Applied Category Theory

The Influence of John and the Evolution of Category Theory

Introduction to Influences

• The speaker reflects on the significant influence John had on their philosophical journey, particularly in mathematics.

• Engaging with John's writings provided confidence to access advanced mathematical and scientific knowledge.

The Nature of Their Collaboration

• The speaker expresses gratitude for John's reciprocal influence, highlighting the excitement of mathematicians engaging with philosophers.

• Focus shifts to applied category theory, emphasizing its historical context rather than just mathematical content.

Traditional Views on Mathematics Disciplines

• A traditional model depicts disciplines communicating primarily with neighboring fields, from pure mathematics to business applications.

• Category theory is often viewed as a highly abstract discipline that only interacts with other mathematicians.

Challenging Conventional Perspectives

• The notion of "applied category theory" was initially met with skepticism due to its perceived contradiction to established views.

• Computer science emerged around 1980 as a pivotal force disrupting traditional information flow between disciplines.

Interconnectedness of Fields

• Computer science's direct interaction across various domains highlights its relevance in both pure and applied mathematics.

• The advent of computers has transformed communication and thought processes across all academic fields.

Reconceptualizing Category Theory's Role

Category Theory's Broader Impact

• Proposes viewing category theory as interconnected across the spectrum from pure to applied mathematics rather than isolated at one end.

• Despite being less recognized than computer science, category theory plays a crucial role in understanding networks and systems.

Practical Applications in Business

• Contrary to initial skepticism, businesses are emerging that utilize principles from category theory effectively.

• The talk aims to explore how category theory is being applied throughout various sectors while addressing challenges faced along the way.

Historical Context and Personal Reflection

Self-Centered Historical Perspective

• Acknowledges that the forthcoming history shared will be subjective and may not encompass all contributors or developments in applied category theory.

Introduction to Functorial Semantics

Concept Overview

• In 1963, Bill of Beer introduced Functorial Semantics, a framework that connects syntax (how expressions are written) with semantics (the meaning of those expressions).

• There is no single functor; various functors can be chosen from category C to other categories, allowing for different interpretations of expressions.

Application in Computer Science

• Programming languages can be viewed as categories where objects represent data types and morphisms represent programs transforming inputs of one type into outputs of another.

• The composition of morphisms indicates that two programs can be combined, where the output of one serves as the input for another.

Lambda Calculus and Categories

• The relationship between Lambda Calculus and Cartesian Closed Categories exemplifies the use of functorial semantics in computer science.

• A functor can map data types to sets representing possible values, ensuring that program compositions correspond to function compositions.

The Role of Functors in Theoretical Computer Science

Importance and Community Reception

• While not widely recognized among all computer scientists, functorial semantics holds significance within certain mathematical communities.

• Languages like Haskell leverage category theory concepts such as monads, which often perplex new computer scientists seeking clarity on their utility.

Quantum Field Theory and Category Theory

Emergence in Particle Physics

• In the 1980s, particle physicists discovered that any Quantum Field Theory defines a category where objects represent particles and morphisms depict interactions between them.

• Feynman diagrams were introduced by Richard Feynman to visualize particle interactions; despite initial opposition from established physicists, they proved effective.

Interpretation through Functorial Semantics

• Feynman diagrams serve as pictorial representations of morphisms within specific categories defined by Quantum Field Theories.

• To compute probabilities related to particle interactions, physicists utilize a functor mapping these diagrams into Hilbert spaces—mathematical structures essential in quantum mechanics.

Conclusion: Advancements Through Category Theory

Integration with Physics

• By applying category theory principles to Quantum Field Theory, physicists have enhanced their ability to tackle complex problems previously challenging without this theoretical framework.

Quantum Teleportation and Category Theory

Introduction to Quantum Processes

• In 2004, Samson Abramsky and Bob Kirka published a significant paper linking quantum teleportation and other processes to category theory using diagrams that resemble Feynman diagrams.

• The edges in these diagrams represent states of arbitrary quantum systems rather than elementary particles, allowing for broader applications beyond just electrons.

Communication in Quantum Teleportation

• Quantum teleportation creates an illusion of faster-than-light communication; however, it relies on prior classical information transfer to facilitate the process.

• This work excited the community by extending category theory's application from elementary particle physics into the emerging fields of quantum information theory and computation.

Development of a Research Group

• Abramsky and Kirka established a large group at Oxford known as the "quantum group," focusing on category theory's foundations in quantum physics and computation.

• Notably, this group operates within the computer science department rather than traditional mathematics or physics departments, highlighting an interdisciplinary approach.

Textbook Contributions

• The team developed diagrammatic methods for explaining quantum physics, leading to textbooks aimed at making category theory more accessible through visual representations.

• These resources avoid heavy jargon while teaching concepts like morphisms through hands-on illustrations, catering to those unfamiliar with advanced mathematics.

Accessibility of Category Theory

• A new generation of applied category theory textbooks aims to reach diverse audiences without requiring extensive mathematical backgrounds first.

• Traditional approaches often treat category theory as a finishing school after mastering other complex subjects; however, this is seen as unnecessarily burdensome for practitioners in fields like computer science or business.

Recommended Readings and Applications

• Eugenia Chang's book "The Joy of Abstraction" is recommended for its approachable introduction to category theory for readers with minimal mathematical exposure.

• In 2021, Kirka transitioned from academia to become chief scientist at Continuum, applying category theory in quantum computing and natural language processing.

Interdisciplinary Connections

• There is notable interaction between ideas from quantum computing and natural language processing; Tai Denae Bradley’s works are highlighted as valuable contributions in this area.

• The use of syntax involving diagrams extends beyond previously mentioned topics into various scientific and engineering disciplines.

Understanding Differential Equations through Category Theory

The Connection Between Diagrams and Mathematics

• The process of converting diagrams into mathematical representations often involves systems of differential equations, which describe changes over time in dynamical systems.

• Composing morphisms within a category allows for the mapping of these diagrams to their corresponding differential equations, revealing an underlying categorical structure.

Examples of Diagrammatic Representations

• Electrical circuit diagrams serve as one of the earliest examples used widely in mathematics and engineering.

• A Petri net model illustrates interactions such as those between the AIDS virus and blood cells, showcasing complex biological processes.

• Control theory employs feedback mechanisms to maintain system stability, with simple examples like thermostats evolving into more complex industrial applications.

Standardization in Molecular Biology

• In molecular biology, various diagram languages have been standardized to facilitate understanding across different research papers, highlighting the complexity and diversity of representation methods.

• Despite being seen as mere tools for understanding differential equations, these diagrams are fundamentally mathematical constructs representing morphisms in categories.

Development of Decorated Co-spans

• Brendan Fong was tasked with studying electrical circuits as a first example; his work led to the development of a general theory called decorated co-spans that applies broadly across various diagram languages.

• This formalism provides a framework for creating categories from diverse types of diagrams, although specific details must be tailored for each case.

Applications and Further Research

• Fong collaborated with Blake Pollard on applying decorated co-spans to Markov processes, expanding its utility beyond electrical circuits.

• After completing his PhD at Oxford under significant mentorship, Fong continued his research at MIT focusing on applied category theory and database language development.

• David Spivak's contributions include creating functorial query language for databases that relies heavily on functorial semantics.

Category Theory and Its Applications

Introduction to Abstract Databases

• The concept of an abstract database is likened to a spreadsheet filled with entries, which can be interpreted without specific names or numbers.

• David's 2014 textbook titled "Category Theory for the Sciences" aims to broaden the audience of category theory beyond pure mathematicians.

Recent Textbooks and Resources

• In 2018, Brendan Fong and David Spivak published "Seven Sketches in Compositionality," focusing on applied category theory.

• The course associated with this textbook is available on YouTube, and the book can be accessed for free online.

Open Source Commitment in Category Theory

• There is a growing commitment among applied category theorists to make resources open source and easily accessible to facilitate learning outside traditional mathematics circles.

Development of Structured Co-spans

• In 2018, Kenny Corser and the speaker developed structured co-spans as a simpler alternative to decorated co-spans, though less flexible.

• Christina Vasilicopolu helped clarify the relationship between structured and decorated co-spans in 2020.

Applications of Co-spans in Open Systems

• Both structured and decorated co-spans are utilized to study open systems characterized by loose ends that can be composed into larger systems.

• Examples include open Petri nets, which model chemical reactions or populations through places (representing entities) and transitions (describing interactions).

Advancements in Scientific Computing

• The integration of rates into open Petri nets leads to more complex models like open chemical reaction networks.

• A vision emerged for a unified language across scientific disciplines through collaboration with scientists.

Algebraic Julia: A New Paradigm

• In 2019, James Fairbanks and Evan Patterson began developing Algebraic Julia, enabling programming using constructs from category theory while leveraging high-performance computing capabilities.

Compositional Epidemiology Model

• In 2020, Evan implemented structured co-spans within Algebraic Julia for modeling COVID-19 dynamics used by the UK government.

• This approach allowed building complex epidemiological models from smaller components rather than constructing them entirely at once.

Establishment of Topos Institute

• In 2021, Brendan Fong and David Spivak left academia to establish the Topos Institute focused on applied mathematics with an emphasis on applied category theory.

Applied Category Theory in Epidemiology

Introduction of Key Personnel and Concepts

• The discussion begins with the hiring of individuals like Evan Patterson and Sophie Lipkin, who contribute to the development of applied category theory within epidemiology.

• Brendan's initiative to start an institute post-postdoc is highlighted, showcasing his bravery and the eventual success of this endeavor.

Development of Models Using Algebraic Julia

• In 2022, a team including Sophie Lipkin and others utilized algebraic Julia to create models for epidemiology through open Petri Nets.

• Open Petri Nets are introduced as a method for modeling; however, their composition differs from traditional end-to-end methods typically seen in category theory.

Operadic Composition in Modeling

• The concept of operadic composition is discussed as a systematic approach incorporated into their software, enhancing model-building capabilities.

• Another team also employed algebraic Julia to develop models using open stock flow diagrams, emphasizing flexibility in diagrammatic representation.

Insights from Epidemiologists

• Conversations with epidemiologists revealed a preference for stock flow diagrams over Petri Nets due to their congeniality for modeling disease spread.

• Co-authors Nathaniel Osgood and Xiaoyen Lee gained significant experience by creating COVID models for the Canadian government using stock flow techniques.

Community-Based Modeling Approach

• The idea of community-based modeling is presented, where local knowledge enhances model accuracy through collaborative efforts involving stock flow diagrams.

• An anecdote illustrates how local insights can lead to critical improvements in models that external experts may overlook.

Limitations of Existing Software

• Current commercial software like AnyLogic has limitations: it does not allow model composition or separate syntax from semantics effectively.

• Issues such as lack of support for stratifying models (e.g., dividing populations by age groups), collaborative building features, and being non-open source are noted as significant drawbacks.

Complexity in Real-world Models

• The complexity inherent in real-world epidemiological models is emphasized; existing software fails to represent these complexities adequately by treating them as monolithic entities rather than modular components.

Stock Flow Diagrams and Applied Category Theory

Introduction to Stock Flow Software

• The software supports compositional modeling using stock flow diagrams, allowing users to create and compose these diagrams.

• It enables the stratification of stocks into smaller components, utilizing techniques from category theory such as pullbacks.

Development of User Interface

• Nate Osgood and Eric Redika developed a graphical user interface called Model Collab for easier interaction with stock flow models.

• This interface operates in web browsers, facilitating collaboration among teams across different locations through drag-and-drop functionality.

Collaborative Features and Training

• Users can save their models online, enabling access for others to integrate their work into larger models.

• Osgood conducts boot camps focused on training epidemiologists in using Model Collab without requiring prior knowledge of category theory or programming languages like Julia.

Challenges in Application

• Transitioning modelers from established software (like AnyLogic) to this new paradigm presents challenges that require significant effort.

• The speaker reflects on the complexities of applying mathematical ideas practically compared to pure mathematics, which often involves less collaborative effort.

Communication and Ethical Considerations

• Applied category theory necessitates new forms of communication due to its interdisciplinary nature, fostering discussions among diverse groups.

• Initiatives like the annual applied category theory conference and open-access journals are emerging to facilitate dialogue within the community.

Ethical Implications of Applied Mathematics

• Engaging with applied category theory raises ethical issues that pure mathematicians may overlook; practitioners must consider the implications of their work more directly.

• Current funding sources for research in this area include military applications, prompting discussions about the moral responsibilities associated with such research.

Ethical Concerns in Applied Category Theory

The Intersection of Ethics and Mathematics

• The speaker expresses concern about the potential for applied category theory to exacerbate existing societal inequalities, making the wealthy even more powerful.

• There is a desire to explore applied category theory in fields like epidemiology and environmental issues, highlighting its versatility beyond traditional applications.

Revitalizing Old Questions

• The discussion emphasizes the intersection of mathematics and ethics, suggesting that category theory can revitalize questions regarding which mathematical structures are best suited for system design.

• With advancements in computer science, there is an opportunity not just to describe but also to create new systems based on clear conceptualizations.

Possibilities for Radical Progress

• The speaker encourages exploration of new ideas within applied category theory that could lead to innovative ways of organizing systems.

Dynamic vs. Static Perspectives in Mathematics

Philosophical Reflections on Time

• A question arises about whether the application of differential equations as flowing entities might challenge static views prevalent in mathematics.

• Diagrams used in this context resemble flow charts, prompting a reflection on how they may represent dynamic processes rather than static functions.

Understanding Dynamics through Category Theory

• The complexity of distinguishing between static and dynamic perspectives is acknowledged; any dynamical system can be perceived as static when viewed at a specific moment.

• Category theory offers a generalization over set theory by introducing morphisms that convey dynamics between objects, thus adding a layer of dynamism to mathematical foundations.

Open Differential Equations

• Open differential equations are introduced as those affected by external factors, contrasting with traditional autonomous systems where all variables are determined internally.

• This perspective allows for treating differential equations as morphisms while acknowledging their non-deterministic nature due to external influences.

Conclusion and Future Directions

Closing Thoughts

• The speaker concludes with gratitude for the discussion and expresses hope for productive future dialogues surrounding these complex topics.

Understanding Causality and Association in Data

The Distinction Between Association and Causation

• The speaker discusses the difference between statistical association and causal connection, emphasizing that understanding causality can provide a better grip on the world.

• Questions arise about how models incorporate assumptions of causation, prompting a discussion on representing these differences in modeling.

Model Building and Inference

• The focus shifts to model building, where opinions about causal relationships (how X affects Y) are integrated into the model.

• In epidemiology, inference is crucial; it involves measuring data to establish causality and numerical quantities for model integration.

Feedback Loops in Modeling

• The speaker highlights a feedback loop between modeling predictions and real-world outcomes, stressing the importance of comparing models with reality for parameter estimation.

• Acknowledgment that only one half of this interaction was discussed, indicating a broader context that includes both modeling and inference.

Causal Modeling Techniques

Diagrammatic Techniques in Category Theory

• Discussion on diagrammatic techniques from category theorists related to causal modeling indicates existing methodologies that could enhance understanding.

Algorithms and Data Structures in Computer Science

• A participant raises questions about algorithms and data structures within computer science, noting their significance for performance optimization.

Category Theory's Role in Programming

Bridging Category Theory with Computer Science

• Inquiry into how recent work links category theory with programming aspects like algorithms suggests an ongoing evolution in applied category theory.

Initiatives Blending Programming with Category Theory

• Mention of the algebraic Julia Community's efforts to integrate category theory into programming practices by defining functors and pre-sheaf categories.

Applications of Category Theory

Database Management Using Categorical Approaches

• Reference to Connexus.ai utilizing categorical ideas for database updates illustrates practical applications of theoretical concepts.

Consulting Work in Database Companies

• Mike Johnson’s consulting role demonstrates the real-world impact of applying category theory principles within database management contexts.