Introduction to Categorical Logic

Abstract. Categorical Logic, and Functorial semantics more specifically, emerged in the 1960s as an alternative framework to capture universal algebra. The framework offers a collection of advantages, including a more flexible and modular approach to semantics, which delivers a perfect correspondence with syntax. These tools offer a more quantitative and conceptual take on completeness results and definability-type theorems.

After a brief introduction to the language of categories, we focus on universal algebra and functorial semantics. We capture the notion of algebraic theory via categories with products (Lawvere theories) and present a syntax-semantics duality between varieties and Lawvere theories. The course ends with some vistas on the theory of sketches which offers a much more general framework, covering the leap from universal algebra to infinitary first order logic.

Structure

Day 1	Categories, functors, natural transformations.	basic definitions and examples
Day 2	Adjunctions, (co)limits	definition and examples of adjunctions and (co)limits
Day 3	Functorial semantics: from posets to Lawvere theories	meet semilattices and their semantics, Lawvere theories and natural operations
Day 4	Dualities: from posets to Gabriel and Ulmer	Algebraic lattices and duality, GU duality, Varieties from Universal algebra
Day 5	Sketches of a broader vision	(co)limits and sketches

Prerequisites. The audience is expected to be familiar and have played with the basic definitions of one of following objects: vector space, monoid, group, ring, module, set equipped with operations.

Exercises and take-home exam. In order to get your final certificate you must (a) attend the course, (b) your problem sets must be graded an overall score of 5/10, (c) pass the take-home exam.

Problem Set I.pdf

Problem Set II.pdf

Problem Set III.pdf

Lecture notes by Lingyuan (English)

Take-home exam

Solutions are kindly provided by Ansi Bai.

Solution 1.pdf

Solution 2.pdf

Solution 3.pdf

Notes taken by Jinyi Wang (Chinese)

Day 3

Day 4

Day 5

Literature on general category theory.

Leinster, Basic Category Theory. Quite gentle introduction aimed at people with a mathematical background.
Awodey, Category Theory. Gentle introduction aimed at people with a more logical background.
Riehl, Category Theory in context. Introduction aimed at more mature students with a mathematical background.

Literature on Categorical Logic.

Awodey and Bauer, ongoing book.

On sketches.

Barr and Wells, Category Theory for Computing Science. Chap. 7.

Further suggested readings.

Click here.