nLab
model structure on simplicial sets

under construction

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 $(\mathrm{\infty },1)$-category $(\mathrm{\infty },1)\mathrm{Cat}$.

Classical Model Structure

The classical model structure – or Quillen model structure ${\mathrm{sSet}}_{\mathrm{Quillen}}$ on sSet has the following distinguished classes of morphisms:

Definition

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

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

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

  $$\begin{array}{ccc}{\Lambda }^k[n]& \to & X\\ \downarrow & {}^{\exists }\nearrow & {\downarrow }^f\\ \Delta [n]& \to & Y\end{array}.$$\array{ \Lambda^k[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.

• A morphism $f:X\to Y$ 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 $f:X\to Y$ that have the right lifting property with respect to all inclusions $\partial \Delta [n]\hookrightarrow \Delta [n]$ of boundaries of $n$-simplices into their $n$-simplices

$$\begin{array}{ccc}\delta \Delta [n]& \to & X\\ \downarrow & {}^{\exists }\nearrow & {\downarrow }^f\\ \Delta [n]& \to & Y\end{array}.$$\array{ \delta \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

${\mathrm{sSet}}_{\mathrm{Quillen}}$ is a cofibrantly generated model category with

• generating cofibrations the horn inclusions ${\Lambda }^i[n]\to \Delta [n]$;

• generatic acyclic cofibrations the boundary inclusions $\partial \Delta [n]\to \Delta [n]$.

Theorem

The singular simplicial complex-functor and geometric realization

$$(\mid -\mid ⊣\mathrm{Sing}):{\mathrm{Top}}_{\mathrm{Quillen}}\stackrel{\stackrel{\mid -\mid }{\leftarrow }}{\underset{\mathrm{Sing}}{\to }}{\mathrm{sSet}}_{\mathrm{Quillen}}$$({\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.

Relation to the model structure on strict $\mathrm{\infty }$-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

$$N:\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}\to \mathrm{Kan}\mathrm{Complx}.$$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 $C,D$ be ordinary groupoids and $N(C)$, $N(D)$ their ordinary nerves. We’d like to show in detail that

Theorem

A functor $F:C\to D$ is

• k-surjective for all $k$ and hence a surjective equivalence of categories precisely if under the nerve $N(F):N(C)\to N(D)$ it induces an acyclic fibration of Kan complexes;

Proof

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

$$\begin{array}{ccc}\delta \Delta [n]& \to & N(C)\\ \downarrow & {}^{\exists }\nearrow & {\downarrow }^{N(F)}\\ \Delta [n]& \to & N(D)\end{array}.$$\array{ \delta \Delta[n] &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ \Delta[n] &\to& N(D) } \,.

We check successively what this means for increasing $n$:

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

  $$\begin{array}{ccc}\varnothing & \to & N(C)\\ \downarrow & {}^{\exists }\nearrow & {\downarrow }^{N(F)}\\ *& \to & N(D)\end{array}\iff \begin{array}{ccc}& & N(C)\\ & {}^{\exists }\nearrow & {\downarrow }^{N(F)}\\ *& \to & N(D)\end{array}.$$\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 $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.

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

• $n=2$. 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 $\partial \Delta [2]\to N(C)$ 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 $a,b\in \mathrm{Obj}(C)$ the map $F_{a,b}:\mathrm{Hom}(a,b)\to \mathrm{Hom}(F(a),F(b))$ is injective. Hence that $F$ is a faithful functor.

  $$\left(\begin{array}{ccc}& & b\\ & {}^{{\mathrm{Id}}_a}\nearrow & & {\searrow }^f\\ a& & \stackrel{g}{\to }& & b\end{array}\right)\stackrel{N(F)}{\mapsto }\left(\begin{array}{ccc}& & b\\ & {}^{{\mathrm{Id}}_a}\nearrow & {\Downarrow }^{=}& {\searrow }^{F(f)}\\ a& & \stackrel{F(g)}{\to }& & b\end{array}\right)$$\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 $\mathrm{sSet}$ – the model structure for quasi-categories ${\mathrm{sSet}}_{\mathrm{Joyal}}$ – which is different (not Quillen equivalent) to the classical one, due to Andre Joyal, with the following distinguished classes of morphisms:

• The cofibrations $C$ are monomorphisms, equivalently, levelwise injections.

• The weak equivalences $W$ are weak categorical equivalences, which are morphisms $u:A\to B$ of simplicial sets such that the induced map $u^{*}:X^B\to X^A$ of internal-homs for all quasi-categories $X$ induces an isomorphism when applying the functor ${\tau }_0$ that takes a simplicial set to the set of isomorphism classes of objects of its fundamental category.

• The fibrations $F$ are called variously isofibrations or quasi-fibration. As always, these are determined by the classes $C$ and $W$. 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 $j_0:*\to J$ where $*$ is the terminal simplicial set and $J$ 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 $I$ 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 ${\mathrm{sSet}}_{\mathrm{Joyal}}$ at the outer horn inclusions.

Fibrant replacement

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

• Kan fibrant replacement.

Related concepts

• model structure on topological spaces, model structure on simplicial presheaves

• model structure on semi-simplicial types?

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 ${\mathrm{Ex}}^{\mathrm{\infty }}$ functor (see Kan fibrant replacement) is given in section 2 of

• Denis-Charles Cisinski, Les préfaisceaux comme type d'homotopie, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (pdf)

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 ${\mathrm{Ex}}^{\mathrm{\infty }}$ without using topological spaces); for two different proofs of this fact using ${\mathrm{Ex}}^{\mathrm{\infty }}$, 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

• Paul Goerss, Rick Jardine, Simplicial homotopy theory (Birkhäuser) (ps).

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