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 the (weak) homotopy type of topological spaces. They can also be taken as models for ∞-groupoids. This is encoded in the model structure on simplicial sets. For more reasons why simplicial sets see MathOverflow here.

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 sSet (also called $\mathrm{SSet}$ or $\mathrm{sSet}$). See there for more details.

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. See quasicategory for a kind of simplicial set which does have such a notion).

Classifying topos

The category of simplicial sets is a presheaf category, and so in particular a Grothendieck topos. In fact, it is the classifying topos of the theory of “intervals”, meaning totally ordered sets equipped with distinct top and bottom elements.

Specifically, if $E$ is a topos containing such an interval $I$, then we obtain a functor $\Delta \to E$ sending $[n]$ to the subobject

$$\{(x_1,x_2,\dots ,x_n)\mid x_1\le x_2\le \dots \le x_n\}\hookrightarrow I^n$$\{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n

The corresponding geometric realization/nerve adjunction $E\leftrightarrows {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$ is the geometric morphism which classifies $I$.

The usual geometric realization into topological spaces cannot be obtained in this way precisely, since Top is not a topos. However, there are Top-like categories which are toposes, such as Johnstone's topological topos.

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.

Related concepts

• simplicial homotopy theory

• simplicial group

• bisimplicial set

• dendroidal set

• semi-simplicial set

References

A pedagogical introduction to simplicial sets is

• Greg Friedman, An elementary illustrated introduction to simplicial sets (arXiv:0809.4221)

A useful (if old) survey article is:

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

More advanced treatments include

• Paul Goerss, Rick Jardine, Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996)