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Stream: community: our work

Topic: Kevin Carlson

Kevin Carlson (Sep 29 2025 at 21:43):

I'll be speaking in the Topos institute colloquium at the following time: 2025-10-02T10:00:00-07:00. If you haven't been, the Topos colloquium is virtual, so if it's not the middle of your night, you should come by!

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 $C$ is to its $C$-sets, so a model of an arbitrary double theory $D$ (that is, a lax double functor $D\to\mathbf{Span}$) is to its instances. The motivating case of the analogy arises precisely when we set $D$ 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 $C$ of "types" and "attributes", or "tables" and "foreign keys", and a database as a $C$-set, the concept of instance encompasses such known extensions as to Baas, Fairbanks, Lynch, and Patterson's attributed $C$-sets, as well as to Schultz, Spivak, Vasilakopoulou, and Wisknesky's algebraic profunctors. From broader motivations, such structures as multifunctors from a multicategory to $\mathbf{Set}$ 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 $C$-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 $X$ and the category of $X$-instances.

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!

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

Valeria de Paiva (Oct 23 2025 at 15:50):

but if one is on the plane, for instance, one can always watch it later!