model structure on reduced simplicial sets



Model category theory

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The model structure on reduced simplicial sets is a presentation of the full sub-(∞,1)-category

∞Grpd 1 */{}^{*/}_{\geq 1} \hookrightarrow ∞Grpd */{}^{*/} \simeq Top */{}^{*/}

of pointed ∞-groupoids on those that are connected.

By the looping and delooping-equivalence, this is equivalent to the (∞,1)-category of ∞-groups and this equivalence is presented by a Quillen equivalence to the model structure on simplicial groups.



A reduced simplicial set is a simplicial set SS with a single vertex, hence with X 0X_0 the singleton set

S 0=*. S_0 \;=\; \ast \,.

Write sSet 0sSet_0 \subset sSet for the full subcategory of the category of simplicial sets on those that are reduced.


There is a model category structure on sSet 0sSet_0 (def. ) whose

  • weak equivalences

  • and cofibrations

are those in the classical model structure on simplicial sets (i.e. the weak homotopy equivalences and the monomorphisms, respectively.)

This appears as (Goerss & Jardine, Ch V, Prop. 6.2).


Relation to ordinary simplicial sets


Under the forgetful functor U:sSet 0sSetU \colon sSet_0 \hookrightarrow sSet

In particular


The first statment appears as (Goerss & Jardine, Ch. V, Lemma 6.6.). The second is an immediate consequence. It appears as (Goerss & Jardine, Ch. V, Corollary 6.8).


Let f:XYf \colon X \longrightarrow Y be a fibration in the model structure on reduced simplicial sets (Prop. ) such that both XX and YY are Kan complexes. Then ff is a Kan fibration precisely if it induces a surjection on the first simplicial homotopy group π 1(f):π 1(X)π 1(Y)\pi_1(f) \colon \pi_1(X) \twoheadrightarrow \pi_1(Y).

(Goerss & Jardine, Ch. V, Cor. 6.9)

As an example:


Let 𝒢 1ϕ𝒢 2\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2 be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces W¯𝒢 1W¯(ϕ)W¯𝒢 2\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2 is a Kan fibration if and only if ϕ\phi is a surjection on connected components: π 0(ϕ):π 0(𝒢 1)π 0(𝒢 1)\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1).

(Goerss & Jardine, Ch. V, Cor. 6.9)

Since W¯()\overline{W}(-) is a right Quillen functor to the model structure on reduced simplicial sets (Prop. ) it follows that W¯(ϕ)\overline{W}(\phi) is in any case a fibration in that model structure. Hence Cor. implies that W¯(ϕ)\overline{W}(\phi) is a Kan fibration precisely of π 1W¯(ϕ)\pi_1 \circ \overline{W}(\phi) is surjective. But π 1W¯=π 0\pi_1 \circ \overline{W} = \pi_0, by this Prop.

Relation to pointed simplicial sets


The coreflective embedding

sSet */cnisSet 0 sSet^{\ast/} \underoverset {\underset{cn}{\longrightarrow}} {\overset{i}{\longleftarrow}} {\bot} sSet_{0}

into pointed simplicial sets (where ii the obvious inclusion, and cncn forms the 1st Eilenberg subcomplex over the basepoint) is a Quillen adjunction (with respect to the reduced model structure from prop. and the coslice model structure under the point of the classical model structure on simplicial sets).


By prop. the left adjoint preserves cofibrations and acyclic cofibrations (in fact all weak equivalences).


The operation of reduced suspension Σ *\Sigma_\ast (smash product with the simplicial circle S 1Δ[1]/Δ[1]S^1 \coloneqq \Delta[1]/\partial \Delta[1]) and forming loop space Ω *\Omega_\ast (pointed mapping space out of the circle) constitute a Quillen adjunction

sSet 0Ω Σ *sSet */ sSet_0 \underoverset {\underset{\Omega_\bullet}{\longrightarrow}} {\overset{\Sigma_\ast}{\longleftarrow}} {\bot} sSet^{\ast/}

By the internal hom construction we have the adjunction

sSet */Ω Σ *sSet */ sSet^{\ast/} \underoverset {\underset{\Omega_\bullet}{\longrightarrow}} {\overset{\Sigma_\ast}{\longleftarrow}} {\bot} sSet^{\ast/}

But by the formula at smash product the reduced suspension clearly lands in reduced simplicial sets, so that we do have the restricted adjunction as claimed. Now by the fact that the classical model structure on simplicial sets is a simplicial model category, the above is a Quillen adjunction and Σ *\Sigma_\ast preserves cofibrations and acyclic cofibrations. Hence, by prop. , so does its factorization through the model structure on reduced simplicial sets.


The operations of reduced suspension and loop space (co-)restrict to a Quillen adjunction on the reduced model structure (prop. ) itself, by composition of the Quillen adjunctions from prop. and prop. .

sSet 0Ω Σ *sSet */cnisSet 0 sSet_0 \underoverset {\underset{\Omega_\bullet}{\longrightarrow}} {\overset{\Sigma_\ast}{\longleftarrow}} {\bot} sSet^{\ast/} \underoverset {\underset{cn}{\longrightarrow}} {\overset{i}{\longleftarrow}} {\bot} sSet_{0}

Relation to simplicial groups


(Quillen equivalence between simplicial groups and reduced simplicial sets)

The simplicial loop space functor GG and the simplicial classifying space-construction W¯()\overline{W}(-) constitute a Quillen equivalence

(GW¯):sGrW¯GsSet 0 (G \dashv \overline{W}) \colon sGr \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{G}{\longleftarrow}} {\;\;\;\;\;\;\;\;\bot\;\;\;\;\;\;\;\;} sSet_0

between the model structure on reduced simplicial sets from prop. and model structure on simplicial groups.

This appears as (Goerss-Jardine, ch. V prop. 6.3).


Textbook account:

Last revised on July 5, 2021 at 08:48:38. See the history of this page for a list of all contributions to it.