Uploaded October 2024 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 16th of October 2024.
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It is an elementary fact from real analysis that any monotone bounded sequence of real numbers converges. It turns out that the monotone convergence theorem can be given an equivalent finitary formulation: roughly that any sufficiently long monotone bounded sequence experiences long regions where the sequence is metastable. This so-called "finite convergence principle" is carefully motivated and discussed by Terence Tao in a 2007 blog post ('Soft analysis, hard analysis, and the finite convergence principle'), but was already known to proof theorists, where the use of logical methods to both finitize infinitary statements and provide uniform quantitative information for the finitary versions plays a central role in the so-called proof mining program.
Topos Institute Colloquium, 16th of October 2024.
———
It is an elementary fact from real analysis that any monotone bounded sequence of real numbers converges. It turns out that the monotone convergence theorem can be given an equivalent finitary formulation: roughly that any sufficiently long monotone bounded sequence experiences long regions where the sequence is metastable. This so-called "finite convergence principle" is carefully motivated and discussed by Terence Tao in a 2007 blog post ('Soft analysis, hard analysis, and the finite convergence principle'), but was already known to proof theorists, where the use of logical methods to both finitize infinitary statements and provide uniform quantitative information for the finitary versions plays a central role in the so-called proof mining program.
![[DOTS Lectures] 10. LTL and specifications of behaviours](https://i.ytimg.com/vi/qpWD16mOwr0/mqdefault.jpg "[DOTS Lectures] 10. LTL and specifications of behaviours
*The paper referred to at the end was Temporal Landscapes: A Graphical Logic of Behavior by Brendan Fong, Alberto Speranzon and David I. Spivak.
This talk is part of a lecture series on the Double Operadic Theory of Systems (DOTS) presented by David Jaz Myers.
Some material from these lectures can be found in Davids book on categorical systems theory:
https://www.davidjaz.com/Papers/DynamicalBook.pdf")
![[Oxford Seminar] David Jaz Myers | Composing flavoured Petri nets](https://i.ytimg.com/vi/s793leHjc_4/mqdefault.jpg "[Oxford Seminar] David Jaz Myers | Composing flavoured Petri nets
Oxford seminar, 28th of August 2025
Abstract: Well describe a doctrine of various flavors of Petri nets in the double operadic theory of systems framework.")
![[Oxford Seminar] David Jaz Myers | Compositionality of Flavoured Petri Nets](https://i.ytimg.com/vi/sQW9KI9hdTI/mqdefault.jpg "[Oxford Seminar] David Jaz Myers | Compositionality of Flavoured Petri Nets
Oxford Seminar, 11th of September 2025
In this talk, we will see a natural notion of nesting composition for flavoured Petri nets: Petri nets whose places and transitions come with extra data determined by a symmetric monoidal double category and which determine the intended semantics of the Petri net.")
![[Berkeley Seminar] Owen Lynch | Abstract interpretation for semi-dependent type theories](https://i.ytimg.com/vi/tAKqODzU908/mqdefault.jpg "[Berkeley Seminar] Owen Lynch | Abstract interpretation for semi-dependent type theories
Title: Abstract interpretation for semi-dependent type theories
Abstract: ML-style module systems have a long and rich history which ultimately stems back to Lawvere’s seminal work on algebraic theories. While ML-style module systems have typically been confined to ML-descendants, analogous PL constructs that may be more familiar include Java-style interfaces or Haskell-style type classes. In my research for SGAI, I have adopted the perspective that module signatures are the appropriate programming language analogue for theories in a variety of doctrines of categorical algebra. Classical module signatures as found in ML can be are syntactic presentations of “theories for the doctrine of cartesian closed categories”; it is productive to vary “cartesian closed category” to “cartesian category”, “finite limit category”, “regular hyperdoctrine” (the categorical version of a restricted form of first-order logic), “symmetric monoidal categories”, etc. Then the module language for a doctrine extends the well-known internal language for the doctrine by adding “genericism”. For instance, it is well-known that the internal language of a symmetric monoidal category is given by a “linear do notation” where each variable may only be consumed once. However, typically an implementation of this internal language would fix a specific symmetric monoidal category. The module language allows varying the symmetric monoidal category, and potentially migrating expressions between various symmetric monoidal categories. This is of interest to SGAI because presentations of symmetric monoidal categories (e.g., module signatures for the symmetric monoidal doctrine) are Petri nets, and morphisms between them (e.g., module functors for the symmetric monoidal doctrine) are “hierarchical Petri nets”, where each transition in the codomain Petri net is associated with a process in the domain Petri net. Other applications to the SGAI program include the observation, going back to early work in ACT by Spivak, that databases schemas are productively thought of as theories in certain doctrines, with the choice of doctrine depending on the feature-set of the database engine.
Date: 2025-12-09")
