Uploaded October 2025 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 23rd of October 2025.
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The conventional understanding, which describes symmetry using group theory, is insufficient, because symmetry can be anomalous, higher-group, and/or non-invertible (those symmetries are called algebraic higher symmetries in the paper). We find that all such finite symmetries - when considered up to holo-equivalence - are precisely classified by topological orders in one higher dimension (that can host a gappable boundary). This fundamentally shifts our perspective: symmetry is not inherently about groups, but is more naturally described by the mathematics of braided fusion higher categories.
Topos Institute Colloquium, 23rd of October 2025.
———
The conventional understanding, which describes symmetry using group theory, is insufficient, because symmetry can be anomalous, higher-group, and/or non-invertible (those symmetries are called algebraic higher symmetries in the paper). We find that all such finite symmetries - when considered up to holo-equivalence - are precisely classified by topological orders in one higher dimension (that can host a gappable boundary). This fundamentally shifts our perspective: symmetry is not inherently about groups, but is more naturally described by the mathematics of braided fusion higher categories.
![[Oxford Seminar] Amitai Nachmany | Hyperdoctrines in DOTS](https://i.ytimg.com/vi/ve6Xotcuxbs/mqdefault.jpg "[Oxford Seminar] Amitai Nachmany | Hyperdoctrines in DOTS
Oxford Seminar, August 27 2026
You can view the listing for this talk online at https://topos.institute/events/oxford-seminar/talks/2026-08-27_nachmany_hyperdoctrine.html
Speaker: Amitai Nachmany
Full Title: Hyperdoctrines in DOTS
Abstract: In this talk we motivate and define (regular) hyperdoctrines and sketch how to view them as double-operadic systems theories. We relate this to the notion of /systems with certificates/, which allows us to formally verify a systems theorys properties in a way that is reminiscent of the internal language of a category. Finally, we sketch a program to generalise this procedure to other logics (and functorialise it), and express some of the benefits of this program for strengthening the toolbox of categorical systems theory.
*Assumed knowledge:* General category theory, especially monoidal categories, double categories and adjunctions. Familiarity with functorial semantics and/or DOTS would help but isnt necessary.")
![[Berkeley Seminar] Owen Lynch | Stateful Lenses](https://i.ytimg.com/vi/veQbMfPSi9M/mqdefault.jpg "[Berkeley Seminar] Owen Lynch | Stateful Lenses
Title: Stateful Lenses - A Recipe for Expressive Systems-Theoretic Cartesian Double Categories
Abstract: This is a talk on work-in-progress that was started with David Spivak, and this abstract should be seen as well-motivated conjecture rather than math that has been completely worked out. Double categorical systems theory tells us how to unify various types of system under a single heading: operad algebras of symmetric monoidal double categories. Both resource sharers being composed by undirected wiring diagrams and Moore machines being composed by directed wiring diagrams are examples of double categorical systems theories. However, it turns out that both resource sharers and Moore machines can be found in the same (cartesian!) double category: resource sharers make up the horizontal morphisms into the (vertically) terminal object, and Moore machines make up the horizontal morphisms out of the (vertically) terminal object. Even better, variants of this construction produce both discrete and continuous resource sharers/Moore machines. Another special case of these stateful lenses include the energy-driven open systems from Spivak, Capucci, and _s Organizing Physics paper. Finally, the fact that stateful lenses form a cartesian double category dramatically reduces the amount of structure in formulating them compared to the double operad/operad algebra perspective on Moore machines/resource sharers.
https://topos.site/events/berkeley-seminar/")
![[2-torial] Tim tells Jason about Deformation Theory [3/3]](https://i.ytimg.com/vi/vxceguk8DRY/mqdefault.jpg "[2-torial] Tim tells Jason about Deformation Theory [3/3]
[2-torial] Tim tells Jason about Deformation Theory [3/3]")
![[2-torial] Tim tells Jason about Deformation Theory [2/3]](https://i.ytimg.com/vi/w1tKHbphFlc/mqdefault.jpg "[2-torial] Tim tells Jason about Deformation Theory [2/3]
[2-torial] Tim tells Jason about Deformation Theory [2/3]")
![[Oxford Seminar] David Corfield | Categorical systems theory: control and emergence](https://i.ytimg.com/vi/wSWmHZNjpzg/mqdefault.jpg "[Oxford Seminar] David Corfield | Categorical systems theory: control and emergence
Oxford Seminar, March 5 2026
Speaker: David Corfield
Also appearing in this recording are David Jaz Myers and Matteo Capucci.
Full Title: Categorical systems theory: control and emergence
Abstract: This will be an informal session with plenty of discussion time, investigating a couple of concepts that arise from the category-theoretic treatment of systems.
*(1) Control*
How inputs to a system regulate its behaviour. Starting points: (a) In response to the active inference program, some recent articles (e.g., https://arxiv.org/abs/2406.07577 and https://arxiv.org/abs/2508.06326) have looked to understand autonomous systems as composed of agent and controller subsystems, equipped with dual interfaces; (b) Ordinary Lyapunov functions have been treated category-theoretically (https://arxiv.org/abs/2502.15276), work that should be extendable to variants. Where ISS (input-to-state stability) Lyapunov functions concern stability under any external perturbation, control Lyapunov functions concern stability under a chosen input.
*(2) Emergence*
Phenomena where the composite behaviour of the parts does not equate to the behaviour of the composite. Starting points: (a) Elie Adams thesis, Systems, Generativity and Interactional Effects (https://elieadam.com/eadam_PhDThesis.pdf); (b) Puca et al. on Failures of compositionality (https://arxiv.org/abs/2307.14461) (c) Erik Hoel on causal emergence (e.g., https://arxiv.org/abs/2202.01854). Two relevant CT constructions appear to be laxness of functors and coarse-graining as epimorphisms, potentially fitting well with a double category-theoretic outlook.")
![[TopOx] Tom Leinster: The many faces of magnitude](https://i.ytimg.com/vi/wxqfxRCGoOE/mqdefault.jpg "[TopOx] Tom Leinster: The many faces of magnitude
16th of February 2026. Slides available at https://topos.institute/events/topox/
The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, enables us to define the magnitude of many objects in different contexts across mathematics. All of them can be seen as measures of size. For example, the magnitude of a metric space combines classical quantities such volume, surface area, and dimension. The magnitude of a category is closely related to Euler characteristic. The magnitude of a graph is an invariant sharing features with the Tutte polynomial (but not a specialization of it). Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. And there is now a theory of magnitude homology, which has the same relationship to magnitude as ordinary homology does to Euler characteristic. I will give an aerial view of this landscape.")
