Formal ∞-Category Theory in a Virtual Equipment

2022 ◽  
pp. 341-378
Keyword(s):  
Category Theory ◽  
Formal Category

Elements of ∞-Category Theory

2022 ◽  
Author(s):  
Emily Riehl ◽  
Dominic Verity
Keyword(s):  
First Principles ◽  
Category Theory ◽  
Higher Dimensional ◽  
Background Material ◽  
Model Independent ◽  
The Universe ◽  
Formal Category

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

不定自然変換理論に基づく比喩理解モデルの計算論的実装の試み

2020 ◽  
Author(s):  
Shunsuke Ikeda ◽  
Miho Fuyama ◽  
Hayato Saigo ◽  
Tatsuji Takahashi
Keyword(s):  
Category Theory ◽  
Analogical Reasoning ◽  
Machine Learning Techniques ◽  
Huge Amount ◽  
Metaphor Comprehension ◽  
Weighted Directed Graph ◽  
Learning Techniques ◽  
Descriptive Validity ◽  
Association Data ◽  
Human Participants

Machine learning techniques have realized some principal cognitive functionalities such as nonlinear generalization and causal model construction, as far as huge amount of data are available. A next frontier for cognitive modelling would be the ability of humans to transfer past knowledge to novel, ongoing experience, making analogies from the known to the unknown. Novel metaphor comprehension may be considered as an example of such transfer learning and analogical reasoning that can be empirically tested in a relatively straightforward way. Based on some concepts inherent in category theory, we implement a model of metaphor comprehension called the theory of indeterminate natural transformation (TINT), and test its descriptive validity of humans' metaphor comprehension. We simulate metaphor comprehension with two models: one being structure-ignoring, and the other being structure-respecting. The former is a sub-TINT model, while the latter is the minimal-TINT model. As the required input to the TINT models, we gathered the association data from human participants to construct the ``latent category'' for TINT, which is a complete weighted directed graph. To test the validity of metaphor comprehension by the TINT models, we conducted an experiment that examines how humans comprehend a metaphor. While the sub-TINT does not show any significant correlation, the minimal-TINT shows significant correlations with the human data. It suggests that we can capture metaphor comprehension processes in a quite bottom-up manner realized by TINT.

Category Theory and Foundations

2018 ◽  
Author(s):  
Michael Ernst
Keyword(s):  
Category Theory ◽  
Mathematical Thinking ◽  
Mathematical Practice ◽  
Foundations Of Mathematics ◽  
Ongoing Debate ◽  
Technical Adequacy ◽  
Theoretical Foundations

In the foundations of mathematics there has been an ongoing debate about whether categorical foundations can replace set-theoretical foundations. The primary goal of this chapter is to provide a condensed summary of that debate. It addresses the two primary points of contention: technical adequacy and autonomy. Finally, it calls attention to a neglected feature of the debate, the claim that categorical foundations are more natural and readily useable, and how deeper investigation of that claim could prove fruitful for our understanding of mathematical thinking and mathematical practice.

Enriched Meanings

2020 ◽  
Author(s):  
Ash Asudeh ◽  
Gianluca Giorgolo
Keyword(s):  
Cognitive Science ◽  
Programming Languages ◽  
Category Theory ◽  
Worst Case ◽  
Disciplinary Boundaries ◽  
Semantics Of Programming Languages ◽  
Basic Meaning ◽  
Language Interpretation ◽  
Major Branch ◽  
Compositional Properties

This book presents a theory of enriched meanings for natural language interpretation. Certain expressions that exhibit complex effects at the semantics/pragmatics boundary live in an enriched meaning space while others live in a more basic meaning space. These basic meanings are mapped to enriched meanings just when required compositionally, which avoids generalizing meanings to the worst case. The theory is captured formally using monads, a concept from category theory. Monads are also prominent in functional programming and have been successfully used in the semantics of programming languages to characterize certain classes of computation. They are used here to model certain challenging linguistic computations at the semantics/pragmatics boundary. Part I presents some background on the semantics/pragmatics boundary, informally presents the theory of enriched meanings, reviews the linguistic phenomena of interest, and provides the necessary background on category theory and monads. Part II provides novel compositional analyses of the following phenomena: conventional implicature, substitution puzzles, and conjunction fallacies. Part III explores the prospects of combining monads, with particular reference to these three cases. The authors show that the compositional properties of monads model linguistic intuitions about these cases particularly well. The book is an interdisciplinary contribution to Cognitive Science: These phenomena cross not just the boundary between semantics and pragmatics, but also disciplinary boundaries between Linguistics, Philosophy and Psychology, three of the major branches of Cognitive Science, and are here analyzed with techniques that are prominent in Computer Science, a fourth major branch. A number of exercises are provided to aid understanding, as well as a set of computational tools (available at the book's website), which also allow readers to develop their own analyses of enriched meanings.

scholarly journals An Essential, Hyperconnected, Local Geometric Morphism that is not Locally Connected

2021 ◽  
Author(s):  
Jens Hemelaer ◽  
Morgan Rogers
Keyword(s):  
Category Theory ◽  
Mailing List ◽  
Geometric Morphism

AbstractThomas Streicher asked on the category theory mailing list whether every essential, hyperconnected, local geometric morphism is automatically locally connected. We show that this is not the case, by providing a counterexample.

scholarly journals Category Theory for Autonomous and Networked Dynamical Systems

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 302 ◽  
Author(s):  
Jean-Charles Delvenne
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Ergodic Theory ◽  
Category Theory ◽  
Open Systems ◽  
Topological Dynamics ◽  
Control Problems ◽  
Natural Conditions ◽  
Discussion Paper ◽  
Open Systems Theory

In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.

scholarly journals Reflective subcategories

2000 ◽  
Vol 42 (1) ◽  
pp. 97-113 ◽  
Author(s):  
Juan Rada ◽  
Manuel Saorín ◽  
Alberto del Valle
Keyword(s):  
Category Theory ◽  
Full Subcategory ◽  
Subject Classification ◽  
Adjoint Functor ◽  
Direct Limits ◽  
The Relationship ◽  
Good Behaviour

Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory.In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories.In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R}, and has some connections with the “pure-semisimple conjecture”.1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

THE PROSPECTS OF UNLIMITED CATEGORY THEORY: DOING WHAT REMAINS TO BE DONE

2015 ◽  
Vol 8 (2) ◽  
pp. 306-327 ◽  
Author(s):  
MICHAEL ERNST
Keyword(s):  
Category Theory

AbstractThe big question at the end of Feferman (2013) is: Is it possible to find a foundation for unlimited category theory? I show that the answer is no by showing that unlimited category theory is inconsistent.

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