nLab
model structure on reduced simplicial sets

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Contents

Idea

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

∞Grpd ≥1 */↪ ∞Grpd */ ≃ 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 S with a single vertex:

S 0=*.S_0 = * \,.

Write sSet 0⊂ 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 0 whose

  • weak equivalences

  • and cofibrations

are those in the standard model structure on simplicial sets.

This appears as (GoerssJardine, ch V, prop. 6.2).

Properties

Proposition

The simplicial loop space functor G and the delooping functor W¯(−) (discussed at simplicial group) constitute a Quillen equivalence

(G⊣W¯):sGr→W¯←GsSet 0(G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0

with the model structure on simplicial groups.

This appears as (GoerssJardine, ch. V prop. 6.3).

Proposition

Under the forgetful functor U:sSet 0↪sSet

  • a fibration f:X→Y maps to a fibration precisely if it has the right lifting property against *→S 1:=Δ[1]/∂Δ[1];

In particular

  • every fibrant object maps to a fibrant object.

The first statment appears as (GoerssJardine, ch. V, lemma 6.6.). The second (an immediate consequence) appears as (GoerssJardine, ch. V, corollary 6.8).

References

A standard textbook reference is chapter V of

Revised on April 14, 2012 10:23:31 by Urs Schreiber (82.113.106.163)
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