nLab
model structure on simplicial sets

under construction

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

Simplicial sets are the archetypical combinatorial “model” for the (∞,1)-category of (compactly generated weakly Hausdorff) topological spaces and equivalently that of ∞-groupoids, as well as a standard model for the (∞,1)-category of (∞,1)-categories (∞,1)Cat itself.

This statement is made precise by the existence of the structure of a model category on sSet, called the classical model structure that is a presentation for the (infinity,1)-category Top, as well as the Joyal model structure which similarly is a presentation of the (,1)(\infty,1)-category (,1)Cat(\infty,1)Cat.

Classical Model Structure

The classical model structure – or Quillen model structure sSet QuillensSet_{Quillen} on sSet has the following distinguished classes of morphisms:

Definition
  • The cofibrations CC are simply the monomorphisms f:XYf : X \to Y which are precisely the levelwise injections, i.e. the morphisms of simplicial sets such that f n:X nY nf_n : X_n \to Y_n is an injection of sets for all nn \in \mathbb{N}.

  • The weak equivalences WW are weak homotopy equivalences, i.e., morphisms whose geometric realization is a weak homotopy equivalence of topological spaces.

  • The fibrations FF are the Kan fibrations, i.e., maps f:XYf : X \to Y which have the right lifting property with respect to all horn inclusions.

    Λ k[n] X f Δ[n] Y. \array{ \Lambda^k[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.
  • A morphism f:XYf : X \to Y of fibrant simplicial sets / Kan complexes is a weak equivalence precisely if it induces an isomorphism on all simplicial homotopy groups.

  • All simplicial sets are cofibrant with respect to this model structure.

  • The fibrant objects are precisely the Kan complexes.

Proposition

The acyclic fibrations (i.e. the maps that are both fibrations as well as weak equivalences) between Kan complexes are precisely the morphisms f:XYf : X \to Y that have the right lifting property with respect to all inclusions Δ[n]Δ[n]\partial \Delta[n] \hookrightarrow \Delta[n] of boundaries of nn-simplices into their nn-simplices

Δ[n] X f Δ[n] Y. \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.

This appears spelled out for instance as (Goerss-Jardine, theorem 11.2).

In fact:

Proposition

sSet QuillensSet_{Quillen} is a cofibrantly generated model category with

  • generating cofibrations the boundary inclusions Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n];

  • generating acyclic cofibrations the horn inclusions Λ i[n]Δ[n]\Lambda^i[n] \to \Delta[n].

Theorem

The singular simplicial complex-functor and geometric realization

(||Sing):Top QuillenSing||sSet Quillen ({\vert -\vert}\dashv Sing) : Top_{Quillen} \stackrel{\overset{{\vert -\vert}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}

constitutes a Quillen equivalence with the standard model structure on topological spaces.

For more on this see homotopy hypothesis.

Characterisations of weak homotopy equivalences

Theorem

Let WW be the smallest class of morphisms in sSetsSet satisfying the following conditions:

  1. The class of monomorphisms that are in WW is closed under pushout, transfinite composition, and retracts.
  2. WW has the two-out-of-three property in sSetsSet and contains all the isomorphisms.
  3. For all natural numbers nn, the unique morphism Δ[n]Δ[0]\Delta [n] \to \Delta [0] is in WW.

Then WW is the class of weak homotopy equivalences.

Proof
  • First, notice that the horn inclusions Λ 0[1]Δ[1]\Lambda^0 [1] \hookrightarrow \Delta [1] and Λ 1[1]Δ[1]\Lambda^1 [1] \hookrightarrow \Delta [1] are in WW.
  • Suppose that the horn inclusion Λ k[m]Δ[m]\Lambda^k [m] \hookrightarrow \Delta [m] is in WW for all m<nm \lt n and all 0km0 \le k \le m. Then for 0ln0 \le l \le n, the horn inclusion Λ l[n]Δ[n]\Lambda^l [n] \hookrightarrow \Delta [n] is also in WW.
  • Quillen’s small object argument then implies all the trivial cofibrations are in WW.
  • If p:XYp : X \to Y is a trivial Kan fibration, then its right lifting property implies there is a morphism s:YXs : Y \to X such that ps=id Yp \circ s = id_Y, and the two-out-of-three property implies s:YXs : Y \to X is a trivial cofibration. Thus every trivial Kan fibration is also in WW.
  • Every weak homotopy equivalence factors as pip \circ i where pp is a trivial Kan fibration and ii is a trivial cofibration, so every weak homotopy equivalence is indeed in WW.
  • Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the smallest such class.

As a corollary, we deduce that the classical model structure on sSetsSet is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.

Proposition

Let π 0:sSetSet\pi_0 : sSet \to Set be the connected components functor, i.e. the left adjoint of the constant functor cst:SetsSetcst : Set \to sSet. A morphism f:ZWf : Z \to W in sSetsSet is a weak homotopy equivalence if and only if the induced map

π 0K f:π 0K Wπ 0K Z\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z

is a bijection for all Kan complexes KK.

Proof

One direction is easy: if KK is a Kan complex, then axiom SM7 for simplicial model categories implies the functor K ():sSet opsSetK^{(-)} : sSet^{op} \to sSet is a right Quillen functor?, so Ken Brown’s lemma implies it preserves all weak homotopy equivalences; in particular, π 0K ():sSet opSet\pi_0 K^{(-)} : sSet^{op} \to Set sends weak homotopy equivalences to bijections.

Conversely, when KK is a Kan complex, there is a natural bijection between π 0K X\pi_0 K^X and the hom-set Ho(sSet)(X,K)Ho (sSet) (X, K), and thus by the Yoneda lemma, a morphism f:ZWf : Z \to W such that the induced morphism π 0K Wπ 0K Z\pi_0 K^W \to \pi_0 K^Z is a bijection for all Kan complexes KK is precisely a morphism that becomes an isomorphism in Ho(sSet)Ho (sSet), i.e. a weak homotopy equivalence.

Relation to the model structure on strict \infty-groupoids

Urs it would be nice to eventually have a discussion of the following

Recall the model structure on strict omega-groupoids and the omega-nerve operation

N:StrGrpdKanComplx. N : Str \infty Grpd \to Kan Complx \,.

this ought to be a Quillen functor, but is it? is there a reference?

As a warmup, let C,DC, D be ordinary groupoids and N(C)N(C), N(D)N(D) their ordinary nerves. We’d like to show in detail that

Theorem

A functor F:CDF : C \to D is

Proof

We know that both N(C)N(C) and N(D)N(D) are Kan complexes. By the above theorem it suffices to show that N(f)N(f) being a surjective equivalence is the same as having all lifts

Δ[n] N(C) N(F) Δ[n] N(D). \array{ \partial \Delta[n] &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ \Delta[n] &\to& N(D) } \,.

We check successively what this means for increasing nn:

  • n=0n= 0. In degree 0 the boundary inclusion is that of the empty set into the point *\emptyset \hookrightarrow {*}. The lifting property in this case amounts to saying that every point in N(D)N(D) lifts through N(F)N(F).

    N(C) N(F) * N(D) N(C) N(F) * N(D). \array{ \emptyset &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ {*} &\to& N(D) } \Leftrightarrow \array{ && N(C) \\ &{}^\exists\nearrow& \downarrow^{N(F)} \\ {*} &\to& N(D) } \,.

    This precisely says that N(F)N(F) is surjective on 0-cells and hence that FF is surjective on objects.

  • n=1n=1. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval {,}{}\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}. The lifting property here evidently is equivalent to saying that for all objects a,bObj(C)a,b \in Obj(C) all elements in Hom(F(a),F(b))Hom(F(a),F(b)) are hit. Hence that FF is a full functor.

  • n=2n=2. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map Δ[2]N(C)\partial \Delta[2] \to N(C) sends the top left edge of the triangle to an identity. Then the lifting property here evidently is equivalent to saying that for all objects a,bObj(C)a,b \in Obj(C) the map F a,b:Hom(a,b)Hom(F(a),F(b))F_{a,b} : Hom(a,b) \to Hom(F(a),F(b)) is injective. Hence that FF is a faithful functor.

    ( b Id a f a g b)N(F)( b Id a = F(f) a F(g) b) \left( \array{ && b \\ & {}^{Id_a}\nearrow && \searrow^{f} \\ a &&\stackrel{g}{\to}&& b } \right) \stackrel{N(F)}{\mapsto} \left( \array{ && b \\ & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)} \\ a &&\stackrel{F(g)}{\to}&& b } \right)

Joyal’s Model Structure

There is a second model structure on sSetsSet – the model structure for quasi-categories sSet JoyalsSet_{Joyal} – which is different (not Quillen equivalent) to the classical one, due to Andre Joyal, with the following distinguished classes of morphisms:

  • The cofibrations CC are monomorphisms, equivalently, levelwise injections.

  • The weak equivalences WW are weak categorical equivalences, which are morphisms u:ABu : A \rightarrow B of simplicial sets such that the induced map u *:X BX Au^* : X^B \rightarrow X^A of internal-homs for all quasi-categories XX induces an isomorphism when applying the functor τ 0\tau_0 that takes a simplicial set to the set of isomorphism classes of objects of its fundamental category.

  • The fibrations FF are called variously isofibrations or quasi-fibration. As always, these are determined by the classes CC and WW. Quasi-fibrations between weak Kan complexes have a simple description; they are precisely the inner Kan fibrations, the maps that have the right lifting property with respect to the inner horn inclusions and also the inclusion j 0:*Jj_0 : * \rightarrow J where ** is the terminal simplicial set and JJ is the nerve of the groupoid on two objects with one non-trivial isomorphism.

All objects are cofibrant. The fibrant objects are precisely the quasi-categories.

This model structure is cofibrantly generated. The generating cofibrations are the set II described above. There is no known explicit description for the generating trivial cofibrations.

Importantly, this model structure is Quillen equivalent to several alternative model structures for the ‘’homotopy theory of homotopy theories“ such as that on the category of simplicially enriched categories.

Comparison

Every weak categorical equivalence is a weak homotopy equivalence. Since both model structures have the same cofibrations, it follows that the classical model structure is a Bousfield localization of Joyal’s model structure.

The Quillen model structure is the left Bousfield localization of sSet JoyalsSet_{Joyal} at the outer horn inclusions.

Fibrant replacement

Fibrant replacement in sSet QuillensSet_{Quillen} models the process of \infty-groupoidification, of freely inverting all k-morphisms in a simplicial set. Techniques for fibrant replacements sSet QuillensSet_{Quillen} are discussed at

References

Dan Quillen’s original proof in

  • Dan Quillen, Homotopical Algebra, LNM 43, Springer, (1967)

is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. This proof has been rewritten several times in the literature: at the end of

  • S.I. Gelfand and Yu. I. Manin, Methods of Homological Algebra, Springer, 1996

as well as in

  • Andre Joyal and M. Tierney An introduction to simplicial homotopy theory (web)

A proof (in fact two variants of it) using Kan’s Ex Ex^\infty functor (see Kan fibrant replacement) is given in section 2 of

which discusses the topic as a special case of a Cisinski model structure.

The fun part is not that much about the existence of model structure, but to prove that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of Ex Ex^\infty without using topological spaces); for two different proofs of this fact using Ex Ex^\infty, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.

Another standard textbook reference for the classical model structure is

For references on the Joyal model structure see model structure for quasi-categories.

Revised on May 6, 2014 06:59:16 by Adeel Khan (132.252.137.197)