nLab model structure on simplicial sets

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 on simplicial sets that is a presentation for the (infinity,1)-category Top, as well as the Joyal model structure which similarly is a presentation of the $(\infty,1)$-category $(\infty,1)Cat$.

Classical Model Structure

The classical model structure on simplicial sets, $sSet_{Quillen}$, 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.

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

• 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

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

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

In fact:

Proposition

$sSet_{Quillen}$ is a cofibrantly generated model category with

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

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

Theorem

The singular simplicial complex-functor and geometric realization

$$({\vert -\vert}\dashv Sing) : Top_{Quillen} \stackrel{\overset{{\vert -\vert}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}$$

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

For more on this see homotopy hypothesis.

Characterisations of weak homotopy equivalences

Theorem

Let $W$ be the smallest class of morphisms in $sSet$ satisfying the following conditions:

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

Then $W$ is the class of weak homotopy equivalences.

Proof

• First, notice that the horn inclusions $\Lambda^0 [1] \hookrightarrow \Delta [1]$ and $\Lambda^1 [1] \hookrightarrow \Delta [1]$ are in $W$.
• Suppose that the horn inclusion $\Lambda^k [m] \hookrightarrow \Delta [m]$ is in $W$ for all $m \lt n$ and all $0 \le k \le m$. Then for $0 \le l \le n$, the horn inclusion $\Lambda^l [n] \hookrightarrow \Delta [n]$ is also in $W$.
• Quillen’s small object argument then implies all the trivial cofibrations are in $W$.
• If $p : X \to Y$ is a trivial Kan fibration, then its right lifting property implies there is a morphism $s : Y \to X$ such that $p \circ s = id_Y$, and the two-out-of-three property implies $s : Y \to X$ is a trivial cofibration. Thus every trivial Kan fibration is also in $W$.
• Every weak homotopy equivalence factors as $p \circ i$ where $p$ is a trivial Kan fibration and $i$ is a trivial cofibration, so every weak homotopy equivalence is indeed in $W$.
• 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 $sSet$ 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 $\pi_0 : sSet \to Set$ be the connected components functor, i.e. the left adjoint of the constant functor $cst : Set \to sSet$. A morphism $f : Z \to W$ in $sSet$ is a weak homotopy equivalence if and only if the induced map

$$\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z$$

is a bijection for all Kan complexes $K$.

Proof

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

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

Relation to the model structure on strict $\infty$-groupoids

under construction

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

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

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

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

Proposition

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

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

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

• $n= 0$. 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 $N(D)$ lifts through $N(F)$.

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

  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 $\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}$. The lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ 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 $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a faithful functor.

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

Constructive version

The original proofs of the existence of the classical model structure on simplicial sets are based in classical mathematics as they use the principle of excluded middle and the axiom of choice, and are hence not valid in constructive mathematics. This becomes more than a philosophical issue with the relevance of this model category-structure in homotopy type theory, where internalization into the type theory requires constructive methods for interpreting proofs as programs.

A constructively valid model structure on simplicial sets and coinciding with the classical model structure if excluded middle and axiom of choice are assumed was found in Henry 19. Alternative simpler proofs were found in Gambino-Sattler-Szumiło 19.

See at constructive model structure on simplicial sets.

Joyal’s Model Structure

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

• The cofibrations of $sSet_{Joy}$ are the monomorphisms, hence the degreewise injections,

• The weak equivalences in $sSet_{Joy}$ are weak categorical equivalences,

  namely those maps $u \colon A \rightarrow B$ of simplicial sets such that the induced map $u^* \colon X^B \rightarrow X^A$ of internal homs (mapping complexes) 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 class of acyclic cofibrations already defined.

  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 $j_0 : * \rightarrow J$ 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 simplicial weak homotopy equivalence (but not conversely). Since both model structures have the same class of cofibrations (namely the monomorphisms), it follows that the classical model structure on simplicial sets is a Bousfield localization of Joyal's model structure? (namely at the outer horn-inclusions):

$$sSet_{KQ} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\;\; \bot \;\;} sSet_{Joy} \,.$$

Fibrant replacement

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

• Kan fibrant replacement.

Related concepts

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

• model structure on semi-simplicial sets

Properness

The Quillen model structure is both left and right proper. Left properness is automatic since all objects are cofibrant. Right properness follows from the following argument: it suffices to show that there is a functor $R$ which (1) preserves fibrations, (2) preserves pullbacks of fibrations, (3) preserves and reflects weak equivalences, and (4) lands in a category in which the pullback of a weak equivalence along a fibration is a weak equivalence. For if so, we can apply $R$ to the pullback of a fibration along a weak equivalence to get another such pullback in the codomain of $R$, which is a weak equivalence, and hence the original pullback was also a weak equivalence. Two such functors $R$ are

• geometric realization $sSet \to Top$, where $Top$ denotes a sufficiently convenient category of topological spaces (e.g. the category of k-spaces suffices) and
• $Ex^\infty : sSet \to Kan$, where $Kan$ is the category of Kan complexes.

This can be found, for instance, in Goerss-Jardine, Cor. II.9.6. Another proof can be found in Moss, and a different proof of properness can be found in Cisinski, Prop. 2.1.5.

References

Classical model structure on simplicial sets

On the classical model structure on simplicial sets:

The original proof is due to

• Daniel Quillen, Section II.3 of: Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

This proof is purely combinatorial (i.e. does not pass through geometric realization of simplicial sets as topological spaces): Quillen uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the simplicial classifying space of a simplicial group is a Kan complex.

Other proofs are were given in:

• Sergei Gelfand, Yuri Manin, Sections V.1-2 of: Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (doi:10.1007/978-3-662-12492-5)

• Paul Goerss, J. F. Jardine, Section I.11 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

• André Joyal, Myles Tierney, Notes on simplicial homotopy theory, Lecture at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf)

• André Joyal, Myles Tierney An introduction to simplicial homotopy theory, 2005 (web, pdf)

A proof (in fact two variants of it) using the Kan fibrant replacement $Ex^\infty$ functor is given (in the context of_Cisinski model structure) in:

• Denis-Charles Cisinski, Section 2 of: Les préfaisceaux comme type d'homotopie, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (pdf)

The crucial step is the proof that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of $Ex^\infty$ without using topological spaces); for two different proofs of this fact using $Ex^\infty$, 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 approach using $Ex^\infty$ is:

• Sean Moss, Another approach to the Kan-Quillen model structure, Journal of Homotopy and Related Structures volume 15, pages 143–165 (2020) (arXiv:1506.04887, doi:10.1007/s40062-019-00247-y)

A proof of the model structure not relying on the classical model structure on topological spaces nor on explicit models for Kan fibrant replacement is in

• Christian Sattler, The Equivalence Extension Property and Model Structures (arXiv:1704.06911)

Proofs valid in constructive mathematics are given in:

• Simon Henry, A constructive account of the Kan-Quillen model structure and of Kan’s Ex∞ functor, arXiv:1905.06160.

• Nicola Gambino, Simon Henry, Christian Sattler, Karol Szumiło, The effective model structure and ∞-groupoid objects, arXiv:2102.06146.

As a categorical semantics for homotopy type theory, the model structure on simplicial sets is considered in

• Chris Kapulkin, Peter LeFanu Lumsdaine, Vladimir Voevodsky, (arXiv:1203.2553)

• Chris Kapulkin, Peter LeFanu Lumsdaine, The Simplicial Model of Univalent Foundations (after Voevodsky), Journal of the European Mathematical Society (arXiv:1211.2851,doi:10.4171/jems/1050)

Joyal model structure on simplicial sets

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