Sorry, there’s a few too many things here which need names. The pattern (original category + copy + etc) is the query; the target is whatever category we like. We’re looking for the original category and a model of it, all embedded within some other category. I should probably clarify that in the OP; part what makes it interesting is that we’re looking for a map-territory pair all embedded in one big system.
UPDATE: changed the source-target language to system-model for natural transformations.
Good question, I spent a while chewing on that myself.
Vertical composition is relatively simple—we’re looking for two natural transformations, where pattern-target of one is pattern-source of the other. Equivalently, we’re pattern-matching on a pattern which has three copies of the original category stacked one-on-top-the-other, rather than just two copies.
Horizontal composition is trickier. We’re taking the pattern for a natural transformation (i.e. two copies of the original category) and using that as the original category for another natural transformation. So we end up with a pattern containing four copies of the original category, connected in a square shape, with arrows going from one corner (the pattern-source for the composite transformation) to the opposite corner (the pattern-target for the composite transformation).
Sorry, there’s a few too many things here which need names. The pattern (original category + copy + etc) is the query; the target is whatever category we like. We’re looking for the original category and a model of it, all embedded within some other category. I should probably clarify that in the OP; part what makes it interesting is that we’re looking for a map-territory pair all embedded in one big system.
UPDATE: changed the source-target language to system-model for natural transformations.
Natural transformations can be composed (in two ways) - how does your formulation express this?
Good question, I spent a while chewing on that myself.
Vertical composition is relatively simple—we’re looking for two natural transformations, where pattern-target of one is pattern-source of the other. Equivalently, we’re pattern-matching on a pattern which has three copies of the original category stacked one-on-top-the-other, rather than just two copies.
Horizontal composition is trickier. We’re taking the pattern for a natural transformation (i.e. two copies of the original category) and using that as the original category for another natural transformation. So we end up with a pattern containing four copies of the original category, connected in a square shape, with arrows going from one corner (the pattern-source for the composite transformation) to the opposite corner (the pattern-target for the composite transformation).