![[Berkeley Seminar] CB Aberle: All Concepts are Essentially Algebraic @ToposInstitute](https://i.ytimg.com/vi/RmiTOa4b0bA/maxresdefault.jpg)
Uploaded August 2024 | Updated September 2026, 2 weeks ago
Title: All Concepts are Essentially Algebraic
Abstract: Lawvere's categorical formulation of algebraic theories enables one to study some of the most common structures found in mathematics – e.g. groups, rings, etc. – at a high level of precision and generality. However, many significant mathematical concepts, including categories, topological spaces, etc., turn out not to be algebraic, in this sense. Notably, the very framework used by Lawvere to describe algebraic theories and their models – categories with finite products and product-preserving functors between them – cannot be described as an algebraic theory, and so it seems that algebra alone cannot encompass the whole of mathematics (nor even itself). There is, however, a deeper sense in which all of mathematics is essentially algebraic. What is needed to reveal this fact is to adapt the classical notion of algebraic theories, which are fundamentally simply typed, to an appropriate notion of dependently typed algebraic theories. At this level of generality, one is capable of defining not only individual mathematical structures, but structures that themselves encompass whole universes of mathematics, including topoi, models of type theory, etc. In particular, the theory of dependently-typed algebraic theories is itself describable as a dependently-typed algebraic theory. This fact has many profound consequences, of which I shall highlight just one: using the framework of dependently-typed algebraic theories, one can construct a type theory whose types themselves correspond to type theories, with functions between these types corresponding to translations between the corresponding type theories.
https://topos.site/events/berkeley-seminar/
Title: All Concepts are Essentially Algebraic
Abstract: Lawvere's categorical formulation of algebraic theories enables one to study some of the most common structures found in mathematics – e.g. groups, rings, etc. – at a high level of precision and generality. However, many significant mathematical concepts, including categories, topological spaces, etc., turn out not to be algebraic, in this sense. Notably, the very framework used by Lawvere to describe algebraic theories and their models – categories with finite products and product-preserving functors between them – cannot be described as an algebraic theory, and so it seems that algebra alone cannot encompass the whole of mathematics (nor even itself). There is, however, a deeper sense in which all of mathematics is essentially algebraic. What is needed to reveal this fact is to adapt the classical notion of algebraic theories, which are fundamentally simply typed, to an appropriate notion of dependently typed algebraic theories. At this level of generality, one is capable of defining not only individual mathematical structures, but structures that themselves encompass whole universes of mathematics, including topoi, models of type theory, etc. In particular, the theory of dependently-typed algebraic theories is itself describable as a dependently-typed algebraic theory. This fact has many profound consequences, of which I shall highlight just one: using the framework of dependently-typed algebraic theories, one can construct a type theory whose types themselves correspond to type theories, with functions between these types corresponding to translations between the corresponding type theories.
https://topos.site/events/berkeley-seminar/
![[Berkeley Seminar] CB Aberle: All Concepts are Essentially Algebraic](https://i.ytimg.com/vi/RmiTOa4b0bA/mqdefault.jpg "[Berkeley Seminar] CB Aberle: All Concepts are Essentially Algebraic
Title: All Concepts are Essentially Algebraic
Abstract: Lawveres categorical formulation of algebraic theories enables one to study some of the most common structures found in mathematics – e.g. groups, rings, etc. – at a high level of precision and generality. However, many significant mathematical concepts, including categories, topological spaces, etc., turn out not to be algebraic, in this sense. Notably, the very framework used by Lawvere to describe algebraic theories and their models – categories with finite products and product-preserving functors between them – cannot be described as an algebraic theory, and so it seems that algebra alone cannot encompass the whole of mathematics (nor even itself). There is, however, a deeper sense in which all of mathematics is essentially algebraic. What is needed to reveal this fact is to adapt the classical notion of algebraic theories, which are fundamentally simply typed, to an appropriate notion of dependently typed algebraic theories. At this level of generality, one is capable of defining not only individual mathematical structures, but structures that themselves encompass whole universes of mathematics, including topoi, models of type theory, etc. In particular, the theory of dependently-typed algebraic theories is itself describable as a dependently-typed algebraic theory. This fact has many profound consequences, of which I shall highlight just one: using the framework of dependently-typed algebraic theories, one can construct a type theory whose types themselves correspond to type theories, with functions between these types corresponding to translations between the corresponding type theories.
https://topos.site/events/berkeley-seminar/")
![[Berkeley Seminar] David Espinosa: Monad translations compose](https://i.ytimg.com/vi/Snvwc4JVIYc/mqdefault.jpg "[Berkeley Seminar] David Espinosa: Monad translations compose
Title: Monad Translations Compose
Abstract: Many people have observed that monad translations compose. We give this idea a try using the ML module system and see what mileage we can get out of it.
https://topos.site/events/berkeley-seminar/")
![[2-torial] Categorical algebraic geometry, Part 1](https://i.ytimg.com/vi/SwnZ_i0t86g/mqdefault.jpg "[2-torial] Categorical algebraic geometry, Part 1
2-torial, July 28 2026
Speaker: Tim Hosgood
There are many approaches to algebraic geometry, and many different ways to arrive at the subject. Here were going to take an incredibly specific and biased approach: what if you already love 2-categories and want to be able to say the phrase fpqc sheaf as quickly as possible, but not /too/ quickly? In this series of exercises we will build towards an understanding of *relative algebraic geometry*, which allows us to work in arbitrary (nice) symmetric monoidal categories.
*Prerequisites.* Quotients of rings, Yoneda embedding, 2-limits, symmetric monoidal categories (cosmoi).
*Key concepts.* Functor of points, affine scheme, quasi-coherent sheaf, Grothendieck pseudofunctor, pre-topology, faithfully flat topology, algebra over a commutative monoid, sheaf, affine scheme (again).
*Further reading.*
- Bertrand Toën, Michel Vaquié, Under Spec Z. [arXiv:math/0509684]
- Bertrand Toen, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications. [arXiv:math/0404373]
[arXiv:math/0509684] https://arxiv.org/abs/math/0509684
[arXiv:math/0404373] https://arxiv.org/abs/math/0404373")
![Rory Lucyshyn-Wright: V-graded categories [...] for enrichment and actions of monoidal categories](https://i.ytimg.com/vi/UR9LUUfaJiM/mqdefault.jpg "Rory Lucyshyn-Wright: V-graded categories [...] for enrichment and actions of monoidal categories
Topos Institute Colloquium, 24th of April 2025.
———
Enriched categories have hom-objects in a monoidal category V, but their theory is usually formulated under the assumption that V is biclosed and so is enriched in itself. Categories equipped with an action of V (or V-actegories) provide a related setting with the advantage that an arbitrary monoidal category V can always be regarded as a V-actegory. Richard Wood delineated a setting subsuming both V-enriched categories and V-actegories by considering V-graded categories, which are categories enriched in a monoidal category of presheaves on V but admit also a direct and elementary definition in terms of a notion of morphism with an additional parameter in V. Graded categories have also been called procategories (by Kelly-Labella-Schmitt-Street) and locally V-graded categories (by Levy).")
![[2-torial] José tells Jason about coalgebraic-modal extensions of logic](https://i.ytimg.com/vi/UVj3BDy0iaU/mqdefault.jpg "[2-torial] José tells Jason about coalgebraic-modal extensions of logic
[2-torial] José tells Jason about coalgebraic-modal extensions of logic")
![[Oxford Seminar] Paolo Perrone | Descent in Probability Theory: the first steps downward](https://i.ytimg.com/vi/VG2RTE1R0BY/mqdefault.jpg "[Oxford Seminar] Paolo Perrone | Descent in Probability Theory: the first steps downward
Oxford Seminar, October 16 2025
Speaker: Paolo Perrone
Full Title: Descent in Probability Theory: the first steps downward
Abstract: Coarse-graining, forming quotients by dropping distinctions, is a unifying idea across mathematics: identifying the endpoints of an interval yields a circle; groups are conveniently presented as quotients of free ones; sheaves and stacks emerge from gluing local data. This idea is also central to probability, where “observing” a random variable similarly quotients a sample space via the sigma-algebra it generates (and is crucial for modeling randomness as ignorance). Yet this quotienting procedure, in probability, has so far lacked a systematic categorical treatment.
We develop a descent theory for probability that makes this intuition precise, while respecting probabilistic practice as much as possible. On the category theory side, the theory parallels classical descent, but diverges in a few ways due to the presence of stochastic dependence (correlations). On the probability side, it unifies the three core concepts of measurability, disintegration and stochastic dominance, within a single framework, providing conceptual understanding of the relationships between random variables, statistical experiments, and inference procedures.")
![[Oxford Seminar] B. Scot Rousse | Who cares about values?](https://i.ytimg.com/vi/VJZUZ37gj2Y/mqdefault.jpg "[Oxford Seminar] B. Scot Rousse | Who cares about values?
Oxford Seminar, 19th of June 2025
Today it is common to hear about “human values” and the importance of designing technologies that “align” with our values. But where does this notion of “human values” come from? In this talk I trace the history of the concept of human values. I connect this notion with an evolution in our understanding of human autonomy, and argue that both are inadequate abstractions for the challenges of being human in our technological age. Finally, I introduce the notion of care as an alternative to “values,” showing how it furnishes a subtler map for imagining and shaping our relationship to technology today.")
