Model Space as a Diagram

Kris Brown, Tyler Hanks, and James Fairbanks reframe scientific model selection categorically. A search space of models is a single diagram in a category of models, so exploring candidates becomes walking that diagram rather than enumerating a list.

A model is a C-Set, a copresheaf , equivalently a categorical relational database on the schema . This is the same instance-as-functor that functorial data migration transports between schemas. A model space is a diagram whose objects are candidate models and whose arrows are structure-preserving relationships between them. Exploration is traversal of the diagram, and selection is choosing one of its objects.

Since is bicomplete, (co)limits of these diagrams exist, and larger model spaces are built from primitive ones by gluing along shared structure. A colimit assembles independently specified components into a composite model, so a space of compound models is generated compositionally rather than written out by hand.

Example. Petri-net epidemiology by grey-boxing

Compartmental epidemic models are Petri nets (a species/transition C-Set). A functor assigns to each open sub-net its tunable rate parameters, so grey-boxing exposes exactly the pieces a search may vary while keeping the surrounding net fixed.

Property. Composite spaces via colimits

Because bicompleteness gives pushouts, two model spaces sharing a sub-model glue along it by a colimit; the resulting diagram’s objects are the coherent combinations of the parts. Model construction and model search are thereby the same operation performed on a diagram.

References

• K. Brown, T. Hanks, J. Fairbanks, Compositional Exploration of Combinatorial Scientific Models (2023)