34 research outputs found
Displayed Categories
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.Comment: v3: Revised and slightly expanded for publication in LMCS. Theorem
numbering change
Displayed Categories
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
The local universes model: an overlooked coherence construction for dependent type theories
We present a new coherence theorem for comprehension categories, providing
strict models of dependent type theory with all standard constructors,
including dependent products, dependent sums, identity types, and other
inductive types.
Precisely, we take as input a "weak model": a comprehension category,
equipped with structure corresponding to the desired logical constructions. We
assume throughout that the base category is close to locally Cartesian closed:
specifically, that products and certain exponentials exist. Beyond this, we
require only that the logical structure should be *weakly stable* --- a pure
existence statement, not involving any specific choice of structure, weaker
than standard categorical Beck--Chevalley conditions, and holding in the now
standard homotopy-theoretic models of type theory.
Given such a comprehension category, we construct an equivalent split one,
whose logical structure is strictly stable under reindexing. This yields an
interpretation of type theory with the chosen constructors.
The model is adapted from Voevodsky's use of universes for coherence, and at
the level of fibrations is a classical construction of Giraud. It may be viewed
in terms of local universes or delayed substitutions.Comment: 36 pages. Definition of "pseudo-stable" corrected from earlier
version. To appear in ACM Transactions on Computational Logi