Simplicial sets which are not Kan complexes

Question

A Kan complex is a simplicial set satisfying the horn-filler condition.

What examples are there of simplicial sets which do not satisfy the horn-filler condition? In particular, what simplicial sets are not Kan complexes?

Answer 1

If you write down some small simplicial set with an easy geometric description (e.g., the simplicial set corresponding to some simplicial complex), it will almost never be a Kan complex. This should be evident if you get some practice working with Kan complexes and using the horn filling condition: it implies your simplicial set must be very "rich" in simplices and that you can "combine" simplices in it similar to the way you can combine singular simplices in a topological space. This is highly incompatible with how simplices behave in a typical nice triangulation of, say, a manifold.

For a really simple example, consider the $1$-simplex $\Delta^1$. Note that there is a horn $\Lambda^2_0\to\Delta^1$ given by mapping the vertices $0,1,2$ to $0,1,0$, respectively ($\Lambda^2_0$ only has the edges $0,1$ and $0,2$, and this map sends them the the edge $0,1$ and the degenerate edge $0,0$). This horn cannot be filled, since there is no edge from $1$ to $0$ in $\Delta^1$.

This example more generally shows that in any Kan complex, edges must be "reversible": if there is an edge $e$ beween two vertices, there is another edge $\bar{e}$ between the same vertices in the opposite order, such that $e$ and $\bar{e}$ together form a 2-simplex where the third edge is degenerate. This property can never hold in a $\Delta$-complex with any edges, since if $e$ was nondegenerate, the 2-simplex will also be nondegenerate, but a nondegenerate simplex in a $\Delta$-complex cannot have a degenerate face. So no non-discrete $\Delta$-complex is a Kan complex. On the other hand, this property is familiar from the context of arbitrary continuous paths in a topological space, which can be reversed.

Answer 2

Any finite, not contractible, simply connected simplicial set is not a Kan complex.

It is a result of Serre that any space satisfying these properties has infinitely many nontrivial homotopy groups. Suppose we have a Kan complex. Using the fact that the naive homotopy set of a Kan complex is the homotopy group of the realization, we can deduce that if we have a nontrivial homotopy group in dimension $n$ we must have a nondegenerate simplex in degree $n$, otherwise upon realization all the maps from the simplicial sphere would be constant since the geometric realization quotients out by degenerate simplices.

So we deduce that anything with infinite homotopy (either in infinite dimensions or even a single dimension with infinitely many elements) cannot be both finite dimensional and a Kan complex. Combined with Serre’s result, we have the claim.

But it’s important to note even contractible simplicial sets won’t necessarily be Kan. I think not even the n-simplex is Kan for most n.

Answer 3

For another wealth of examples: take the nerve of any category that is not a groupoid. Roughly speaking, filling outer horns corresponds to inverse morphisms.