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Stream: community: our work

Topic: Kevin Carlson


view this post on Zulip Kevin Carlson (Sep 29 2025 at 21:43):

I'll be speaking in the Topos institute colloquium at the following time: . If you haven't been, the Topos colloquium is virtual, so if it's not the middle of your night, you should come by!

view this post on Zulip Kevin Carlson (Sep 29 2025 at 21:43):

My title and abstract:
Instances of models of double theories

I'll introduce the notion of instance of a model of a double theory. I'm motivated by the analogy that, as a category CC is to its CC-sets, so a model of an arbitrary double theory DD (that is, a lax double functor DSpanD\to\mathbf{Span}) is to its instances. The motivating case of the analogy arises precisely when we set DD to be the terminal double theory.

Thus the theory of instances can be seen as a chapter of categorical database theory. Beyond the most fundamental case, introduced by Spivak and Kent, of a database schema as a small category CC of "types" and "attributes", or "tables" and "foreign keys", and a database as a CC-set, the concept of instance encompasses such known extensions as to Baas, Fairbanks, Lynch, and Patterson's attributed CC-sets, as well as to Schultz, Spivak, Vasilakopoulou, and Wisknesky's algebraic profunctors. From broader motivations, such structures as multifunctors from a multicategory to Set\mathbf{Set} and models of a Lawvere theory are also instances of "instances." 

I'll describe the category of instances of models of simple double theories, as well, hopefully, as introducing the story for models of the modal double theories recently introduced into our CatColab tool as our main approach for handling categorical structures including multi-ary operations. I'll aim to explain how quite a lot of the standard theory of CC-sets generalizes beyond the terminal double theory. For instance, I'll give a comprehensive factorization system on the category of models of a double theory, together with a Grothendieck construction giving an equivalence between the category of discrete opfibrations over a model XX and the category of XX-instances.

view this post on Zulip Kevin Carlson (Sep 29 2025 at 21:44):

This is the first public talk on the paper on the same talk Evan and I have been preparing for the Paré birthday issue of ACS. The paper had better appear between now and the colloquium, since the issue's deadline is tomorrow!

view this post on Zulip Kevin Carlson (Oct 02 2025 at 16:57):

My talk will be starting in about 3 minutes here: https://topos-institute.zoom.us/j/84392523736?pwd=bjdVS09wZXVscjQ0QUhTdGhvZ3pUdz09