### Simplicial Sets vs. Simplicial Complexes

#### Posted by John Baez

I’m looking for a reference. Homotopy theorists love simplicial sets; certain other topologists love simplicial complexes; they are related in various ways, and I’m interested in one such relation.

Let me explain…

There’s a category $SSet$ of simplicial sets and a category $SCpx$ of simplicial complexes.

There is, I believe, a functor

$F: SCpx \to SSet$

that takes the simplices in a simplicial complex, which have unordered vertices, and creates a simplicial set in which the vertices of each simplex are given all possible orderings.

There is a geometric realization functor

$| \cdot | : SSet \to Top$

There’s also a geometric realization functor for simplicial complexes. I’d better give it another name… let’s say

$[ \cdot ] : SCpx \to Top$

Here’s what I want a reference for, assuming it’s true: there’s a natural transformation $\alpha$ from the composite

$SCpx \stackrel{F}{\longrightarrow} SSet \stackrel{|\cdot|}{\longrightarrow} Top$

to

$SCpx \stackrel{[\cdot]}{\longrightarrow} Top$

and for any simplicial complex $X$

$\alpha_X : |F(X)| \to [X]$

is a homotopy equivalence. (I don’t think it’s a homeomorphism; a bunch of simplices need to be squashed down.)

Since this field is fraught with confusion, I’d better say exactly what I mean by some things. By a **simplicial set** I mean a functor $X : \Delta^{op} \to Set$, and a morphism between simplicial sets to be a natural transformation between them. That seems pretty uncontroversial. By a **simplicial complex** I mean what Wikipedia calls an abstract simplicial complex, to distinguish them from simplicial complexes that are made of concrete and used as lawn ornaments… or something like that. Namely, I mean a set $X$ equipped with a family $U_X$ of non-empty finite subsets that is downward closed and contains all singletons. By a map of simplicial complexes from $(X,U_X)$ to $(Y,U_Y)$ I mean a map $f: X \to Y$ such that the image of any set in $U_X$ is a set in $U_Y$.

On MathOverflow the topologist Allen Hatcher wrote:

In some areas simplicial sets are far more natural and useful than simplicial complexes, in others the reverse is true. If one drew a Venn diagram of the people using one or the other structure, the intersection might be very small.

Perhaps this cultural divide is reflected in the fact that the usual definition of an abstract simplicial complex involves a set that *contains* the set of vertices, whose extra members are *entirely irrelevant* when it comes to defining morphisms! It’s like defining a vector space to consist of two sets, one of which is a vector space in the usual sense and the other of which doesn’t show up in the definition of linear map. Someone categorically included would instead take the approach described on the nLab.

Ultimately you get equivalent categories either way. And as Alex Hoffnung and I explain, the resulting category of simplicial complexes is a quasitopos of concrete sheaves:

- John Baez and Alex Hoffnung, Convenient categories of smooth spaces, Def. 25 — Prop. 27.

A quasitopos has many, but not quite all, of the good properties of a topos. Simplicial sets are a topos of presheaves, so from a category-theoretic viewpoint they’re more tractable than simplicial complexes.

## Re: Simplicial Sets vs. Simplicial Complexes

The claim is not true. While there is always a map $|F(X)| \to [X]$ that forgets the ordering of vertices, it is almost never a homotopy equivalence. The first counterexample is when $X$ is the 1-simplex! In this case $[X]$ is an interval and $|F(X)|$ is a circle. (The two orderings of the 1-simplex produce two different 1-simplices that combine to form the circle.)

The functor $F$ is much-studied. If $X$ is the $n$-simplex with vertices $0,1,\ldots,n$ then $|F(X)|$ is called the “complex of injective words”. It is homotopy equivalent to a wedge of $n$-spheres, and the number of spheres in the wedge is equal to the number of derangements of the set $\{0,1,...,n\}$. (A derangement is a permutation with no fixed points.)

The complex of injective words is an important tool in proofs of homological stability for the symmetric group and for various types of configuration spaces. That’s why I’m interested in it. It is also of interest in combinatorial topology, though I don’t know why, and there’s apparently also a connection to knot theory. I’ll try to provide references later.

A web search for “complex of injective words” should prove helpful!

Brief moment of pickiness: It is most common to define $F(X)$ as a semi-simplicial set. In this case it is easy to make your definition of $F(X)$ precise: an $n$-simplex of $F(X)$ is an $n$-simplex of $X$, together with an ordering of its vertices, and the $i$-th face of such a simplex is obtained by erasing the $i$-th vertex. If you wanted to do it all simplicially then you would have to talk about degenerate sinplices and so on. Anyhow, the good news is that you could rephrase everything you wrote to be semi-simplicial, and that in the end $|F(X)|$ doesn’t change!