nLab
model structure on reduced simplicial sets

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

Contents

Idea

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.

Definition

Definition

A reduced simplicial set is a simplicial set SS with a single vertex:

S 0=*. S_0 = * \,.

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

Proposition

There is a model category structure on sSet 0sSet_0 (def. 1) 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).

Properties

Relation to ordinary simplicial sets

Proposition

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

In particular

  • every fibrant object maps to a fibrant object.
Proof

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).

Relation to pointed simplicial sets

Proposition

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. 1 and the coslice model structure under the point of the classical model structure on simplicial sets).

Proof

By prop. 1 the left adjoint preserves cofibrations and acyclic cofibrations (in fact all weak equivalences).

Proposition

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/}
Proof

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. 1, so does its factorization through the model structure on reduced simplicial sets.

Corollary

The operations of reduced suspension and loop space (co-)restrict to a Quillen adjunction on the reduced model structure (prop. 1) itself, by composition of the Quillen adjunctions from prop. 3 and prop. 4.

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

Proposition

The simplicial loop space functor GG and the delooping functor W¯() \overline{W}(-) (discussed at simplicial group) constitute a Quillen equivalence

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

between the model structure on reduced simplicial sets from prop. 1 and model structure on simplicial groups.

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

References

A standard textbook reference is chapter V of

Revised on March 6, 2017 11:48:53 by Urs Schreiber (185.25.95.132)