\(i) Joyal, [@joyal], uses the term *inner horn* for any $(n,k)$-horn in $K$ with $0< k < n$. The two remaining cases are then conveniently called *outer horns*.

\(ii) For any space $X$, its singular complex, $Sing(X)$ is given by $Sing(X)_n = Top(\Delta^n,X)$ with the well known face and degeneracy maps. This simplicial set is always a Kan complex as is the related $\mathcal{T}op(X,Y)$ as mentioned above.

Nerves
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The categorical analogue of the singular complex is, of course, the nerve: if $\mathbb{C}$ is a category, its *nerve*, $Ner(\mathbb{C})$, is the simplicial set with $Ner(\mathbb{C})_n = Cat([n],\mathbb{C})$, where $[n]$ is the category associated to the finite ordinal $[n] = \{ 0 < 1< \ldots < n\}$. The face and degeneracy maps are the obvious ones using the composition and identities in $\mathbb{C}$.

The following is well known and easy to prove.

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(i) $Ner(\mathbb{C})$ is always weakly Kan.\
(ii) $Ner(\mathbb{C})$ is Kan if and only if $\mathbb{C}$ is a groupoid.$\blacksquare$

Of course more is true. Not only does any inner horn in $Ner(\mathbb{C})$ have a filler, it has exactly one filler. To express this with maximum force the following idea, often attributed to Graeme Segal or to Grothendieck, is very 