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.

Algebraic theories :

  • 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 :

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


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 )


  • 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)