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 \(B\) and related structures: in the “functorial” style, as functors out of \(B\) into a fixed large category, and in the “fibrational” style, as functors into \(B\), i.e., objects of the slice category \(\mathbf{Cat}/B\).
| Functorial name | Functorial definition | Fibrational name |
|---|---|---|
| Copresheaf over \(B\) | functor \(B \to \mathbf{Set}\) | Discrete opfibration over \(B\) |
| Presheaf over \(B\) | functor \(B^{\mathrm{op}} \to \mathbf{Set}\) | Discrete fibration over \(B\) |
| Displayed category over \(B\) with UFL | pseudofunctor \(B \to \mathbf{Span}\) | Discrete Conduché fibration over \(B\) |
| Displayed category over \(B\) | lax functor \(B \to \mathbf{Span}\) | Category over \(B\) |
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 \(B \to \mathbf{Prof}\).
- Ahrens & Lumsdaine, 2019: Displayed categories (doi, arxiv)
- Pavlović & Abramsky, 1997: Specifying interaction categories (doi)
- Proposition 4 records the equivalence \(\mathbf{Cat}/B \simeq [B,\mathbf{Span}]_{\mathrm{lax}}\)
- 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 fibrational 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