Displayed categories

Displayed categories axiomatize categories over a base category in a generalized algebraic way, offering a different perspective on discrete fibered category theory.

There are two ways to think about displayed categories over BB and related structures: in the “indexed” style, as functors out of BB into a fixed large category, and in the “fibered” style, as functors into BB, i.e., objects of the slice category Cat/B\mathbf{Cat}/B.

Indexed nomenclature Indexed definition Fibered nomenclature
Copresheaf over BB functor B→SetB \to \mathbf{Set} Discrete opfibration over BB
Presheaf over BB functor Bop→SetB^{\mathrm{op}} \to \mathbf{Set} Discrete fibration over BB
Displayed category over BB with UFL pseudofunctor B→SpanB \to \mathbf{Span} Discrete Conduché fibration over BB
Displayed category over BB lax functor B→SpanB \to \mathbf{Span} Category over BB

In the table, “UFL” is short for “unique factorization of lifting” (see below).

Literature

Displayed categories

The name “displayed category” is recent but the idea is much older, going back to Bénabou in the guise of normal lax functors B→ProfB \to \mathbf{Prof}.

  • Ahrens & Lumsdaine, 2019: Displayed categories (doi, arxiv)
  • Pavlović & Abramsky, 1997: Specifying interaction categories (doi)
    • Proposition 4 records the equivalence Cat/B≃[B,Span]lax\mathbf{Cat}/B \simeq [B,\mathbf{Span}]_{\mathrm{lax}}
  • Manuell, 2021: Monoid extensions and the Grothendieck construction (arxiv), Sec. 2: The Grothendieck–Bénabou correspondence

UFL functors

Unique factorization lifting (UFL) functors, also known as strict Conduché functors and discrete Conduché fibrations, are (the fibered analogue) of a special kind of displayed category that nevertheless generalizes both presheaves and copresheaves.

  • Kock & Spivak, 2020: Decomposition-space slices are toposes (doi, arxiv)
  • Bunge & Fiore, 2000: Unique factorisation lifting functors and categories of linearly-controlled processes (doi)
    • A good place to start: fairly readable and motivated by CS applications
  • Bunge & Niefield, 2000: Exponentiability and single universes (doi)
  • Johnstone, 1999: A note on discrete Conduché fibrations (pdf)
  • Street, 1996: Categorical structures, in Handbook of Algebra, Vol 1
    • Brief mention of UFL on pp. 532-3
  • Lawvere, 1986: State categories and response functors (pdf)
    • UFL categories mainly over monoids, loosely motivated by physics