Internal category theory

Internal category theory , not to be confused with enriched category theory, interprets the (essentially algebraic) theory of categories in any category with enough pullbacks.


  • A (small) category is a category in \(\mathbf{Set}\)
  • A topological category is a category in \(\mathbf{Top}\); a smooth category is a category in \(\mathbf{Diff}\) (early examples studied by Ehresmann)
  • A double category is a category in \(\mathbf{Cat}\) (also introduced by Ehresmann)
  • A triple category is a category in \(\mathbf{DblCat}\)
  • More generally, an n-fold category is an internal category in \(\mathbf{Set}\) iterated \(n\) times
  • A 2-vector space is a category in \(\mathbf{Vect}\) (Baez & Crans, 2004)


Books and surveys

  • Borceux, 1994: Handbook of Categorical Algebra, Vol. 1, Ch. 8: Internal category theory
  • Jacobs, 1999: Categorical Logic and Type Theory, Ch. 7: Internal category theory
  • Johnstone, 1977: Topos theory, Ch. 2: Internal category theory
  • Johnstone, 2002: Sketches of an Elephant, Sec B2.3: Internal categories and diagram categories & Sec B2.7: Internal profunctors
  • Pradines, 2007: In Ehresmann’s footsteps: from group geometries to groupoid geometries (doi, arxiv)
    • Recommended on nLab as “a survey [of internal categories] with an eye towards Lie groupoids”
    • Also in response to my MO question about Cencov’s categories of figures
    • Unfortunately, I cannot make heads or tails of it


  • Baez & Lauda, 2004: Higher-dimensional algebra V: 2-Groups (pdf, arxiv), Sec. 7: Internalization
    • Good summary of main definitions of internal category theory
  • Baez & Crans, 2004: Higher-dimensional algebra VI: Lie 2-algebras (pdf, arxiv), Sec. 2: Internal categories & Sec. 3: 2-Vector spaces
    • Introduces 2-vector spaces, or categories internal to \(\mathbf{Vect}\)

Non-strict versions

Monoids are to pseudomonoids as categories are to pseudocategories: categories internal to a 2-category with 2-pullbacks, where associativity and unitality hold only up to coherent invertible 2-cells. The most important example is a pseudocategory in \(\mathbf{Cat}\), which is a pseudo double category.

  • Martins-Ferreira, 2006: Pseudo-categories (pdf, arxiv)
  • Grandis & ParĂ©, 2015: Intercategories (pdf, arxiv)
    • Considers not just pseudocategories but double pseudocategories!
    • An intercategory is a 3-dimensional categorical structure defined to be a double pseudocategory in \(\mathsf{Cat}\)