Fibered and indexed category theory

Fibered categories and indexed categories are families of categories indexed by another category. Fibered categories take the “bundle viewpoint,” based on maps into a category. Indexed categories take the “(pre)sheaf viewpoint,” based on maps out of a category. Specifically, indexed categories are 2-presheaves, being to categories what presheaves are to sets. Fibered categories and indexed categories are equivalent via the Grothendieck construction.

Fibered and indexed categories provided the categorical semantics of dependent types, as in for example generalized algebraic theories.

General references

Fibered categories

Books and surveys

  • Loregian & Riehl, 2019: Categorical notions of fibration (doi, arxiv)
    • Survey of fibrations in categories, 2-categories, and bicategories
  • Streicher, 2018: Fibred categories a la Jean Bénabou (arxiv)
  • Jacobs, 1999: Categorical Logic and Type Theory, Ch. 1: Introduction to fibred category theory & Ch. 9: Advanced fibred category theory
  • Borceux, 1994: Handbook of Categorical Algebra, Vol. 2, Ch. 8: Fibred categories
  • Bénabou, 1985: Fibered categories and the foundations of naive category theory (doi, pdf)

Monoidal fibrations

Monoidal fibrations are the analogue of fibered categories for monoidal categories.

  • Shulman, 2008: Framed bicategories and monoidal fibrations (pdf, arxiv)
  • Moeller & Vasilakopoulou, 2019: Monoidal Grothendieck construction (arxiv, Azimuth 1 ,2 , nLab )

Enriched fibrations

  • Vasilakopoulou, 2018: On enriched fibrations (pdf, arxiv)
    • Develops a theory of a fibration enriched in a monoidal fibration
    • Section 4.3 compares this theory with Bunge’s and Shulman’s theories of enriched indexed categories, cited below

Indexed categories

Books and surveys

  • Johnstone and Paré, eds., 1978: Indexed categories and their applications (doi)
    • Standard reference on indexed categories is the chapter: Paré & Schumacher, 1978: Abstract families and the adjoint functor theorems (doi)

Indexed monoidal categories

Indexed monoidal categories are to monoidal fibrations as indexed categories are to (Grothendieck) fibrations.

  • Hofstra & De Marchi, 2006: Descent for monads (pdf), Sec 3: Indexed monoidal categories
  • Ponto & Shulman, 2012: Duality and traces for indexed monoidal categories (arxiv, nCat Cafe )
    • Sec 9: String diagrams for indexed objects, inspired by: Brady & Trimble, 1998: A string diagram calculus for predicate logic (pdf, nLab )
    • Sec 10: String diagrams for indexed morphisms, conceptualized as three dimensional but drawn in two dimensions
  • Moeller & Vasilakopoulou, 2019: Monoidal Grothendieck construction

Enriched indexed categories

  • Bunge, 2013: Tightly bounded completions (pdf), Section 2: Indexed enriched category theory
  • Shulman, 2013: Enriched indexed categories (arxiv, pdf)
    • Blog posts at n-Cat Cafe (1 , 2 )
    • As the title suggests, enriched category theory + indexed category theory
    • “Includes classical enriched categories, indexed and fibered categories, and internal categories as special cases”

Applications

  • Tarlecki, Burstall, Goguen, 1991: Some fundamental algebraic tools for the semantics of computation: Part 3. Indexed categories (doi, tech report )
  • Rosebrugh & Wood, 1992: Relational databases and indexed categories (pdf)