# Algebraic theories

Algebraic theories are logical theories that are purely equational, with no quantifiers permitted. They are among the simplest logical systems in the hierarchy of categorical logic.

## Notions of algebraic theory

Diagrammatic theories or preheaf theories (nonstandard terminology, not to be confused with theories of presheaf type ):

• specifications are sketches (with no cones or cocones)
• syntactic categories are just the (small) categories, with no extra structure
• model categories are categories of presheaves

Marta Bunge characterized, up to equivalence, categories of presheaves (“diagrammatic categories”) as the cocomplete, atomic , regular categories.

• specifications are finite product sketches
• syntactic categories are the (multi-sorted) Lawvere theories (aka finite product theories)
• model categories are the “algebraic categories” (terminology varies)
• syntax-semantics are related by Lawvere duality (Awodey, 2009)

Essentially algebraic theories are closely related to generalized algebraic theories, a kind of dependently typed algebraic theory.

## Literature

Lecture notes and introductory papers

• Awodey, 2009, lecture notes: Introduction to categorical logic , Ch. 1: Algebraic theories (pdf)
• My preferred introduction: very clear with minimal prerequisites
• Hyland & Power, 2007: The category-theoretic understanding of universal algebra: Lawvere theories and monads (doi, pdf, nCat Cafe )

Books

• Manes, 1976: Algebraic theories (doi)
• Barr & Wells, 1985: Toposes, triples, and theories, Ch. 4: Theories
• Based on finite product sketches
• Crole, 1993: Categories for types, Ch. 3: Algebraic Type Theory
• The best book to start with, providing details omitted in other sources
• Borceux, 1994: Handbook of categorical algebra, Vol 2, Ch. 3: Algebraic theories
• Theorem 3.9.1: Characterization of algebraic categories (for single-sorted theories)
• Adamek & Rosicky, 1994: Locally presentable and accessible categories, Ch. 3: Algebraic categories
• Adamek, Rosicky, Vitale, 2011: Algebraic theories: A categorical introduction to general algebra
• Its title notwithstanding, this book is for specialists only
• Several characterizations of algebraic categories (single-sorted and multi-sorted)