737 research outputs found
Categorical Structures for Type Theory in Univalent Foundations
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently. Specifically, we construct maps between the different structures and show that these maps are equivalences under suitable assumptions.
We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure.
We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library
Heterogeneous substitution systems revisited
Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical
description of substitution systems capable of capturing syntax involving
binding which is independent of whether the syntax is made up from least or
greatest fixed points. We extend this work in two directions: we continue the
analysis by creating more categorical structure, in particular by organizing
substitution systems into a category and studying its properties, and we
develop the proofs of the results of the cited paper and our new ones in
UniMath, a recent library of univalent mathematics formalized in the Coq
theorem prover.Comment: 24 page
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
Fibred Fibration Categories
We introduce fibred type-theoretic fibration categories which are fibred
categories between categorical models of Martin-L\"{o}f type theory. Fibred
type-theoretic fibration categories give a categorical description of logical
predicates for identity types. As an application, we show a relational
parametricity result for homotopy type theory. As a corollary, it follows that
every closed term of type of polymorphic endofunctions on a loop space is
homotopic to some iterated concatenation of a loop
Towards a constructive simplicial model of Univalent Foundations
We provide a partial solution to the problem of defining a constructive
version of Voevodsky's simplicial model of univalent foundations. For this, we
prove constructive counterparts of the necessary results of simplicial homotopy
theory, building on the constructive version of the Kan-Quillen model structure
established by the second-named author. In particular, we show that dependent
products along fibrations with cofibrant domains preserve fibrations, establish
the weak equivalence extension property for weak equivalences between
fibrations with cofibrant domain and define a univalent classifying fibration
for small fibrations between bifibrant objects. These results allow us to
define a comprehension category supporting identity types, -types,
-types and a univalent universe, leaving only a coherence question to be
addressed.Comment: v3: changed the definition of the type Weq(U) of weak equivalences to
fix a problem with constructivity. Other Minor changes. 31 page
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
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