Uploaded May 2026 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 28th of May 2026.
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This talk is motivated by trying to understand how closed monoidal categories fit into higher categorical frameworks, in particular to understand why, in higher categorical terms, the projection formula -- f_!(a x f^*b) = f_!(a) x b -- holds for an adjunction f_! -| f^* when f^* strong closed monoidal. The circle of ideas involves the notion of extranatural transformation, which is key to a formal definition of closed monoidal category. These can be represented graphically using 'surface diagrams' and it transpires that these actually have a natural interpretation in terms of the monoidal double category of categories, functors and profunctors. Along the way we will see the notion of conjugation for adjunctions of two variables which will help formalize the projection formula result.
Topos Institute Colloquium, 28th of May 2026.
———
This talk is motivated by trying to understand how closed monoidal categories fit into higher categorical frameworks, in particular to understand why, in higher categorical terms, the projection formula -- f_!(a x f^*b) = f_!(a) x b -- holds for an adjunction f_! -| f^* when f^* strong closed monoidal. The circle of ideas involves the notion of extranatural transformation, which is key to a formal definition of closed monoidal category. These can be represented graphically using 'surface diagrams' and it transpires that these actually have a natural interpretation in terms of the monoidal double category of categories, functors and profunctors. Along the way we will see the notion of conjugation for adjunctions of two variables which will help formalize the projection formula result.
![[Berkeley Seminar] Kevin Carlson | Free cocompletions made friendly?](https://i.ytimg.com/vi/EwtMivSKIYE/mqdefault.jpg "[Berkeley Seminar] Kevin Carlson | Free cocompletions made friendly?
Title: Free cocompletions made friendly?
Abstract: Lots of us are familiar with computations in Poly, some of which are made really nice by the fact that Poly is the category of families in Set^op, more formally, the free cocompletion of Set^op under coproducts. Free cocompletions under all colimits are also really important; theyre presheaf categories! However, presheaf categories are kind of annoying to actually calculate colimits in (coequalizers of sets are scary), in contrast to how trivial it is to calculate coproducts of polynomials. Free cocompletions are supposed to be closing a category up under formal colimits in some sense, so shouldnt taking the colimits also be formal? Actually, it can be! There is a category structure on the class of all small diagrams in C that models the free cocompletion of C in an extremely elementary way with lots of nice properties. This has been known to a few people, such as Andrée Ehresmann, for decades, and in the case of free cocompletions under filtered colimits to a lot more people, such as Grothendieck, but has only been known to me for a couple of weeks. Id like to make it known to you too. Im hoping this will be helpful both for generalizations of Poly beyond mere sets of positions and for better expressing data migrations, which is all about writing down functors from a category into a free cocompletion.
https://topos.site/events/berkeley-seminar/")
![[Berkeley Seminar] Keri DAngelo | Composing Instantaneous Machines](https://i.ytimg.com/vi/FevtvBgF1RU/mqdefault.jpg "[Berkeley Seminar] Keri DAngelo | Composing Instantaneous Machines
Title: Composing Instantaneous Machines
Abstract: In this talk, I’ll discuss recent progress Sophie and I have made on composing instantaneous machines. Instantaneous machines means that at any point in time, input can be given to the machine and the machine will give output based on this input and its current state. In this talk, I’ll show how we can compose such machines. We first create an extended category of directed wiring diagrams accounting for the dependency between input and output, and then define an operad algebra giving us the semantics that defines composition. Depending on where the interest lies, we can delve into some details including that since the output now depends on the input, we come into the “problem” that every time the output changes, the input may also change. This can be accounted for by giving a fixed point that shows after finite time, our input and output will stabilize.
https://topos.site/events/berkeley-seminar/")
![[2-torial] Kevin tells Jason and David about instances of models of double theories](https://i.ytimg.com/vi/GJMBFPe7T6I/mqdefault.jpg "[2-torial] Kevin tells Jason and David about instances of models of double theories
21st of November 2025
Kevin Carlson has recently written a paper, Instances of models of double-categorical theories (https://arxiv.org/abs/2510.08861), with Evan Patterson. Here he explains to David and Jason what instances are and why theyre interesting.")
![[DOTS Lectures] 7. Symmetric monoidal double categories of systems](https://i.ytimg.com/vi/GrGJ58O1NKg/mqdefault.jpg "[DOTS Lectures] 7. Symmetric monoidal double categories of systems
Part of a lecture series on the Double Operadic Theory of Systems (DOTS) presented by David Jaz Myers.")
![[Oxford Seminar] Joanna Ko | Models of Enhanced 2-sketches & Algebras over Enhanced 2-monads](https://i.ytimg.com/vi/HDrcpYeSJhU/mqdefault.jpg "[Oxford Seminar] Joanna Ko | Models of Enhanced 2-sketches & Algebras over Enhanced 2-monads
Oxford Seminar, June 18 2026
You can view the listing for this talk online at https://topos.institute/events/oxford-seminar/talks/2026-06-18_ko_models.html
Speaker: Joanna Ko
Full Title: Models of Enhanced 2-sketches & Algebras over Enhanced 2-monads
Abstract: We study the enhanced 2-category of models of enhanced limit 2-sketches with tight weighted cones. We show that for any enhanced limit 2-sketch (mathbb{T}) with tight cones, the enhanced 2-category (mathbb{M}mathrm{od}_{s, w}(mathbb{T}, mathbb{K})) of models of (mathbb{T}) in a locally presentable enhanced 2-category (mathbb{K}), in which the tight and the loose morphisms are the (mathscr{F})-natural transformations and the loose (w)-natural transformations, respectively, is equivalent to the enhanced 2-category ({mathrm{T}text{-}mathbb{A}mathrm{lg}}_{s, w}) of algebras over an enhanced 2-monad (T) on the models (mathbb{M}mathrm{od}(mathcal{T}_tau, mathbb{K})) restricted to the tight morphisms in (mathbb{T}) with strict (T)-morphisms and (w)-(T)-morphisms.
Along the way, we establish an enriched analogue of the Orthogonal Sub-category Theorem, and generalise results on the reflectivity and the monadicity of models of enriched limit sketches in the base of enrichment to any arbitrary locally presentable enriched category.")
![[Berkeley Seminar] Kevin Carlson | What is it like to be a lax double functor?](https://i.ytimg.com/vi/IFQgqey_388/mqdefault.jpg "[Berkeley Seminar] Kevin Carlson | What is it like to be a lax double functor?
Title: What is it like to be a lax double functor?
Abstract: I will attempt to impart the vibe that the models of double theories that are the basis of CatColab are certain kinds of families of categories, even though the definition looks like a generalization of the families of sets we know and love as presheaves. Furthermore, CatColab does (on a sufficiently bleeding-edge branch) contain families of sets that live over these families of categories in an appropriate way, which we call instances of models of double theories. (Deep breath.) Probably only two people know what those are, yet, so let me try to tell you, because theyre going to be important; its not all that bad, just new.
https://topos.site/events/berkeley-seminar/")
