch 02, theorem 1.3).

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 simplicial abelian group and is necessarily a Kan complex.


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


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. I believe this is the Badzioch paper cited above - JB


Simplicial associative 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 (Toën-Vezzosi, section 2.2.1). According to (Schwede 97, Lemma 3.1.3), this model structure is proper.

Simplicial commutative kk-algebras

The following is a variant of the model structure on simplicial commutative algebras that is implied by the above general theorem.


(second model structure)

Let TT be the Lawvere theory for (associative and) commutative algebras over a ring kk. Then (cAlg k) Δ op(cAlg_k)^{\Delta^{op}} becomes a simplicial model category with

  • weak equivalences the morphisms whose underlying morphism of simplicial sets are weak equivalences in the classical model structure on simplicial sets;

  • 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 (Goerss-Schemmerhorn, theorem 4.17).

Simplicial Lie algebras

See at model structure on simplicial Lie algebras.

Simplicial complete Hopf algebras

Complete rational Hopf algebras are not the models of a Lawvere theory, even though in some sense they are close (Quillen 67, bottom of p. 265 (61 of 92)).

But they do satisfy the assumptions of proposition (Quillen 69, appendix B, prop. 2.24). Hence simplicial rational complete Hopf algebras form a simplicial model category.


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

with some further remarks in

  • Dan Quillen, Section II.3 of: Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR, pdf)

The generalization to the case that the theory TT itself is allowed to be simplicial is due to

  • Christopher Reedy, Homology of algebraic theories, Ph.D. Thesis, University of California, San Diego, 1974

and the topological version is due to

More is in

  • Stefan Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1-41 (pdf)


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

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

and the multi-sorted version is in

The simplicial model structure on ordinary simplicial algebras (i.e. simplicial associative algebras) is discussed in

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

Discussion of divided power operations on simplicial algebras is in

  • Benoit Fresse, On the homotopy of simplicial algebras over an operad, Transactions of the AMS, volume 352, number 9 (jstor)

Last revised on August 19, 2023 at 15:33:54. See the history of this page for a list of all contributions to it.