Is there a simple description of those colimits that exist in the simplex category $\mathbf{\Delta}$ (of finite linear orders and non decreasing maps) ? It is easy to find examples of diagrams for which a colimit exists or not, but I would be interested in a concise and exhaustive description of exactly which diagrams $I \to \mathbf{\Delta}$ admit a colimit. The corresponding question about limits would be interesting too, but colimits are more important for me at the moment.
One sees easily that coproducts don't exist, and that for a colimit to exist the indexing category must be connected. A span $[m] \leftarrow [k] \to [n]$ in $\mathbf{\Delta}$ has a colimit only if the initial and terminal objects of $[n]$ are in the image of $[k]\to [n]$ (or the similar statement with $n$ replaced by $m$). Analogous conditions can be found for multispans $$ [k_1] \leftarrow [k_2] \to \cdots \leftarrow [k_{n-1}] \to [k_n] $$ but what about more general diagrams? My intuition would be that any existing colimit can be reduced to such a multispan, something like : if $I\to \mathbf{\Delta}$ has a colimit then there exist a functor $J\to I$ such that $J$ is a multispan category $$ (1) \leftarrow (2) \to \cdots \leftarrow (n-1) \to (n) $$ and $\mathrm{colim}(J\to I \to \mathbf{\Delta}) \simeq \mathrm{colim}(I \to \mathbf{\Delta})$, but I would be interested to hear anyone's thoughts on this.