From Sets to Types to Categories to Sets
Steve Awodey
Online link
Prerequisites: Category theory, set theory, type theory.
A short preprint, exploring the links between set theory, type theory, and category theory. A must-read for anyone interested in the topic.
Topoi: The Categorial Analysis of Logic
Robert Goldblatt
Prerequisites: None.
This text is a comprehensive introduction to category and topos theory in the context of logic. While technically all category theory required is covered in this text, I would still recommend learning some category theory elsewhere first, as the ordering/pacing of the material is somewhat dubious. For instance, the Yoneda lemma is one of the first results covered in most category theory texts, but is – for some reason – deferred to the final chapter here, making many of the proofs throughout the book extremely difficult to follow for little reason. The typesetting in the book is also noticably outdated, as the book is relatively old, predating the common usage of TeX.
Sheaves in Geometry and Logic
Saunders Mac Lane, Ieke Moerdijk
Online link
Prerequisites: In general, a strong background in category theory; for the geometry side, a strong background in commutative algebra and algebraic geometry; for the logic side, some experience with point-set topology, set theory, type theory, and model theory.
A Simple Introduction to Computable Analysis
Klaus Weihrauch
Online link
Prerequisites: Basic real analysis, point-set topology, and complexity theory; a first course on computability theory.
This short book introduces the fundamentals of computable analysis of the real numbers. An interesting read for logicians and computer scientists that enjoy analytical flavours of mathematics.