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It may be helpful to consider the geometric join operation for compact topological spaces or simplicial complexes. If $A$ and $B$ are discrete sets, then their join $A \star B$ is the complete bipartite graph connecting $A$ to $B$. The join in general is the natural generalization of that to topological spaces, or simplicial complexes, or simplicial sets, or (if you like) CW complexes. In fact it always is the complete bipartite graph, but with a topology or a simplicial geometry on the set of line segments. If $A$ and $B$ are simplices, their join is another simplex, whose vertices are the disjoint union of the vertices of $A$ and $B$.

A simplicial set $S$ has a small geometric realization consisting of the non-degenerating simplices glued together. In many early examples, such as for instance horns, the small geometric realization is just a finite simplicial complex with locally ordered vertices. (That is, the vertices of each simplex are compatibly ordered.) For instance, the horn $\Lambda_j^2$ is a V, and the horn $\Lambda_j^3$ is a triangular hat (or horn, hence the name and the clever symbol). (As best I can tell, the subscript referring to the apex of the horn $\Lambda^n_j$ is not intrinsic to it as a simplicial set, but rather comes from its inclusion into the simplex $\Delta^n$.) (Edit: Even though a horn is hollow, the local orderings of its faces induce a total ordering of its vertices when $n \ge 2$, or a partial ordering when $n=1$. The subscript indicates the position of the apex.) The join of $A$ with point $\Delta^0$ is also a cone with base $A$. So as Reid said, the cone $\Delta^0 \star \partial \Delta^n$ over a hollow simplex is a horn. It's also easy to see, by drawing a picture, that the cone over a horn is the next horn.

(I apologize if this geometric discussion is too close to what Reid already said somewhat more algebraically.)

It may be helpful to consider the geometric join operation for compact topological spaces or simplicial complexes. If $A$ and $B$ are discrete sets, then their join $A \star B$ is the complete bipartite graph connecting $A$ to $B$. The join in general is the natural generalization of that to topological spaces, or simplicial complexes, or simplicial sets, or (if you like) CW complexes. In fact it always is the complete bipartite graph, but with a topology or a simplicial geometry on the set of line segments. If $A$ and $B$ are simplices, their join is another simplex, whose vertices are the disjoint union of the vertices of $A$ and $B$.

A simplicial set $S$ has a small geometric realization consisting of the non-degenerating simplices glued together. In many early examples, such as for instance horns, the small geometric realization is just a finite simplicial complex with locally ordered vertices. (That is, the vertices of each simplex are compatibly ordered.) For instance, the horn $\Lambda_j^2$ is a V, and the horn $\Lambda_j^3$ is a triangular hat (or horn, hence the name and the clever symbol). (As best I can tell, the subscript referring to the apex of the horn $\Lambda^n_j$ is not intrinsic to it as a simplicial set, but rather comes from its inclusion into the simplex $\Delta^n$.) (Edit: Even though a horn is hollow, the local orderings of its faces induce a total ordering of its vertices when $n \ge 2$, or a partial ordering when $n=1$. The subscript indicates the position of the apex.) The join of $A$ with point $\Delta^0$ is also a cone with base $A$. So as Reid said, the cone $\Delta^0 \star \partial \Delta^n$ over a hollow simplex is a horn. It's also easy to see, by drawing a picture, that the cone over a horn is the next horn. [Edit: It's a good idea to draw the picture of the cone over of a horn, to see that it isn't a horn. :-)]

(I apologize if this geometric discussion is too close to what Reid already said somewhat more algebraically.)

It may be helpful to consider the geometric join operation for compact topological spaces or simplicial complexes. If $A$ and $B$ are discrete sets, then their join $A \star B$ is the complete bipartite graph connecting $A$ to $B$. The join in general is the natural generalization of that to topological spaces, or simplicial complexes, or simplicial sets, or (if you like) CW complexes. In fact it always is the complete bipartite graph, but with a topology on the set of line segments. If $A$ and $B$ are simplices, their join is another simplex, whose vertices are the disjoint union of the vertices of $A$ and $B$.

A simplicial set $S$ has a small geometric realization consisting of the non-degenerating simplices glued together. In many early examples, such as for instance horns, the small geometric realization is just a finite simplicial complex with locally ordered vertices. (That is, the vertices of each simplex are compatibly ordered.) For instance, the horn $\Lambda_j^2$ is a V, and the horn $\Lambda_j^3$ is a triangular hat (or horn, hence the name). (As best I can tell, the subscript referring to the apex of the horn $\Lambda^n_j$ is not intrinsic to it as a simplicial set, but rather comes from its inclusion into the simplex $\Delta^n$.) (Edit: Even though a horn is hollow, the local orderings of its faces induce a total ordering of its vertices when $n \ge 2$, or a partial ordering when $n=1$. The subscript indicates the position of the apex.) The join of $A$ with point $\Delta^0$ is also a cone with base $A$. So as Reid said, the cone $\Delta^0 \star \partial \Delta^n$ over a hollow simplex is a horn. It's also easy to see, by drawing a picture, that the cone over a horn is the next horn.

(I apologize if this geometric discussion is too close to what Reid already said somewhat more algebraically.)

It may be helpful to consider the geometric join operation for compact topological spaces or simplicial complexes. If $A$ and $B$ are discrete sets, then their join $A \star B$ is the complete bipartite graph connecting $A$ to $B$. The join in general is the natural generalization of that to topological spaces, or simplicial complexes, or simplicial sets, or (if you like) CW complexes. In fact it always is the complete bipartite graph, but with a topology or a simplicial geometry on the set of line segments. If $A$ and $B$ are simplices, their join is another simplex, whose vertices are the disjoint union of the vertices of $A$ and $B$.

A simplicial set $S$ has a small geometric realization consisting of the non-degenerating simplices glued together. In many early examples, such as for instance horns, the small geometric realization is just a finite simplicial complex with locally ordered vertices. (That is, the vertices of each simplex are compatibly ordered.) For instance, the horn $\Lambda_j^2$ is a V, and the horn $\Lambda_j^3$ is a triangular hat (or horn, hence the name and the clever symbol). (As best I can tell, the subscript referring to the apex of the horn $\Lambda^n_j$ is not intrinsic to it as a simplicial set, but rather comes from its inclusion into the simplex $\Delta^n$.) (Edit: Even though a horn is hollow, the local orderings of its faces induce a total ordering of its vertices when $n \ge 2$, or a partial ordering when $n=1$. The subscript indicates the position of the apex.) The join of $A$ with point $\Delta^0$ is also a cone with base $A$. So as Reid said, the cone $\Delta^0 \star \partial \Delta^n$ over a hollow simplex is a horn. It's also easy to see, by drawing a picture, that the cone over a horn is the next horn.

(I apologize if this geometric discussion is too close to what Reid already said somewhat more algebraically.)

It may be helpful to consider the geometric join operation for compact topological spaces or simplicial complexes. If $A$ and $B$ are discrete sets, then their join $A \star B$ is the complete bipartite graph connecting $A$ to $B$. The join in general is the natural generalization of that to topological spaces, or simplicial complexes, or simplicial sets, or (if you like) CW complexes. In fact it always is the complete bipartite graph, but with a topology on the set of line segments. If $A$ and $B$ are simplices, their join is another simplex, whose vertices are the disjoint union of the vertices of $A$ and $B$.

A simplicial set $S$ has a small geometric realization consisting of the non-degenerating simplices glued together. In many early examples, such as for instance horns, the small geometric realization is just a finite simplicial complex with locally ordered vertices. (That is, the vertices of each simplex are compatibly ordered.) For instance, the horn $\Lambda_j^2$ is a V, and the horn $\Lambda_j^3$ is a triangular hat (or horn, hence the name). (As best I can tell, the subscript referring to the apex of the horn $\Lambda^n_j$ is not intrinsic to it as a simplicial set, but rather comes from its inclusion into the simplex $\Delta^n$.) (Edit: Even though a horn is hollow, the local orderings of its faces induce a total ordering of its vertices. Hence the subscript to indicate the position of the apex.) The join of $A$ with point $\Delta^0$ is also a cone with base $A$. So as Reid said, the cone $\Delta^0 \star \partial \Delta^n$ over a hollow simplex is a horn. It's also easy to see, by drawing a picture, that the cone over a horn is the next horn.

(I apologize if this geometric discussion is too close to what Reid already said somewhat more algebraically.)

It may be helpful to consider the geometric join operation for compact topological spaces or simplicial complexes. If $A$ and $B$ are discrete sets, then their join $A \star B$ is the complete bipartite graph connecting $A$ to $B$. The join in general is the natural generalization of that to topological spaces, or simplicial complexes, or simplicial sets, or (if you like) CW complexes. In fact it always is the complete bipartite graph, but with a topology on the set of line segments. If $A$ and $B$ are simplices, their join is another simplex, whose vertices are the disjoint union of the vertices of $A$ and $B$.

A simplicial set $S$ has a small geometric realization consisting of the non-degenerating simplices glued together. In many early examples, such as for instance horns, the small geometric realization is just a finite simplicial complex with locally ordered vertices. (That is, the vertices of each simplex are compatibly ordered.) For instance, the horn $\Lambda_j^2$ is a V, and the horn $\Lambda_j^3$ is a triangular hat (or horn, hence the name). (As best I can tell, the subscript referring to the apex of the horn $\Lambda^n_j$ is not intrinsic to it as a simplicial set, but rather comes from its inclusion into the simplex $\Delta^n$.) (Edit: Even though a horn is hollow, the local orderings of its faces induce a total ordering of its vertices when $n \ge 2$, or a partial ordering when $n=1$. The subscript indicates the position of the apex.) The join of $A$ with point $\Delta^0$ is also a cone with base $A$. So as Reid said, the cone $\Delta^0 \star \partial \Delta^n$ over a hollow simplex is a horn. It's also easy to see, by drawing a picture, that the cone over a horn is the next horn.

(I apologize if this geometric discussion is too close to what Reid already said somewhat more algebraically.)