eng
episciences.org
Logical Methods in Computer Science
1860-5974
2019-03-05
Volume 15, Issue 1
10.23638/LMCS-15(1:20)2019
4308
journal article
Displayed Categories
Benedikt Ahrens
Peter LeFanu Lumsdaine
We introduce and develop the notion of *displayed categories*.
A displayed category over a category C is equivalent to "a category D and
functor F : D --> C", but instead of having a single collection of "objects of
D" with a map to the objects of C, the objects are given as a family indexed by
objects of C, and similarly for the morphisms. This encapsulates a common way
of building categories in practice, by starting with an existing category and
adding extra data/properties to the objects and morphisms.
The interest of this seemingly trivial reformulation is that various
properties of functors are more naturally defined as properties of the
corresponding displayed categories. Grothendieck fibrations, for example, when
defined as certain functors, use equality on objects in their definition. When
defined instead as certain displayed categories, no reference to equality on
objects is required. Moreover, almost all examples of fibrations in nature are,
in fact, categories whose standard construction can be seen as going via
displayed categories.
We therefore propose displayed categories as a basis for the development of
fibrations in the type-theoretic setting, and similarly for various other
notions whose classical definitions involve equality on objects.
Besides giving a conceptual clarification of such issues, displayed
categories also provide a powerful tool in computer formalisation, unifying and
abstracting common constructions and proof techniques of category theory, and
enabling modular reasoning about categories of multi-component structures. As
such, most of the material of this article has been formalised in Coq over the
UniMath library, with the aim of providing a practical library for use in
further developments.
https://lmcs.episciences.org/4308/pdf
Mathematics - Category Theory
Mathematics - Logic
18A15, 03B15, 03B70
F.4.1