Thanks for another great article. These are really helping me ground my intuitions in Category Theory and its application to programming. Btw. exploring co-monads does help shed a lot of light on what Monads are.
Since you are working with graphs, you may be interested in the semantic web which is based completely on graphs (and named graphs). The graphs need not be connected as they seem to have to be with Quiver. The key intuition one needs to develop there is one of semantics that stems from logic, ie notions of meaning and reference, etc... (it has many echoes in philosophy) . The aim there is to create a global distributed database on-top of the web. There are a couple of talks online on the banana-rdf wiki including one I gave in December 2014 at Scala eXchange, which develops a use case for the semantic web, with scala examples.
Hi there, just a note for those who, like me just until a few minutes ago, where wondering if this exists in Haskell land: almost. Quiver is in fact based on fgl (as the Decomp and GDecomp give away) with small differences I can spot (like Decomp is done on an unlabelled Context) and a big difference which is that fgl doesn't have a (or rather neither of the) comonad instance(s).
Regarding my comment above from 5 years ago, I have since found one very nice explanation of RDF in the 2017 paper Knowledge Representation in Bicategories of Relations, which very helpfully places RDF in the space of bicategories rather than categories, that is where there are arrows between arrows too. But only 1 relation type: the sub property relation. That does not take away from the comonadic view of graphs presented in the blog post.