![[Berkeley Seminar] Evan Patterson | A uniform type theory for internal languages of cat. structures @ToposInstitute](https://i.ytimg.com/vi/mknZxNcn0Ho/maxresdefault.jpg)
Uploaded July 2026 | Updated September 2026, 2 weeks ago
Title: A uniform type theory for internal languages of categorical structures
Abstract: A hallmark of categorical logic is the interplay between category theory and type theory. A categorical structure can provide the semantics of a type theory; in the other direction, type theory can be used to describe the internal language of a categorical structure. The internal language is an ergonomic notation, based on elements and function application, to specify morphisms in the category. Typically, discovering an internal language is a bespoke process, performed anew for each categorical structure of interest. In this talk, we propose a uniform type theory giving internal languages for much of the “substructural” hierarchy of categorical logic, including such structures as categories, multicategories (planar, symmetric, and cartesian), PROPs, Lawvere theories, (symmetric) monoidal categories, cartesian categories, and Markov categories. The framework builds on the speaker’s work in double-categorical logic, and is described in preliminary form at: next.catcolab.org/rfc/0004
Date: May 26, 2026
Title: A uniform type theory for internal languages of categorical structures
Abstract: A hallmark of categorical logic is the interplay between category theory and type theory. A categorical structure can provide the semantics of a type theory; in the other direction, type theory can be used to describe the internal language of a categorical structure. The internal language is an ergonomic notation, based on elements and function application, to specify morphisms in the category. Typically, discovering an internal language is a bespoke process, performed anew for each categorical structure of interest. In this talk, we propose a uniform type theory giving internal languages for much of the “substructural” hierarchy of categorical logic, including such structures as categories, multicategories (planar, symmetric, and cartesian), PROPs, Lawvere theories, (symmetric) monoidal categories, cartesian categories, and Markov categories. The framework builds on the speaker’s work in double-categorical logic, and is described in preliminary form at: next.catcolab.org/rfc/0004
Date: May 26, 2026
![[Berkeley Seminar] Evan Patterson | A uniform type theory for internal languages of cat. structures](https://i.ytimg.com/vi/mknZxNcn0Ho/mqdefault.jpg "[Berkeley Seminar] Evan Patterson | A uniform type theory for internal languages of cat. structures
Title: A uniform type theory for internal languages of categorical structures
Abstract: A hallmark of categorical logic is the interplay between category theory and type theory. A categorical structure can provide the semantics of a type theory; in the other direction, type theory can be used to describe the internal language of a categorical structure. The internal language is an ergonomic notation, based on elements and function application, to specify morphisms in the category. Typically, discovering an internal language is a bespoke process, performed anew for each categorical structure of interest. In this talk, we propose a uniform type theory giving internal languages for much of the “substructural” hierarchy of categorical logic, including such structures as categories, multicategories (planar, symmetric, and cartesian), PROPs, Lawvere theories, (symmetric) monoidal categories, cartesian categories, and Markov categories. The framework builds on the speaker’s work in double-categorical logic, and is described in preliminary form at: https://next.catcolab.org/rfc/0004
Date: May 26, 2026")
![[Oxford Seminar] Matteo Capucci | 2-classifiers for 2-algebras](https://i.ytimg.com/vi/mpHnD7peDS8/mqdefault.jpg "[Oxford Seminar] Matteo Capucci | 2-classifiers for 2-algebras
Oxford Seminar, 26th of June 2025
In this talk I will report on work in progress, joint with David Jaz Myers, about lifting discrete opfibration classifiers (2-classifiers, i.e. a Set-like object) from a 2-category K to the 2-category of algebras of a 2-monad T. In the setting of DOTS, we often construct behaviour functors as representables, but without a 2-classifier one cant really call these representables. Moreover, there is a strong connection between compositionality of such functors, the properties of the algebra they map out of, and the properties of the object(s) that represents them. These phenomena are in fact completely general, so we set out to better understand the situation and found some frankly interesting notions and results, chiefly a tight result on the existence of 2-classifiers for 2-algebras.")
![[Oxford Seminar] Nathan Haydon | Peirce’s Existential Graphs](https://i.ytimg.com/vi/oMDDqZsVlJE/mqdefault.jpg "[Oxford Seminar] Nathan Haydon | Peirce’s Existential Graphs
Oxford Seminar, 3rd of July 2025
Title: The second (graphical) calculus of relations: Peirce’s Existential Graphs
Abstract: C.S. Peirce’s Existential Graphs are a precursor to string diagrams as we know them today. In this talk I’ll discuss the assumptions that led Peirce to develop the graphs and give an overview of how Peirce’s work inspired recent developments in categorical logic.")
![[Berkeley Seminar] Hugo Paquet | Lazy categorical semantics of discrete probabilistic programming](https://i.ytimg.com/vi/oVsGsyDRFyQ/mqdefault.jpg "[Berkeley Seminar] Hugo Paquet | Lazy categorical semantics of discrete probabilistic programming
Title: Lazy categorical semantics of discrete probabilistic programming
Abstract: A lazy program interpreter postpones computation until the result is actually needed. This is typically more efficient than an eager (or call-by-value) interpreter, but the semantics is not generally the same.
In this talk I will discuss a new categorical semantics of lazy evaluation for algebraic effects, that relies on a subtle combination of name generation and read-only state. This semantic model suggests better intermediate representations of sum and product types in a lazy interpreter.
The practical motivation for this work is a real-world application of probabilistic programming, in which large algebraic data types cause significant performance issues. As I will explain, since probabilistic programming is described by an affine monad, one can use lazy evaluation to speed up the computation without affecting the semantics.
This is joint work with Simon Castellan (Inria, France).
Date: May 5, 2026")
![[Berkeley Seminar] Benjamin Brast McKie | The Construction of Possible Worlds](https://i.ytimg.com/vi/p-z4bhj7p-g/mqdefault.jpg "[Berkeley Seminar] Benjamin Brast McKie | The Construction of Possible Worlds
Title: The Construction of Possible Worlds
Abstract: Possible worlds are often taken to be complete histories of everything. Insofar as there are temporary sentences that are true at some times and false at other times, evaluating a sentence at a possible world does not fix its truth-value. Moreover, if possible worlds are taken to be primitive, evaluating sentences at world-time pairs invalidates a perpetuity principle that what is necessarily the case is always the case where imposing model constraints cannot validate these principles without undermining the significance of the truth-conditions for the language. Rather, this paper takes world states to be maximal possible ways for things to be at an instant where the task relation encodes the possible transitions between world states. Possible worlds are then defined as functions from times to world states as constrained by the task relation. Since sentences are assigned truth-values at world states, times are exogenous to the truth-conditions for the language, eliminating unnecessary degrees of freedom from the definition of a model. By evaluating sentences at world-time pairs, the resulting semantic theory validates a logic for tense and modality in which the perpetuity principles are theorems, providing a logical foundation for reasoning about future contingency.
Handout: https://benbrastmckie.com/wp-content/uploads/2025/11/talk.pdf
Paper: https://www.benbrastmckie.com/wp-content/uploads/2025/11/possible_worlds.pdf
Date: 11/25/2025")
![[Berkeley Seminar] Kris Brown | Categorical approaches to inferentialist semantics](https://i.ytimg.com/vi/pCUjgC5UKBQ/mqdefault.jpg "[Berkeley Seminar] Kris Brown | Categorical approaches to inferentialist semantics
Title: Categorical approaches to inferentialist semantics
Abstract: There are remarkable similarities between applied category theory and inferentialist semantics in the philosophy of language: a focus on making pre-existing structure explicit, sensemaking without rigid foundations, characterizing content in terms of external structure rather than internal structure, emphasis on open systems, and putting syntax and semantics in the same playing field. I will present some formalizations of inferentialism (a generalization of Girards phase semantics) due to Hlobil and Brandom and show some progress towards understanding what is taking place, categorically.
The accompanying slides are available at https://www.krisb.org/role/role-0034.xml, and a related blog post is available at https://topos.site/blog/2024-10-11-nonlogical-concepts/.
https://topos.site/events/berkeley-seminar/")
![[Oxford Seminar] David Jaz Myers | A modal proof of the nerve theorem](https://i.ytimg.com/vi/pKHjMXSfAX4/mqdefault.jpg "[Oxford Seminar] David Jaz Myers | A modal proof of the nerve theorem
Oxford Seminar, 24th of July 2025
Joint work with Mitchell Riley.
The nerve theorem is a classical result in homotopy theory,
usually attributed to Borsuk, which computes the homotopy type of a
(paracompact) topological space from a good open cover of it. An
open cover is good when finite intersections of opens in the cover
are contractible whenever they contain a point; the nerve of a good
open cover is the simplicial complex with a point for each open in the
cover, and an -simplex for each inhabited intersection of opens in the
cover. The nerve theorem states that the homotopy type of is the same
as the nerve of any good open cover of it.
In this talk, well prove the nerve theorem in the setting of modal
homotopy type theory. The key concepts in the nerve theorem — the
homotopy type of a space, the homotopy type presented by a simplicial
set, and even the Čech nerve itself — may all be understood as
modalities which act on simplicial, spatial homotopy types (that is,
simplicial stacks on a suitably topological site). Once we understand
the idea of modalities, well see that the nerve theorem is a
completely modal statement concerning the commutation of two sorts of
cohesion possible for types (in Lawveres sense): the combinatorial
cohesion of simplices, and the continuous cohesion of topology. As a
result, well actually be proving a nerve theorem for all spatial
stacks in a direct, conceptual way.")
