# 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.

## Examples

- 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)

## Literature

**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- Emphasizes “relation between internal and fibered categories”

- 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

**Papers**

- 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*.