A blog post about the release is coming soon.

Taking another careful read of “Cartesian Double Theories: A Double-Categorical Framework for Categorical Doctrines” by M. Lamberta and @epatters.bsky.social.

I get a 404 ...

What is really nice about the slice category alternative is that there is a recipe for creating deltas for any diagram, not just Set^SGr.

Alternatively, I could slice the functor category Set^SGr over G_BN, as Patterson, Lynch, and Fairbanks showed — by passing to slice categories, one can encode various categories of labelled data.

One approach I came across is to encode additions and removals as ZSets, then let the cumulative interpretation accumulate them. We could change the graph functor from G : SGr → Set to G : SGr → ZSet.

But what is the schema of each delta for a given time interval? Ideally, we should be able to systematically derive it from the snapshot schema.

Exploring an alternative to snapshot-based versioning: instead of capturing full snapshots at each point in time (as the paper describes), I’m proposing to record only the deltas—changes between successive states—and then use their cumulative interpretations to flexibly vary the time intervals.

Then I would like to understand how the Grothendiek may be able to help me deal with models that are also theories, which starts with functors from C to Set that are not constant.

Ideally the same interface should be able to be implemented by CT itself or even by static code.

There is still work to do here to really model a 1-category sketch (product object and pullback diagrams, etc). Next step is to try and come up with an interface/abstraction that hides this machinery away.

And to my suprise, `El(F) : [C,Set] -> Cat/C`(and the related projection functor `El(F) -> C`) is a specific case of the Grothendiek Construction. What a difference it makes to study a subject when there is a very specific usecase or problem in front of us.

Isn’t that analogous to a 1-category where we define objects of type Ob and arrows of type Morphism? In RDF, typing and richer structure come as an additional layer; in category theory, typing — and much more — comes built in.

I can’t help but compare this to RDF’s triples — subject, predicate, object — which are nothing more than an arrow with a domain and codomain. RDF isn’t untyped, but unityped: subjects and objects share a single, catch-all type.

Isn’t that analogous to a 1-category where we define objects of type Ob and arrows of type Morphism? In RDF, typing and richer structure come as an additional layer; in category theory, typing — and much more — comes built in.