nLab
model structure on simplicial sets

Idea

Simplicial sets are the archetypical combinatorial “model” for the (infinity,1)-category of (compactly generated weakly Hausdorff) topological spaces and equivalently that of infinity-groupoids, as well as a standard model for the (infinity,1)-category of (infinity,1)-categories 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 $(\mathrm{\infty }\mathrm{,1})$-category $(\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$.

Classical Model Structure

The classical model structure on the category of simplicial sets has the following distinguished classes of morphisms:

• 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.

It follows from this that

• This model structure is cofibrantly generated by

  • the set $I$ of is the set of boundary inclusions of simplices

    $$i_n:\partial \Delta [n]\hookrightarrow \Delta [n]$$i_n : \partial \Delta[n] \hookrightarrow \Delta[n]

  • the set $J$ of generating cofibrations is the set of horn inclusions

    $$j_n^k:{\Lambda }^k[n]\hookrightarrow {\Delta }^n.$$j^k_n : \Lambda^k[n] \hookrightarrow \Delta^n \,.

• 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.

Theorem

• 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 } \,.

Proof

This is theorem 11.2 on p. 61 of GoerJar (theorem 7.10, page 34 in the online version below).

Importantly, this model structure is Quillen equivalent to the model structure on topological spaces with weak homotopy equivalences as weak equivalences and Serre fibrations as fibrations via the geometric realization and total singular complex functors described here.

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 simplicial sets, which is different (not Quillen equivalent) to the classical one, due to André 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-fibrations. As always, these are determined by the classes $C$ and $W$. Quasi-fibrations between Kan complexes they have a relatively simple description; they are precisely 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 cofibrantly generated. 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 equivalences 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.

Fibrant replacement

Techniques for fibrant replacements $\mathrm{SSet}$ are discussed at

• Kan fibrant replacement

References

A standard reference for the classical model structure is

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

A reference with lots of details on Joyal’s model structure is

• Jacob Lurie, Higher Topos Theory