Introduction to Categories and Categorical Logic

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included. The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what we will need, and it may be useful to review this first. In addition, some prior exposure to abstract algebra - vector spaces and linear maps, or groups and group homomorphisms - would be helpful.

💡 Research Summary

The notes titled “Introduction to Categories and Categorical Logic” constitute a compact yet thorough entry point to the core ideas of category theory and its logical applications, based on a series of Oxford lectures by Samson Abramsky and Nikos Tzevelekos. The authors deliberately keep prerequisites modest: familiarity with sets, relations, functions, and a touch of abstract algebra (vector spaces, groups) suffices. The text is organized into a sequence of self‑contained chapters, each enriched with exercises, diagrams, and a “big picture” perspective that repeatedly emphasizes how categorical notions generalize familiar set‑theoretic concepts.

Chapter 1 begins by reframing ordinary functions as arrows, highlighting that the algebraic laws of composition and identity can be expressed without reference to elements. Injectivity and surjectivity are shown to correspond respectively to monic and epic arrows, establishing the first bridge between element‑based and arrow‑based reasoning. The authors then define a category formally (objects, morphisms, domain/codomain maps, composition, identities) and introduce diagrammatic reasoning, stressing the importance of commuting triangles and squares as visual encodings of equations.

The next sections develop basic constructions: initial and terminal objects (the categorical analogues of empty set and singleton), products and coproducts, pullbacks and equalizers, and finally limits and colimits. Each construction is illustrated in concrete categories such as Set, Mon, Grp, Vectₖ, Pos, Top, and Rel, allowing readers to see how the abstract definitions specialize to familiar mathematical settings.

Functors are presented as structure‑preserving maps between categories, with careful treatment of covariant versus contravariant behavior. Natural transformations are introduced as morphisms between functors, leading to the functor category and the notion of “morphisms of morphisms.” This 2‑dimensional viewpoint prepares the ground for universal arrows and adjunctions, which are explored in depth. The authors explain how adjunctions capture universal properties (e.g., free constructions, product projections) and how they give rise to limits and colimits via unit and counit natural transformations.

A substantial portion of the notes is devoted to the Curry–Howard correspondence. The authors map simply‑typed λ‑calculus to cartesian closed categories, showing that types become objects, terms become arrows, and β‑reduction corresponds to composition with evaluation maps. They discuss logical connectives (∧, →) as categorical products and exponentials, and they raise the question of completeness of the categorical semantics.

The treatment of linear logic follows, with Gentzen sequent calculus introduced first, then the multiplicative fragment (⊗, ⊸) interpreted in monoidal categories. The authors demonstrate how resource‑sensitive reasoning can be modeled categorically, and they hint at extensions beyond the multiplicatives (additives, exponentials).

Monads and comonads are presented as the categorical embodiment of computational effects and context‑dependent computations. The notes explain how every adjunction yields a monad (and a comonad), describe the Kleisli construction, and illustrate the modeling of linear exponentials via a comonad.

Throughout, the authors intersperse exercises that ask the reader to verify that familiar algebraic structures (monoids, groups, posets, topological spaces) indeed form categories, to prove that monic/epic coincide with injective/surjective in Set, and to explore the relationship between monads and algebraic theories. An appendix summarises the set‑theoretic background needed, and a bibliography points to classic texts (Mac Lane, Awodey, Jacobs) and recent research for further study.

Overall, the notes succeed in delivering a concise yet conceptually rich introduction. By constantly linking abstract categorical definitions to concrete mathematical examples and by providing ample practice material, the authors make the “arrow‑centric” viewpoint accessible to mathematicians, computer scientists, logicians, philosophers, and physicists alike. The work serves both as a self‑study guide and as a springboard toward more advanced topics such as topos theory, higher‑category theory, and categorical quantum mechanics.