nLab model structure on reduced simplicial sets

Idea

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

∞Grpd${}^{*/}_{\geq 1} \hookrightarrow$ ∞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 $S$ with a single vertex, hence with $X_0$ the singleton set

$$S_0 \;=\; \ast \,.$$

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

Proposition

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

• weak equivalences

• and cofibrations

are those whose underlying maps are such in the classical model structure on simplicial sets (i.e. the simplicial 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 $U \colon sSet_0 \hookrightarrow sSet$

• a fibration $f \colon X \to Y$ with respect to the reduced model structure (prop. ) maps to a fibration in the classical model structure on simplicial sets (a Kan fibration) precisely if it has the right lifting property against $* \to S^1 \coloneqq \Delta[1]/ \partial \Delta[1]$;

In particular:

• every fibrant object in $sSet_0$ is a fibrant object in $sSet$, hence a Kan complex.

Proof

The first statment is proven in Goerss & Jardine, Ch. V, Lemma 6.6.. The second is an immediate consequence, stated there as as Goerss & Jardine, Ch. V, Corollary 6.8.

Corollary

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

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

As an example:

Proposition

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

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

Proof

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

Relation to pointed simplicial sets

Proposition

The coreflective embedding

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

into pointed simplicial sets (where $i$ the obvious inclusion, and $cn$ 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).

Proof

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

Proposition

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

$$sSet_0 \underoverset {\underset{\Omega_\bullet}{\longrightarrow}} {\overset{\Sigma_\ast}{\longleftarrow}} {\bot} sSet^{\ast/}$$

Proof

By the internal hom construction we have the adjunction

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

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 $\Sigma_\ast$ preserves cofibrations and acyclic cofibrations. Hence, by prop. , 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. ) itself, by composition of the Quillen adjunctions from prop. and prop. .

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

Relation to simplicial groups

Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)

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

$$(G \dashv \overline{W}) \,\colon\, sGr \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{G}{\longleftarrow}} {\;\;\;\;\;\;\;\;\bot\;\;\;\;\;\;\;\;} sSet_0$$

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).

Related concepts

• model structure on simplicial groups

• model structure on reduced simplicial sets

• groupoid object in an (∞,1)-category, ∞-group, looping and delooping

References

Textbook account for the Kan model structure on reduced simplicial sets:

• Paul Goerss, J. F. Jardine, Chapter V in: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

The analog (in fact transfer) of the Joyal model structure for reduced simplicial sets, modelling quasi-categories with a single object:

• Nigel Burke, §2 of: Homotopy Theory of Monoids and Group Completion, PhD thesis, Cambridge (2021) [doi, cam:1810/325358, pdf]