third face $c$ as a composite of $a$ and $b$. This ‘composite’ $c$ is not usually uniquely defined by $a$ and $b$, but is ‘up to homotopy’. If we write $c = ab$ as a shorthand then if $g_1 :\Lambda^0[2]
\rightarrow K$ is a horn, we think of $g_1$ as being $$\xymatrix{ & &\\ \ar[ur]^d\ar[rr]_e&&}$$and to find a filler is to find a diagram $$\xymatrix{ & \ar@{--}[dr]^x&\\ \ar[ur]^d\ar[rr]_e&&}$$ and thus to ‘solve’ the equation $dx = e$ for $x$ in terms of $d$ and $e$. It thus requires in general some approximate inverse for $d$, in fact, taking $e$ to be a degenerate 1-simplex puts one in exactly such a position.

In many useful cases, we do not always have inverses and so want to discard any requirement that would imply they always exist. This leads to the weaker form of the Kan condition in which in each dimension no requirement is made for the existence of fillers on horns that miss out the zeroth or last faces. More exactly:

**Definition**

A simplicial set ${\bf K}$ is *a weak Kan complex* or *quasi-category* if for any $n$ and $0< k < n$, any $(n,k)$-horn in $K$ has a filler.

**Remarks**

