Elements of ∞-Category Theory in nLab

emilyriehl/yoneda

comparative formalizations of the Yoneda lemma for 1-categories and infinity-categories

Infinity Category Theory Offers a Bird's-Eye View of Mathematics

Mathematicians have expanded category theory into infinite dimensions, revealing new connections among mathematical concepts

Categories for the Working Mathematician

Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

Basic Category Theory

This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

F. William Lawvere, Stephen H. Schanuel Conceptual Mathematics A First Introduction To Categories ( 2009, Cambridge University Press) ( 1)

Category Theory first introduction

Introduction To Category Theory

Introduction To Category Theory by Steve Awodey, 2010, Oxford University Press, USA, Oxford University Press edition,

Basic Concepts of Enriched Category Theory

Originally published as: Cambridge University Press, Lecture Notes in Mathematics 64, 1982.

Internal type theory

We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in Martin-Löf's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families.

Computational category theory

Computational category theory by D. E. Rydeheard, 1988, Prentice Hall edition, in English

A categorical manifesto

This paper tries to explain why and how category theory is useful in computing science, by giving guidelines for applying seven basic categorical concepts: category, functor, natural transformation, limit, adjoint, colimit and comma category. Some examples, intuition, and references are given for each concept, but completeness is not attempted. Some additional categorical concepts and some suggestions for further research are also mentioned. The paper concludes with some philosophical discussion.

Horismos: Self-representation and the Derived Constitutional Boundary in Enriched Cognitive Systems

We present a theory of self-representing cognitive systems grounded in $$([0,\infty ],+)$$([0,∞],+)-enriched category theory and the Yoneda lemma. The central object is a self-representing $$([0,\infty ],+)$$([0,∞],+)-enriched category $$\mathcal{C}$$C—a Lawvere metric space whose objects are complete epistemic architectures, whose hom-values record directed informational upgrade costs, and which is separated, closed under internal homs, and bilaterally Cauchy complete—together with a contractive cognitive endofunctor $$F:\mathcal{C}\rightarrow \mathcal{C}$$F:C→Cmodelling iterative self-improvement. We establish eight results in a single logical arc. The Horizon Theorem shows that the Yoneda embedding $$\varphi (A)=\mathcal{C}(-,A)$$φ(A)=C(-,A)is never essentially surjective: $$\mathcal{C}$$C sits strictly inside its own free Cauchy completion $$\mathcal{P}(\mathcal{C})$$P(C), with the non-representable presheaves forming a topologically dense family, proved via a reflexivity argument. The Lawvere–Banach Attractor Theorem shows that every contractive endofunctor on a bilaterally complete, separated $$([0,\infty ],+)$$([0,∞],+)-enriched category converges to a unique fixed point $$\mathbf {\Omega }$$Ωat a geometric rate. The Boundary Derivation Theorem shows that $$\mathbf {\Omega }$$Ωis the minimal F-invariant substructure of $$\mathcal{C}$$C, with all of $$\mathcal{C}$$Cas its basin of attraction—the constitutional boundary, derived rather than postulated. The Horizon Expansion Theorem shows that each strictly ascending self-modification produces a new, quantitatively distinct non-representable witness. Beyond these four central results, we prove that Kleene and Bourbaki–Witt conditions yield only non-expansiveness when metrised, that contractive endofunctors form a monoid, and that the Yoneda horizon admits an observable diagnostic stabilising in finite time. The architectural section derives structural corrigibility and the alignment-incompleteness duality among five implications. The organising duality is exact: the non-surjectivity of $$\varphi $$φ and the existence of $$\mathbf {\Omega }$$Ωare two faces of the same $$([0,\infty ],+)$$([0,∞],+)-enriched structure. $$\mathbf {\Omega }$$Ωinhabits the space between them—not as a postulate, but as a proof. We argue that the eight theorems constitute universal laws of contractive cognitive systems: a stable constitutional boundary is not an engineering design choice but a topological inevitability for any reliably self-improving agent operating within a self-representing enriched metric space. The postulate becomes a theorem. The boundary is not imposed. It emerges.

generalized algebraic theory in nLab

A generalized algebraic theory (GAT, Cartmell 1978/86) is a dependent type theory in the syntactic sense: There are judgments declaring types and terms of types, where, critically, a type AA declared in context Γ\Gamma is allowed to depend on terms of Γ\Gamma. The only other judgments allowed are of equality (judgemental equality) of pairs of types or terms.

Category theory for computing science

Category theory for computing science by Michael Barr, 1990, Prentice Hall edition, in English

Categorical logic and type theory

Categorical logic and type theory by Bart Jacobs, 1999, Elsevier Science edition, in English - 1st ed.

Seven Sketches in Compositionality: An Invitation to Applied Category Theory

This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint functors, enriched categories, or toposes. No prior knowledge of category theory is assumed. A feedback form for typos, comments, questions, and suggestions is available here: https://docs.google.com/document/d/160G9OFcP5DWT8Stn7TxdVx83DJnnf7d5GML0_FOD5Wg/edit