The simplicial $n$-sphere is the unique simplicial set with one non-degenerate 0-simplex, one non-degenerate $n$-simplex, and no other non-degenerate simplices. The $n$-sphere is obtained from $\Delta^n$ by identifying the boundary to a point.

We define reduced as a simplicial set having on one element in $K_0$.

So, we know that the simplicial $n$-sphere is a reduced simplicial set. However, is the simplicial $n$-sphere a Kan complex?

Specifically, I'm looking at the 2-sphere. So, the nondegenerate simplices are $[0]\in X_0$ and $[0,1,2]\in X_2$.

One way to define a simplicial set $X$ as a Kan complex is if it satisfies the following Kan condition: The simplicial set $X$ satisifies the Kan condition if for any collection of $(n-1)$-simplices $x_0,...,x_{k-1},x_{k+1},...,x_n$ in $X$ such that $d_ix_j = d_{j-1}x_i$ for any $i<j$ with $i\neq k$ and $j\neq k$, there is an $n$-simplex $x$ in $X$ such that $d_ix = x_i$ for all $i\neq k$.

Note: Here $d_i$ are face maps.

(The condition on the simplicies $x_i$ of this definition glues them together to form the horn $\Lambda_{k}^{n}$, possibly with degenerate faces, within $X$, and the definition says that we can extend this horn to a (possibly degenerate) $n$-simplex in $X$. )

In our case the only $(n-1)$-simplex or $1$-simplex would be the degenerate simplex $[0,0]\in X_1$.

So, would this be a Kan complex trivially because we only have one $(n-1)$-simplex for the 2-sphere? So, we satisfy the condition trivially.

Or rather, since the only 1-simplex is $[0,0]\in X_1$ and the only 2-simplex is $[0,1,2]$ and we don't satisfy the condition $d_ix=x_i$ (since $d_0[0,1,2] = [1,2]\neq [0,0]$, $d_1[0,1,2] = [0,2]\neq [0,0]$, and $d_2[0,1,2] = [0,1]\neq [0,0]$), then the 2-sphere is not a Kan complex?


The Kan condition has to be satisfied in all degrees, so for all $n$, not just $n=2$ as in your example.

As to your last paragraph, you shouldn't call your $2$-simplex $[0,1,2]$ since its vertices are all $0$. You can't really call it $[0,0,0]$ since this should mean the degenerate $2$-simplex, but both share the same faces (all degenerate).

Let's call your $2$-simplices $a$ and $b$, where $a$ is the non-degenerate one. The Kan condition fails in degree $2$: you can find a horn of $2$-simplices without any filler.

Indeed, any collection of three $2$-simplices (like $a,a,b$, or $b,a,a$) is a horn since all faces are always the degenerate simplex $[0,0]$. But the horn $a,a,a$ has no filler, since all $3$-simplices are degenerate, and degenerate simplices can only have $2$ non-degenerate faces.

Actually, if $X$ is a Kan complex, then you can compute its homotopy groups by looking at its internal spheres (morphisms from your model of sphere to $X$) up to homotopy of simplices (two simplices being homotopic if there are faces of a simplex "going from one to the other", meaning that the other faces are degenerate). It means that if your model were a Kan complex, it would be very easy to compute the homotopy groups of spheres.

Another way to understand the Kan condition is in terms of quasi-categories. Imagine a simplicial set as a kind of higher graph, where you have points, paths between points, paths between paths (called homotopies), etc.

Now intuitively, if you have two composable paths $0 \xrightarrow a 1 \xrightarrow b 2$ then you should have a path directly from $0$ to $2$. It is not the case necessarily, but if you ask that your simplicial set satisfies the horn-filling condition only for inner horns (all horns where $0<k<n$), then you do have such a path, and similarly for higher paths. Such a simplicial set satisfying the inner Kan condition is called a quasi-category. It is a simplicial set with a notion of composition.

For instance, for $n=2$, the horn-filling condition with $k=1$ tells you that given an arrow from $a$ to $b$ and an arrow from $b$ to $c$, then there is an arrow from $a$ to $c$ such that all arrows fit in a triangle $abc$. Therefore, the horn-filler gives you a definition of composition of two arrows as "the third arrow in the horn-filler".

A true Kan complex is a quasi-category where all paths are invertible (it's also called an $\infty$-groupoid). For instance, the arrow $\Delta^1$ is a quasi-category but not a Kan complex because it lacks an inverse arrow. A topological space can be realized as a Kan complex where the points are the $0$-simplices, the paths are the $1$-simplices, the homotopies between paths give $2$-simplices, etc. Here you see that you can compose paths, and you can also invert them.

Your model of the sphere is not even a quasi-category, since you should be able to compose the non-degenerate $2$-simplex (which you can imagine as a homotopy from the constant loop to itself) with itself, but there is no such composition in your model.

In the Kan completion of your model, you should be able to see all the $2$-simplices generated by the non-degenerate one, therefore you must have (at least) $\mathbb Z$ $2$-simplices. The higher simplices are at least as big as the corresponding homotopy groups of the sphere.


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