John Baez: Graphs with Polarities

In fields ranging from business to systems biology, directed graphs with edges labeled by signs are used to model systems in a very simple way: the nodes represent entities of some sort, and an edge indicates that one entity directly affects another either positively or negatively. Multiplying the signs along a directed path of edges lets us determine indirect positive or negative effects, and if the path is a loop we call this a positive or negative feedback loop. Here we generalize this to graphs with edges labeled by a commutative monoid, whose elements represent “polarities” possibly more general than simply “positive” or “negative”. We define a “1st homology monoid” of a directed graph, different from the usual 1st homology group in that it only detects directed loops. Finally, we describe the emergence of new feedback loops when we compose open graphs using a variant of the Mayer-Vietoris exact sequence for homology. This is joint work with Adittya Chaudhuri.

David Corfield: Type-theoretic expressivism

Philosophers have intensively studied the question of what makes logical vocabulary distinctively logical. The thesis of logical expressivism, as proposed by Robert Brandom (2000) and others, is that the role of logical vocabulary is to allow the explicit articulation of inferential relations. This occurs when we endorse instances of reasoning by means of assertions employing logical vocabulary, rather than merely implicitly doing so by reasoning in certain ways. For instance, in the case of propositional logic, implication is understood as allowing the explicit endorsement of the inference of one proposition from another via assertion of the associated hypothetical proposition. Now, from the perspective of dependent type theories, such as homotopy type theory (Corfield 2020), propositional and first-order logic may be considered as mere fragments by restrictions on the dependency structure and on the kinds of types allowed in its judgements. In this talk I will be exploring the extent to which the vocabulary of homotopy type theory may also be given an expressivist reading, touching also on its modal variants. I explore this issue through the lens of Robert Harper’s computational trinitarianism, interpreted via type theory’s Formation-Introduction-Elimination-Computation rules, category theory’s analysis of logic as a “web of adjunctions” (Lawvere); and computation as a means of explicit reasoning.

Igor Bakovic: Comma categories and 2-(co)monads in foundations and theoretical computer science

In my talk I describe a comma 2-comonad on the 2-category whose objects are functors, 1-cell are colax squares and 2-cells are their transformations. I give a complete description of the Eilenberg-Moore 2-category of colax coalgebras, colax morphisms between them and their transformations and I show how many fundamental constructions in formal category theory like adjoint triples, distributive laws, comprehension structures, Frobenius functors etc. naturally fit in this context. Then I proceed to describe various pseudo distributive laws between a comma 2-comonad and its cousins - the associated split fibration 2-monad and the associated split cofibration 2-monad. The former is an instance of a pseudodistributive law which Garner used in his description of Szabo’s polycategories, and the pseudoalgebras for the latter are Beck-Checalley fibrations. This contexts is related to Bunge and Funk admissible 2-monads whose Eilenberg-Moore 2-category of algebras are characterised in terms of (co)completeness. I describe the Kleisi 2-category of the associated split fibration 2-monad by means of its bifibrations which are defined by a certain bicomma object condition and the corresponding comprehensive factorization for those 1-cells which have an admissible domain. Finally I show how the work of Ghani, Fumex and Johann which generalized Hermida and Jacobs approach to induction and conduction in theoretical computer science has a natural interpretation in this context.

Luigi Caputi: Reachability categories and commuting algebras of quivers

In this talk, we will introduce the notion of reachability categories. These categories are obtained from path categories of quivers by taking quotients under the “reachability” relation. We will compare reachability categories to path categories, from both a topological and a categorical viewpoint. Then, we will focus on the category algebras of reachability categories, also known as commuting algebras. As application, we will prove that commuting algebras are Morita equivalent to incidence algebras of posetal reflections of reachability categories, a result previously obtained by E. L. Green and S. Schroll. If time allows it, we shall see further connections to magnitude homology, Hochschild cohomology, and persistent homology of graphs. This is joint work with H. Riihimäki.

Eleftherios Chatzitheodoridis: Rational complete Segal spaces

Complete Segal spaces, as introduced by Rezk, can be interpreted as categories up to homotopy that are enriched in spaces. Drawing motivation from rational homotopy theory, we introduce rational complete Segal spaces, which can be interpreted as categories up to homotopy that are enriched in rational spaces (spaces whose higher homotopy groups are rational vector spaces). In analogy with Rezk’s approach, we produce a model category whose fibrant objects are the rational complete Segal spaces, and we establish that it is compatible with the Cartesian closure of the Reedy model structure on bisimplicial sets.

Amartya Shekhar Dubey: Unital k-Restricted Infinity-Operads

The goal is to understand unital $\infty$-operads by their arity restrictions. Given $k \geq 1$, we develop a model for unital k-restricted $\infty$-operads, which are variants of $\infty$-operads with ($\leq k$)-arity morphisms, as complete Segal presheaves on closed k-dendroidal trees built from corollas with valence \leq k. Furthermore, we prove that the restriction functors from unital $\infty$-operads to unital k-restricted $\infty$-operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying k, the left and right adjoints give a filtration and a co-filtration for any unital $\infty$-operads by k-restricted $\infty$-operads. This is joint work with Yu Leon Liu.

Toby St Clere Smithe: AutoBayes: A Compositional Framework for Generalized Variational Inference

We introduce a new compositional framework for generalized variational inference, clarifying the different parts of a model, how they interact, and how they compose. We explain that both exact Bayesian inference and the loss functions typical of variational inference (such as variational free energy and its generalizations) satisfy chain rules akin to that of reverse-mode automatic differentiation, and we advocate for exploiting this to build and optimize models accordingly. To this end, we construct a series of compositional tools: for building models; for constructing their inversions; for attaching local loss functions; and for exposing parameters. Finally, we explain how the resulting parameterized statistical games may be optimized locally, too. We illustrate our framework with a number of classic examples, pointing to new areas of extensibility that are revealed.

Nathan Haydon: Continued Mathematical Developments in C.S. Peirce’s Philosophy

Authors: David Corfield, Matt Cuffaro, and Nathan Haydon

While philosophical questions often pervade mathematical thought, such themes are usually seen as peripheral to the practice itself. At the same time, mathematical and philosophical investigation sometimes do fruitfully interact. Here we motivate a broader connection between the two fields by taking the work of C.S. Peirce as an example. Peirce’s Existential Graphs have recently inspired developments in categorical logic [1] and we take the opportunity in this talk to provide a summary of these recent developments, and to highlight three areas within Peirce’s work we believe worth future mathematical treatment: further directions in the Existential Graphs, the distinction between abduction, deduction, and induction, and his broader theory of semiotics.

[1] Bonchi, F., Di Giorgio, A., Haydon, N., and Sobocinski, P. (2024). Diagrammatic algebra of first order logic. In Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’24, New York, NY, USA. Association for Computing Machinery.

Nathaniel Osgood: A Stochastic Transition Semantic Domain for Categorical Stock and Flow Models

System Dynamics is a diagram-centric tradition of dynamic modelling for understanding, characterizing, analyzing and managing complex systems. The methodology approaches such complex systems from the perspective of feedbacks and accumulations. To support insight into system structure involving feedbacks and accumulations, System Dynamics employs several types of diagrams that articulate aspects of system structure at successive levels of specificity: causal loop diagrams, system structure diagrams, and stock & flow diagrams. Of these, stock & flow diagrams provide sufficient information to yield well-defined dynamics, including formulas for flows and auxiliary/dynamic variables, and values for initial values of stocks and for constants and parameters. Within System Dynamics practice, the large majority of applications assume that the behaviour of stock & flow diagrams is characterized by ordinary differential equations (ODEs). That ODE interpretation is the sole type of behaviour over time supported by System Dynamics software packages. However, a small but notable subset of applications of stock & flow diagrams force-fit draws from distributions into the formulas associated with the model, so as to allow for stochastic evolution. The resulting formulations are error prone, limit transparency of the formulas, and fragile under changes of integration scheme.

Starting with the work of Baez et al. in 2023, researchers in applied category theory have over the past two years established an increasingly rich categorical foundation for System Dynamics modeling. For stock & flow diagrams, this approach has involved, amongst other factors, a characterization of closed stock & flow diagrams as copresheaves, homomorphisms between such diagrams as natural transformations, open stock & flow diagrams via structured cospans. This work has further mathematically characterized three types of composition of stock & flow diagrams, and formulated stratification of stock & flow diagrams via pullback. Paralleling their work on the underlying mathematics, the researchers have further provided support for such operations with StockFlow.jl, and further contributed ModelCollab, a tool for real-time collaborative creation, composition, browsing, and analysis of System Dynamics models. While these mathematical frameworks, and StockFlow.jl atop them, offer a separation of syntax from semantics, they have heretofore limited themselves to ODE interpretation of stock & flow diagrams. Provision of alternative dynamic interpretations of stock & flow diagrams could offer considerable additional flexibility to System Dynamics modelers using StockFlow.jl -- supporting ready switching between semantics, rather than hardcoding the semantics in the stock & flow model formulation. These tools could further enhance the reach of ModelCollab, and provide additional insight into the dynamics of systems whose states consist of accumulations of discrete quantities of entities -- such as are common in health and social spheres.

We describe here the implementation of a stochastic transition semantics for stock & flow diagrams in StockFlow.jl. This interpretation is appropriate for diagrams where stocks represent discrete collections of entities, which comprise an important subset of such diagrams used in stock & flow practice.

Given such a context, this approach treats stocks as containing a natural number of entities, and flows between such stocks as Poisson processes. The Poisson process for a given flow is associated with a transition rate 𝜆 equal to the rate of that flow, as characterized by its formula. As is typical for Poisson transitions, over any given interval of time during the model time horizon, there is a natural number of transitions that occur. This interpretation allows a given model to be interpreted according to either ODE semantics or state transition semantics.

We are currently seeking to fully integrate this alternative semantics into StockFlow.jl and ModelCollab. We also seek to allow the modeler greater flexibility in designating certain stocks -- and the flows between them, or flows for the stock entering or leaving the model -- as discrete, and others as continuous.

We see this work as bringing categorical modeling of stock & flow diagrams a step closer to exploiting the flexibility offered by the separation of syntax and semantics, and will allow such categorical methods to offer additional value in the context of contemporary System Dynamics process.

Nathaniel Osgood Compositional Public Health: Current Practice and Future Opportunities

Compositional public health is an emerging research field seeking to address the complexity of epidemiological understanding and dynamics, and associated with public health responses. The field lies at the intersection of category theory, epidemiology, systems science, and engineering, and utilizes tools from applied category theory for public health applications. In this talk, we will draw on motivating examples within public health to contextualize the emergence of this area of study and practice, provide an overview of contemporary methods in this space, and potential research directions within this rapidly developing field.

To highlight frontiers of work within this field, we highlight three primary spheres of support and develoment: participatory modeling and systems thinking via categorical System Dynamics (supported by StockFlow.jl and CatCoLab), compositional public health informatics and data science (particularly supported by ACSets.jl, AlgebraicRelations.jl, and DataMigrations.jl), and agent-based and hybrid modeling (AlgebraicABMs.jl, StockFlow.jl, DataMigrations.jl). Most of these lines of work make central use of copresheaves, Catlab, and other elements of the AlgebraicJulia ecosystem.

Within each sphere of work, we note compelling affordances supported by the current state of development, the categorical mechanisms and approaches that make them possible, and ongoing lines of development. We also highlight specific lines of public health research in each area employing these compositional approaches to secure public health insights.

As we conclude our talk, we will present a variety of unmet needs and potential research opportunities within compositional public health. For example, we will discuss the high potential for enhancing integrated understanding of context via sheaves, and the declarative data migration via lifting morphisms. Furthermore, we will discuss additional topics that have emerged which are of distinctive interest to compositional public health, such as the equity implications of preserving relationships across missing data, and the value for public health of capturing metadata or paradata for model quantities and data sources.

We expect that attendees of our talk will come to appreciate the diversity of avenues by which applied category theory can be used in public health contexts, challenges and gaps in the study and practice of public health that motivate such approaches, how compositional public health is emerging as its own field of study, and ways to get involved with efforts around this exciting and rapidly developing new area of applied mathematics research.