Ecosyste.ms: Awesome
An open API service indexing awesome lists of open source software.
awesome-category-theory
A curated list of awesome Category Theory resources.
https://github.com/madnight/awesome-category-theory
Last synced: about 6 hours ago
JSON representation
-
Archive
- Functor theory - Explores the concept of exact categories and the theory of derived functors, building upon earlier work by Buchsbaum. Freyd investigates how properties and statements applicable to abelian groups can extend to arbitrary exact categories. Freyd aims to formalize this observation into a metatheorem, which would simplify categorical proofs and predict lemmas. Peter J. Freyd's dissertation, presented at Princeton University (1960)
- Algebra valued functors in general and tensor products in particular - Discusses the concept of valued functors in category theory, particularly focusing on tensor products. Freyd explores the application of algebraic theories in non-standard categories, starting with the question of what constitutes an algebra in the category of sets, using category predicates without elements. The text outlines the axioms of a group using category theory language, emphasizing objects and maps. Peter Freyd (1966)
- Continuous Yoneda Representation of a small category - Discusses the embedding of a small category A into the category of contravariant functors from A to Set (the category of sets), which preserves inverse limits but does not generally preserve direct limits. Kock introduces a "codensity monad" for any functor from a small category to a left complete category and explores the universal generator for this monad. He demonstrates that the Yoneda embedding followed by this generator provides a full and faithful embedding that is both left and right continuous. Additionally, the relationship with Isbell's adjoint conjugation functors and the definition of generalized (direct and inverse) limit functors are addressed, by Anders Kock (1966).
- Abstract universal algebra - Explores advanced subjects in the realm of universal algebra. The core content is organized into two chapters, each addressing different aspects of universal algebra within the framework of category theory. The first chapter introduces the concept of triplable categories, inspired by the theory of modules over a ring, and explores the equivalence between categories of triples in any given category and theories over that category. In the second chapter, Davis shifts focus to equational systems of functors, a more generalized approach to algebra that encompasses both the triplable and structure category theories. Dissertation by Robert Clay Davis (1967)
- A triple miscellany: some aspects of the theory of algebras over a triple - Explores the field of universal algebra with a particular focus on the concept of algebras over a triple. The work is grounded in the realization that categories of algebras, traditionally defined with finitary operations and satisfying a set of equations, can be extended to include infinitary operations as well, thereby broadening the scope of universal algebra. Manes starts by discussing the conventional understanding of universal algebra, tracing back to G.D. Birkhoff's definition in the 1930s, and then moves to explore how this definition can be expanded by considering sets with infinitary operations. Dissertation by Ernest Gene Manes (1967)
- Algebraic theories - Covers topics such as the fundamentals of algebraic theories, free models, special theories, the completeness of algebraic categories, and extends to more complex concepts like commutative theories, free theories, and the Kronecker product, among others. The notes also touch on the rings-theories analogy proposed by F. W. Lawvere, suggesting an insightful correlation between rings/modules and algebraic theories/models. Gavin C. Wraith (1975)
- Algebra valued functors in general and tensor products in particular - Discusses the concept of valued functors in category theory, particularly focusing on tensor products. Freyd explores the application of algebraic theories in non-standard categories, starting with the question of what constitutes an algebra in the category of sets, using category predicates without elements. The text outlines the axioms of a group using category theory language, emphasizing objects and maps. Peter Freyd (1966)
- Continuous Yoneda Representation of a small category - Discusses the embedding of a small category A into the category of contravariant functors from A to Set (the category of sets), which preserves inverse limits but does not generally preserve direct limits. Kock introduces a "codensity monad" for any functor from a small category to a left complete category and explores the universal generator for this monad. He demonstrates that the Yoneda embedding followed by this generator provides a full and faithful embedding that is both left and right continuous. Additionally, the relationship with Isbell's adjoint conjugation functors and the definition of generalized (direct and inverse) limit functors are addressed, by Anders Kock (1966).
- Abstract universal algebra - Explores advanced subjects in the realm of universal algebra. The core content is organized into two chapters, each addressing different aspects of universal algebra within the framework of category theory. The first chapter introduces the concept of triplable categories, inspired by the theory of modules over a ring, and explores the equivalence between categories of triples in any given category and theories over that category. In the second chapter, Davis shifts focus to equational systems of functors, a more generalized approach to algebra that encompasses both the triplable and structure category theories. Dissertation by Robert Clay Davis (1967)
- A triple miscellany: some aspects of the theory of algebras over a triple - Explores the field of universal algebra with a particular focus on the concept of algebras over a triple. The work is grounded in the realization that categories of algebras, traditionally defined with finitary operations and satisfying a set of equations, can be extended to include infinitary operations as well, thereby broadening the scope of universal algebra. Manes starts by discussing the conventional understanding of universal algebra, tracing back to G.D. Birkhoff's definition in the 1930s, and then moves to explore how this definition can be expanded by considering sets with infinitary operations. Dissertation by Ernest Gene Manes (1967)
- Limit Monads in Categories - The work introduces the notion that the category of complete categories is monadic over the category of all categories, utilizing a family of monads associated with various index categories to define "completeness." A significant portion of the thesis is dedicated to defining associative and regular associative colimits, arguing for their naturalness and importance in category theory. Dissertation by Anders Jungersen Kock (1967)
- V-localizations and V-triples - This work focuses on two primary objectives within category theory. The first goal is to define and study Y-localizations of Y-categories, using a model akin to localizations in ordinary categories, involving certain conditions related to isomorphisms and the existence of unique Y-functors. The second aim is to explore the relationship between Y-localizations and V-triples, presenting foundational theories and examples to elucidate these concepts. Harvey Eli Wolff's dissertation (1970)
- Symmetric closed categories - This work is an in-depth exploration of category theory, focusing on closed categories, monoidal categories, and their symmetric counterparts. It discusses foundational concepts like natural transformations, tensor products, and the structure of morphisms, emphasizing their additional algebraic or topological structures. W. J. de Schipper (1975)
- Limit Monads in Categories - The work introduces the notion that the category of complete categories is monadic over the category of all categories, utilizing a family of monads associated with various index categories to define "completeness." A significant portion of the thesis is dedicated to defining associative and regular associative colimits, arguing for their naturalness and importance in category theory. Dissertation by Anders Jungersen Kock (1967)
- On the concreteness of certain categories - This work discusses the concept of concreteness in categories, stating that a concrete category is one with a faithful functor to the category of sets, and must be locally-small. He highlights the homotopy category of spaces as a prime example of a non-concrete category, emphasizing its abstract nature due to the irrelevance of individual points within spaces and the inability to distinguish non-homotopic maps through any functor into concrete categories. Peter Freyd (1969)
- V-localizations and V-triples - This work focuses on two primary objectives within category theory. The first goal is to define and study Y-localizations of Y-categories, using a model akin to localizations in ordinary categories, involving certain conditions related to isomorphisms and the existence of unique Y-functors. The second aim is to explore the relationship between Y-localizations and V-triples, presenting foundational theories and examples to elucidate these concepts. Harvey Eli Wolff's dissertation (1970)
- Symmetric closed categories - This work is an in-depth exploration of category theory, focusing on closed categories, monoidal categories, and their symmetric counterparts. It discusses foundational concepts like natural transformations, tensor products, and the structure of morphisms, emphasizing their additional algebraic or topological structures. W. J. de Schipper (1975)
- Algebraic theories - Covers topics such as the fundamentals of algebraic theories, free models, special theories, the completeness of algebraic categories, and extends to more complex concepts like commutative theories, free theories, and the Kronecker product, among others. The notes also touch on the rings-theories analogy proposed by F. W. Lawvere, suggesting an insightful correlation between rings/modules and algebraic theories/models. Gavin C. Wraith (1975)
- On the concreteness of certain categories - This work discusses the concept of concreteness in categories, stating that a concrete category is one with a faithful functor to the category of sets, and must be locally-small. He highlights the homotopy category of spaces as a prime example of a non-concrete category, emphasizing its abstract nature due to the irrelevance of individual points within spaces and the inability to distinguish non-homotopic maps through any functor into concrete categories. Peter Freyd (1969)
-
Articles
- A Categorical Foundation for Bayesian probability - Bayesian inference and decision making on measurable spaces with countably generated σ-algebras, using regular conditional probabilities and Eilenberg--Moore algebras by Jared Culbertson, Kirk Sturtz (2013)
- A Channel-Based Perspective on Conjugate Priors - Introduces channels in a graphical language to define and study conjugate priors in Bayesian probability, and shows how they ensure the same class of distributions for prior and posterior by Bart Jacobs (2017)
- A Type Theory for Probabilistic and Bayesian Reasoning - A new type theory and logic for probabilistic reasoning with fuzzy predicates and normalisation and conditioning of states by Robin Adams, Bart Jacobs (2015)
- A category theory framework for Bayesian Learning - Drawing from Spivak, Fong, and Cruttwell et al.'s foundational works, this study establishes a categorical framework for Bayesian inference, incorporating concepts of Bayesian inversions by Kotaro Kamiya, John Welliaveetil (2021)
- Automatic Backward Filtering Forward Guiding for Markov processes and graphical models - Describing a Backward Filtering Forward Guiding (BFFG) paradigm for inference on latent states and parameters, and transforms a forward generative model into a data-guided pre-conditional mode by Frank van der Meulen, Moritz Schauer (2020)
- Bayesian Open Games - Extends compositional game theory to handle stochasticity and incomplete information using category theory and coend optics by Joe Bolt, Jules Hedges, Philipp Zahn (2019)
- Bayesian Updates Compose Optically - Utilizing lens pattern and a fibred category to model the compositional structure of Bayesian inversion by Toby St. Clere Smithe (2020)
- Bayesian machine learning via category theory - Using categorical methods, the study frames machine learning concepts within the realm of conditional probabilities, building models for both parametric and nonparametric Bayesian reasoning on function spaces, and exemplifies the Kalman filter's relation to the hidden Markov model by Jared Culbertson, Kirk Sturtz (2013)
- Categorical Stochastic Processes and Likelihood - Explores the link between probabilistic modeling and function approximation, introduces two extensions of function composition related to stochastic processes by Dan Shiebler (2020)
- Causal Inference by String Diagram Surgery - Extract causal relationships from correlations, using string diagram syntax and semantics, and showcases a method to compute causal effects by Bart Jacobs, Aleks Kissinger, Fabio Zanasi (2018)
- Causal Theories: A Categorical Perspective on Bayesian Networks - This dissertation introduces a graphical framework for causal reasoning, based on monoidal categories, and presents a new structure called a causal theory by Brendan Fong (2013)
- A Categorical Foundation for Bayesian probability - Bayesian inference and decision making on measurable spaces with countably generated σ-algebras, using regular conditional probabilities and Eilenberg--Moore algebras by Jared Culbertson, Kirk Sturtz (2013)
- A Channel-Based Perspective on Conjugate Priors - Introduces channels in a graphical language to define and study conjugate priors in Bayesian probability, and shows how they ensure the same class of distributions for prior and posterior by Bart Jacobs (2017)
- A Type Theory for Probabilistic and Bayesian Reasoning - A new type theory and logic for probabilistic reasoning with fuzzy predicates and normalisation and conditioning of states by Robin Adams, Bart Jacobs (2015)
- A category theory framework for Bayesian Learning - Drawing from Spivak, Fong, and Cruttwell et al.'s foundational works, this study establishes a categorical framework for Bayesian inference, incorporating concepts of Bayesian inversions by Kotaro Kamiya, John Welliaveetil (2021)
- Automatic Backward Filtering Forward Guiding for Markov processes and graphical models - Describing a Backward Filtering Forward Guiding (BFFG) paradigm for inference on latent states and parameters, and transforms a forward generative model into a data-guided pre-conditional mode by Frank van der Meulen, Moritz Schauer (2020)
- Bayesian Open Games - Extends compositional game theory to handle stochasticity and incomplete information using category theory and coend optics by Joe Bolt, Jules Hedges, Philipp Zahn (2019)
- Bayesian Updates Compose Optically - Utilizing lens pattern and a fibred category to model the compositional structure of Bayesian inversion by Toby St. Clere Smithe (2020)
- Bayesian machine learning via category theory - Using categorical methods, the study frames machine learning concepts within the realm of conditional probabilities, building models for both parametric and nonparametric Bayesian reasoning on function spaces, and exemplifies the Kalman filter's relation to the hidden Markov model by Jared Culbertson, Kirk Sturtz (2013)
- Categorical Stochastic Processes and Likelihood - Explores the link between probabilistic modeling and function approximation, introduces two extensions of function composition related to stochastic processes by Dan Shiebler (2020)
- Causal Inference by String Diagram Surgery - Extract causal relationships from correlations, using string diagram syntax and semantics, and showcases a method to compute causal effects by Bart Jacobs, Aleks Kissinger, Fabio Zanasi (2018)
- Causal Theories: A Categorical Perspective on Bayesian Networks - This dissertation introduces a graphical framework for causal reasoning, based on monoidal categories, and presents a new structure called a causal theory by Brendan Fong (2013)
- Compositionality in algorithms for smoothing - Backward Filtering Forward Guiding with optics and prove that different ways of composing the building blocks of BFFG correspond to equivalent optics by Moritz Schauer, Frank van der Meulen (2023)
- Denotational validation of higher-order Bayesian inference - Introduces Bayesian inference algorithms in probabilistic programming using higher-order functions and quasi-Borel spaces (2017)
- Compositionality in algorithms for smoothing - Backward Filtering Forward Guiding with optics and prove that different ways of composing the building blocks of BFFG correspond to equivalent optics by Moritz Schauer, Frank van der Meulen (2023)
- Denotational validation of higher-order Bayesian inference - Introduces Bayesian inference algorithms in probabilistic programming using higher-order functions and quasi-Borel spaces (2017)
- Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion - Extends the concept of Bayesian Lenses to include cases where one object depends on another, providing a framework to study specific stochastic maps by Dylan Braithwaite, Jules Hedges (2022)
- Disintegration and Bayesian Inversion via String Diagrams - Abstract graphical representations of disintegration and Bayesian inversion concepts in conditional probability by Kenta Cho, Bart Jacobs (2017)
- Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus - A guarded λ-calculus with discrete probabilities and a program logic to understand relational aspects of probabilistic computations, such as Markov chains (2018)
- Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion - Extends the concept of Bayesian Lenses to include cases where one object depends on another, providing a framework to study specific stochastic maps by Dylan Braithwaite, Jules Hedges (2022)
- Disintegration and Bayesian Inversion via String Diagrams - Abstract graphical representations of disintegration and Bayesian inversion concepts in conditional probability by Kenta Cho, Bart Jacobs (2017)
- Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus - A guarded λ-calculus with discrete probabilities and a program logic to understand relational aspects of probabilistic computations, such as Markov chains (2018)
- The Compositional Structure of Bayesian Inference - Examines how Bayes' rule, which updates beliefs based on new evidence, can be applied piecewise to complex processes, linking it to the lens pattern in programming by Dylan Braithwaite, Jules Hedges, Toby St Clere Smithe (2023)
- The Geometry of Bayesian Programming - A geometric interaction model for a typed lambda-calculus equipped with tools for continuous sampling and soft conditioning by Ugo Dal Lago, Naohiko Hoshino (2019)
- The Compositional Structure of Bayesian Inference - Examines how Bayes' rule, which updates beliefs based on new evidence, can be applied piecewise to complex processes, linking it to the lens pattern in programming by Dylan Braithwaite, Jules Hedges, Toby St Clere Smithe (2023)
- The Geometry of Bayesian Programming - A geometric interaction model for a typed lambda-calculus equipped with tools for continuous sampling and soft conditioning by Ugo Dal Lago, Naohiko Hoshino (2019)
Categories
Sub Categories