sequence of colimit diagrams

Question

Suppose we have a finite diagram with a colimit. Is it possible to then take the colimit diagram as a new base diagram and then have a new colimit of this new diagram? We could build up diagrams from, say, just a simple two object, single arrow diagram with a colimit to a three object, two arrow diagram with a new colimit. This seems like a natural process. Is there a name for this kind of construction? This would continue for all finite chains, where every chain has its own colimit.

Answer 1

This process stabilizes imediately. To see this, note a somewhat more general result: if a diagram $D$ has an initial object $i$, then every functor $F : D → \mathscr C$ has a limit $Fi$ (with the obvious limiting cone). Given another cone $ΔC ⇒ F$, the component $C → Fi$ at $i$ is the required unique factorization through the limiting cone. In your case, the diagram you get is a cone, but every cone has a unique initial object: it's tip. Dually, if $D$ has a terminal object $t$, then $Ft$ is a colimit of $F$.

This is perhaps best understood when $D$ is a directed set. If $D$ has a maximum, then intuitively the last element of a direct system ought to be it's direct limit (categorical colimit), and similarly should inverse systems with the smallest element have trivial inverse limits (categorical limits).