Speakers

Title: A Compositional Framework for System Dynamics Modeling using Category Theory

Abstract: This presentation will introduce a modular and compositional framework for system dynamics, developed using category theory. System dynamics is a computer-aided approach for decision and policy making in complex dynamical systems. By leveraging categorical constructions, we establish a separation between syntax—the diagrammatic languages such as stock-and-flow diagrams—and semantics, such as ordinary differential equations (ODEs). Stock-and-flow diagrams (closed) are represented categorically as C-Sets (also known as copresheaves), enabling structure-preserving mappings via homomorphisms and functorial translations across different diagrammatic tools.

We further extend closed dynamical systems to open ones using decorated/structured cospans, which allows large, complex systems to be built compositionally from smaller components applying pushouts. Pullbacks in the category of closed diagrams are applied to support stratification, enabling the construction of refined model structures.

Finally, I will demonstrate applications of this framework to real-world problems in the public health domain, using the Julia package StockFlow.jl, which is implemented within the AlgebraicJulia ecosystem and built on Catlab.jl. This work illustrates how categorical methods provide a principled foundation for modular, compositional, and scalable modeling in system dynamics.

Title: The dimension of the tangent bundle and the universality of the vertical lift
Abstract: Tangent categories are a categorical generalization of the differentiation structure in the category of smooth manifolds. Part of the definition of a tangent category is a condition called the universality of the vertical lift. It is often the most difficult and hardest to check condition when showing that something is a tangent category.
We will define a notion of monoid valued dimension to give an intuition for the consequences of the universality of the vertical lift. Examples for such dimensions are the dimension of smooth manifolds, the cardinality of sets (in the opposite category) and the Betti numbers of CW complexes (in the opposite category).
Using this notion of dimension we will see that the universality of the vertical lift restricts the possibilities for tangent structures on a given category.

Title: Sheaf theoretic structure of contextuality

Abstract: In this talk, I will present Abramsky and Brandenburger’s sheaf-theoretic framework of non-locality and contextuality in quantum theory. We will begin by setting up the mathematical framework of sheaf theory and empirical models. I will present in detail the central theorem, which gives a correspondence between the existence of local hidden variables, at the ontological model and the empirical model having a global section, at the operational level.

We will use Bell’s model as our running example to exhibit the non-existence of the global section using linear algebraic methods. We will see how this framework leads to a hierarchy of no-go theorems in quantum theory. The formalism presented here applies to any compatible empirical model (quantum theory being one of them), and thus has applications in topics beyond quantum. Time permitting, we will comment on a couple of them. 

Title: Universally Quantified Type Inference

Abstract: Haskell’s type system is based on the Hindley–Milner type system, which supports only rank-1 types. As functional languages have evolved, higher-rank types have been recognized as being useful for expressing more general abstract functions. Peyton Jones et al. introduced a type inference algorithm for universally quantified types in Haskell. However, full type inference for higher-ranked types is undecidable and consequently often requires explicit type annotations to facilitate type inference. In this work, we propose a simplified algorithm for inferring these types. Our approach is demonstrated using a minimal programming language syntax and provides a modular method for inferring higher-ranked types. The inference process is organized into two distinct steps: generating type equations and then solving them. By separating the collection of equations from their resolution, this work gives a more modular inference algorithm, which is slightly more general than the bidirectional system introduced by Peyton Jones.

Title: Actegories and Copowers with Application to Message Passing Semantics

Abstract: Reversible computing considers those computational operations which are reversible over a state space. 

Generalized reversible computing aims to generalize the notion of reversible computing (closer to engineering) by considering a probability distribution over the initial set of states and considering reversibility of only those states with non-zero probability. 

Not long ago, Michael P.Frank proposed a mathematical framework based on set theory for setting up the fundamental theorem generalized reversible computing relating (conditional) reversibility of an operation and entropy ejection by the operation. My colleague Clemence Chanavat and I realized that the framework has a categorical flavor.   

In this talk, I aim to set up generalized reversible computing starting with partial Markov categories, especially, the Kleisli category of sub-distribution monads over partitioned sets. Once this is done, the fundamental theorem of generalized reversible computing can be set up as a functor of resource theories between the above mentioned Kleisli category into the indiscrete category of positive reals, [0,\infty]. 

This is a research in progress and would very much appreciate audience interaction! 

Title: Actegories and Copowers with Application to Message Passing Semantics

Abstract: In this talk we prove that giving a right actegory with hom-objects is equivalent to giving a right-enriched category with copowers. While this result is known in the closed symmetric setting, our contribution extends the equivalence to non-closed and non-symmetric monoidal bases. This generalization is motivated by the semantics of higher-order message passing in the \textbf{Categorical Message Passing Language (CaMPL)}, a concurrent language whose semantics is given by a linear actegory. A desirable feature for this language is the support of higher-order processes: processes that are passed as first class citizens between processes. While this ability is already present in any closed linear type systems — such as {\bf CaMPL}’s — to support arbitrary recursive process definitions requires the ability to reuse passed processes. Concurrent resources in {\bf CaMPL}, however, cannot be duplicated, thus, passing processes as linear closures does not provide the required flexibility. This means processes must be passed as sequential data and the concurrent side must be {\em enriched} in the sequential side, motivating the technical result of this paper.

Title: Cartesian Linearly Distributive Categories: Revisited

Abstract: Linearly distributive categories (LDC) were introduced by Cockett and Seely as alternative categorical semantics for multiplicative linear logic, taking conjunction and disjunction as primitive notions. Given that a LDC has two monoidal products, it is natural to ask when these coincide with categorical products and coproducts. Such LDCs, known as cartesian linearly distributive categories (CLDC), were introduced alongside LDCs. Initially, it was believed that CLDCs and distributive categories would coincide, but this was later found not to be the case. Consequently, the study of CLDCs was largely abandoned. In this talk, we will revisit the notion of CLDCs, demonstrating strong structural properties they all satisfy and investigating two key classes of examples: bounded distributive lattices and semi-additive categories. Additionally, we re-examine a previously assumed class of CLDCs, the Kleisli categories of exception monads of distributive categories, and show that they do not, in fact, form CLDCs.

Title: More on the Categorical and Differential Algebra of Commutative Rigs

Abstract: In this talk we’ll continue our friendly introduction to the differential algebraic theory of commutative rings by moving towards answering the question: “In what sense are the Kahler differentials functorial in the first place?” We’ll start by introducing fibrations and their morphisms, move on to briefly discussing the Grothendieck Construction (which allows one to move between fibrations and pseudofunctors, depending on taste),  and then explain the sense in which taking the module of relative Kahler differentials is functorial in CRig. 

The scope and vibes of this talk are meant to be more relaxed and informal, so if you’ve never met rigs before and want to get to know how their differential algebra works this is a fantastic opportunity (which of course I’d say, for I am the speaker) to see. Alternatively, this makes precise many constructions involving Kahler differentials which are at best ad-hoc in traditional commutative ring theory.

Title: The Differential Algebra of Commutative Rigs: Derivations and Differentials

Abstract: In the last talk I gave on commutative rigs, we got to know a bit about how the categories of commutative rig algebras and their modules interact together. In this talk we’ll continue towards our ultimate goal of understanding the differential/algebraic geometry of rigs by getting to know how the theory of derivations and differentials interacts with rigs. I’ll start by recalling our main cast of characters, define what it means to be a derivation, define the rig module of Kahler differentials, and then start to show some of the properties these modules have. Depending on time we’ll discuss how to encode derivations as solutions to specific lifting problems and/or in what sense the Kahler differentials construction is functorial. This talk will be relatively gentle in delivery, so all are encouraged to come and see how cool rigs are! 

Title: Generalising the Coecke-Pavlovic-Vicary Correspondence

Abstract: 

In their paper [1], Coecke-Palvovic-Vicary (CPV) gives a correspondence between orthogonal bases in finite-dimensional Hilbert Spaces (FdHilb) and Commutative-Dagger Frobenius algebras in FdHilb. In the first part of my talk, I will go over this correspondence and the associated categorical statements.

In the second part of my talk, I will give an outline of the programme about how this correspondence can be generalised to arbitrary dimensions. I will first introduce the ingredients involved- Finiteness spaces, Lefschetz Spaces and time-permitting linear monoids and then will give a few results connecting these mathematical objects. 

Reference:

[1] Coecke, B., Pavlovic, D., & Vicary, J. (2012). A new description of orthogonal bases. Mathematical Structures in Computer Science, 23(3).