nLab
simplicial set

Idea

Simplicial sets generalize the idea of simplicial complexes: a simplicial set is like a combinatorial space built up out of gluing abstract simplices to each other. Equivalently, it is an object equipped with a rule for how to consistently map the objects of the simplex category into it.

More concretely, a simplicial set $S$ is a collection of sets $S_n$ for $n\in \mathbb{N}$, so that elements in $S_n$ are to be thought of as $n$-simplices, equipped with a rule that says:

• which $(n-1)$-simplices in $S_{n-1}$ are faces of which elements of $S_n$;
• which $(n+1)$-simplices are thin in that they are really just $n$-simplices regarded as degenerate $(n+1)$-simplices.

One of the main uses of simplicial sets is as combinatorial models for topological spaces. They can also be taken as models for ∞-groupoids. This is encoded in the model structure on simplicial sets.

Definition

A simplicial set is a presheaf on the simplex category $\Delta$, that is, a functor $X:{\Delta }^{\mathrm{op}}\to \mathrm{Sets}$.

This is, of course, a simplicial object in the category Set of sets.

With the standard morphisms of presheaves as morphisms, simplicial sets form the category SimpSet (also called $\mathrm{SSet}$ or $\mathrm{sSet}$).

Remarks

simplicial sets as spaces built of simplices

• The definition is to be understood from the point of view of space and quantity: a simplicial set is a space characterized by the fact that and how it may be probed by mapping standard simplices into it: the set $S_n$ assigned by a simplicial set to the standard $n$-simplex $[n]$ is the set of $n$-simplices in this space, hence the way of mapping a standard $n$-simplex into this spaces.

• For $S$ a simplicial set, the face map

  $$d_i:=S({\delta }^i):S_n\to S_{n-1}$$d_i := S(\delta^i): S_n \rightarrow S_{n-1}

  is dual to the unique injection ${\delta }^i:[n-1]\to [n]$ in the category $\Delta$ whose image omits the element $i\in [n]$.

• Similarly, the degeneracy map

  $$s_i:=S({\sigma }^i):S_n\to S_{n+1}$$s_i := S(\sigma^i) : S_n \rightarrow S_{n+1}

  is dual to the unique surjection ${\sigma }^i:[n+1]\to [n]$ in $\Delta$ such that $i\in [n]$ has two elements in its preimage.

• The maps ${\delta }^i$ and ${\sigma }^i$ satisfy certain obvious relations – the simplicial identities – dual to those spelled out at simplex category.

Visualisation

(based on cubical set)

The face maps go from sets $S_{n+1}$ of $(n+1)$-dimensional simplices to the corresponding set $S_n$ of $n$-dimensional simplices and can be thought of as sending each simplex in the simplicial set to one of its faces, for instance for $n=1$ the set $S_2$ of 2-simplices would be sent in three different ways by three different face maps to the set of $1$-simplices, for instance one of the face maps would send

$$\left(\begin{array}{ccc}& & b\\ & \nearrow & {\Downarrow }^F& \searrow \\ a& & \to & & c\end{array}\right)\mapsto \left(\begin{array}{ccc}& & b\\ & \nearrow \\ a\end{array}\right)$$\left( \array{ & & b \\ & \nearrow & \Downarrow^F & \searrow \\ a & & \rightarrow & & c } \right) \;\; \mapsto \;\; \left( \array{ & & b \\ & \nearrow \\ a } \right)

another one would send

$$\left(\begin{array}{ccc}& & b\\ & \nearrow & {\Downarrow }^F& \searrow \\ a& & \to & & c\end{array}\right)\mapsto \left(\begin{array}{ccccc}a& & \to & & c\end{array}\right).$$\left( \array{ & & b \\ & \nearrow & \Downarrow^F & \searrow \\ a & & \rightarrow & & c } \right) \;\; \mapsto \;\; \left( \array{ a & & \rightarrow & & c } \right) \,.

On the other hand, the degeneracy maps go the other way round and send sets $S_n$ of $n$-simplices to sets $S_{n+1}$ of $(n+1)$-simplices by regarding an $n$-simplex as a degenerate or “thin” $(n+1)$-simplex in the various different ways that this is possible. For instance again for $n=1$ a degeneracy map may act by sending

$$\left(\begin{array}{ccc}a& \stackrel{f}{\to }& b\end{array}\right)\mapsto \left(\begin{array}{ccc}& & b\\ & {\nearrow }_f& {\Downarrow }^{\mathrm{Id}}& {\searrow }^{\mathrm{Id}}\\ a& & \stackrel{f}{\to }& & b\end{array}\right).$$\left( \array{ a &\stackrel{f}{\to}& b } \right) \;\; \mapsto \;\; \left( \array{ & & b \\ & \nearrow_f & \Downarrow^{Id} & \searrow^{Id} \\ a & & \stackrel{f}\to & & b } \right) \,.

Notice the $\mathrm{Id}$-labels, which indicate that the edges and faces labeled by them are “thin” in much the same way as an identity morphism is thin (notice however that a simplicial set by itself is not equipped with a notion of composition of simplices. If it were, we’d call it a simplicial category).

Examples

• Let $[n]$ denote the object of $\Delta$ corresponding to the totally ordered set $\{0,1,2,\dots ,n\}$. Then the represented presheaf $\Delta (-,[n])$, which we typically write as $\Delta [n]$ is an example of a simplicial set. By the Yoneda lemma, the $n$-simplices of a simplicial set $X$ are in natural bijective correspondence to maps $\Delta [n]\to X$ of simplicial sets.

• If $C$ is a small category, the nerve of $C$ is a simplicial set which we denote $\mathrm{NC}$. If we intepret the poset $[n]$ defined above as a category, we define the $n$-simplices of $\mathrm{NC}$ to be the set of functors $[n]\to C$. Equivalently, the $0$-simplices of $\mathrm{NC}$ are the objects of $C$, the $1$-simplices are the morphisms, and the $n$-simplices are strings of $n$ composable arrows in $C$. Face maps are given by composition (or omission, in the case of $d_0$ and $d_n$) and degeneracy maps are given by inserting identity arrows.

• Recall from simplex category or geometric realization the standard functor $\Delta \to \mathrm{Top}$ which sends $[n]\in \Delta$ to the standard topological $n$-simplex ${\Delta }^n$. This functor induces for every topological space $X$ the simplicial set

  $$SX:[n]\mapsto {\mathrm{Hom}}_{\mathrm{Top}}({\Delta }^n,X)$$S X : [n] \mapsto Hom_{Top}(\Delta^n, X)

  called the simplicial singular complex of $X$. This simplicial set is always a Kan complex and may be regarded as the fundamental ∞-groupoid of $X$.

• A symmetric set is a simplicial set equipped with additional transposition maps $t_i^n:X_n\to X_n$ for $i=0,\dots ,n-1$. These transition maps generate an action of the symmetric group on $X_n$ and satisfy certain commutation relations with the face and degeneracy maps.

The category of simplicial sets

Like all categories of presheaves on a small category, the category SimpSet of simplicial sets is complete and cocomplete (with limits and colimits constructed levelwise) and cartesian closed. In fact, like all presheaf categories, it is a topos.

monoidal structure

As described at closed monoidal structure on presheaves the cartesian tensor product $S\otimes T=S\times T$ of simplicial sets $S$ and $T$ is the simplicial set

$$(S\otimes T):[n]\mapsto S_n\times T_n,$$(S \otimes T) : [n] \mapsto S_n \times T_n \,,

where the product on the right is the cartesian product in Set.

One cental reason why simplicial sets are useful and important is that this simple monoidal structure (“disturbingly simple minded” in the words of Friedman08, p. 24) actually does fully capture the standard monoidal structure on topological spaces under geometric realization $\mid \cdot \mid :\mathrm{SSet}\to \mathrm{Top}$

Proposition

For $S$ and $T$ simplicial sets, we have

$$\mid S\times T\mid \simeq \mid S\mid \times \mid T\mid ,$$|S \times T| \simeq |S| \times |T| \,,

where on the right the cartesian product is in the nice category of compactly generated Hausdorff spaces.

closed structure

As described at closed monoidal structure on presheaves the internal hom $[S,T]$ of simplicial sets is the simplicial set

$$[S,T]:[n]\mapsto {\mathrm{Hom}}_{\mathrm{SSet}}(S\times {\Delta }^n,T),$$[S,T] : [n] \mapsto Hom_{SSet}(S \times \Delta^n, T) \,,

where ${\Delta }^n=Y([n])$ is the standard simplicial $n$-simplex, the image of $[n]\in \Delta$ under the Yoneda embedding.

Adjunctions

The maps $N:Cat\to SimpSet$ and $S:Top\to SimpSet$ described in the examples are actually functors, both of which have left adjoints. These adjoint pairs are examples of a very general sort of adjunction involving simplicial sets, of which there are many examples.

Let $E$ be any cocomplete category and let $F:\Delta \to E$ be a functor. We define the right adjoint $R:E\to SimpSet$ as follows. Given an object $e\in E$ the $n$-simplices of $\mathrm{Re}$ are defined to be the set $E(F[n],e)$ of morphisms in $E$ from $F[n]$ to $e$. Face and degeneracy maps are given by precomposition by the appropriate (dual) maps in the image of $F$. $R$ is defined on morphisms by postcomposition.

The left adjoint $L$ is defined to be the left Kan extension of $F$ along the Yoneda embedding $y:\Delta \to SimpSet$. Because the $y$ is full and faithful, we will have $\mathrm{Ly}=F$, i.e., $L(\Delta [n])=F[n]$. By specifying $F$, we have already defined a functor to $E$ on the represented simplicial sets; $L$ is the unique cocontinuous extension of this functor to $SimpSet$. It can be described explicitly on objects as a coend, or as a weighted colimit.

(Easy) abstract nonsense shows that $L$ and $R$ form an adjoint pair $L⊣R$.

Here are some examples:

• Let $E=Cat$ and $F$ be the functor $[n]\mapsto [n]$ (the inclusion of posets into categories). The right adjoint is the nerve functor $N$ described above. The left adjoint ${\tau }_1$ takes a simplicial set to its fundamental category.

• Let $E=Top$ and $F$ be the functor $[n]\mapsto {\Delta }_n$. The right adjoint is the total singular complex functor $S$ described above. The left adjoint $\mid -\mid$ is called geometric realization. As a consequence of the Kan extension construction, the geometric realization of the represented simplicial set $\Delta [n]$ is the standard $n$-simplex ${\Delta }_n$.

• (Barycentric) subdivision and extension $sd:SimpSet\leftrightarrow SimpSet:ex$.

• The homotopy coherent nerve functor and its left adjoint $SimpSet\leftrightarrow SimpCat$ where SimpCat? denotes the category of simplicially enriched categories, i.e., categories enriched in $SimpSet$.

Tim: What is the reference for this simplicial nerve? (I do not like that terminology.) Is it what I would call the homotopy coherent nerve (as explicitly first introduced by Cordier)? If so it needs an entry for itself.

Urs: I guess it should be the notion introduced in

• Cordier, J. M. Sur la notion de diagramme homotopiquement coh´erent. Cahiers Top. et Geom. Diff. XXIII 1, 1982, 93-112.

Over at geometric ∞-function theory I am asking for an entry titled simplicial nerve of simplicial categories. Likely not a good term either. I took “simplicial nerve” from section 1.1.5, p. 26 of HTT.

• The adjunction $-\times X:SimpSet\leftrightarrow SimpSet:(-{)}^X$ between the product with a simplicial set $X$ and the internal-hom, which makes $SimpSet$ into a cartesian closed category.

Related concepts

• dendroidal set

References

A pedagogical introduction to simplicial sets is

• Greg Friedman, An elementary illustrated introduction to simplicial sets (arXiv)

A useful (if old) survey article is:

• E. Curtis, Simplicial Homotopy Theory, Advances in Math., 6, (1971), 107 – 209.

More advanced treatments include

• P. G. Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)

Tim: We seem to have managed to have two notations for the basic simplices in simplicial sets. There are pages with $\Delta [n]$ and those with ${\Delta }^n$. (I prefer the former.) This is confusing!

My preferred notation is $\Delta [n]$ for the representable simplicial set $\Delta (-,[n])$, with its geometric realisation being ${\Delta }^n$. These are quite different beasties and it seems a shame to use the same notation for both. Should we rationalise/standardise the notation?

Urs: would be fine with me (I am probably responsible for much of the ${\Delta }^n$s) – on the other hand, globally consistent notation over the $n$Lab will generally be a challange. But we should try. Yes.