41 research outputs found
Constructing categories and setoids of setoids in type theory
In this paper we consider the problem of building rich categories of setoids,
in standard intensional Martin-L\"of type theory (MLTT), and in particular how
to handle the problem of equality on objects in this context. Any
(proof-irrelevant) family F of setoids over a setoid A gives rise to a category
C(A, F) of setoids with objects A. We may regard the family F as a setoid of
setoids, and a crucial issue in this article is to construct rich or large
enough such families. Depending on closure conditions of F, the category C(A,
F) has corresponding categorical constructions. We exemplify this with finite
limits. A very large family F may be obtained from Aczel's model construction
of CZF in type theory. It is proved that the category so obtained is isomorphic
to the internal category of sets in this model. Set theory can thus establish
(categorical) properties of C(A, F) which may be used in type theory. We also
show that Aczel's model construction may be extended to include the elements of
any setoid as atoms or urelements. As a byproduct we obtain a natural extension
of CZF, adding atoms. This extension, CZFU, is validated by the extended model.
The main theorems of the paper have been checked in the proof assistant Coq
which is based on MLTT. A possible application of this development is to
integrate set-theoretic and type-theoretic reasoning in proof assistants.Comment: 14 page
On Equality of Objects in Categories in Constructive Type Theory
In this note we remark on the problem of equality of objects in categories
formalized in Martin-L\"of's constructive type theory. A standard notion of
category in this system is E-category, where no such equality is specified. The
main observation here is that there is no general extension of E-categories to
categories with equality on objects, unless the principle Uniqueness of
Identity Proofs (UIP) holds. We also introduce the notion of an H-category, a
variant of category with equality on objects, which makes it easy to compare to
the notion of univalent category proposed for Univalent Type Theory by Ahrens,
Kapulkin and Shulman.Comment: 7 page
Formalizing of Category Theory in Agda
The generality and pervasiness of category theory in modern mathematics makes
it a frequent and useful target of formalization. It is however quite
challenging to formalize, for a variety of reasons. Agda currently (i.e. in
2020) does not have a standard, working formalization of category theory. We
document our work on solving this dilemma. The formalization revealed a number
of potential design choices, and we present, motivate and explain the ones we
picked. In particular, we find that alternative definitions or alternative
proofs from those found in standard textbooks can be advantageous, as well as
"fit" Agda's type theory more smoothly. Some definitions regarded as equivalent
in standard textbooks turn out to make different "universe level" assumptions,
with some being more polymorphic than others. We also pay close attention to
engineering issues so that the library integrates well with Agda's own standard
library, as well as being compatible with as many of supported type theories in
Agda as possible
Proof-relevance in Bishop-style constructive mathematics
Bishop's presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory (BST), our minimal extension of Bishop's theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas f of BISH a set Prf(f) of "proofs" or witnesses of f. Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of aMartin-Lof set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional MartinLof type theory (MLTT). Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky's axiom of univalence) can be translated into BISH. While Bishop's theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction
W-types in setoids
We present a construction of W-types in the setoid model of extensional
Martin-L\"of type theory using dependent W-types in the underlying intensional
theory. More precisely, we prove that the internal category of setoids has
initial algebras for polynomial endofunctors. In particular, we characterise
the setoid of algebra morphisms from the initial algebra to a given algebra as
a setoid on a dependent W-type. We conclude by discussing the case of free
setoids. We work in a fully intensional theory and, in fact, we assume identity
types only when discussing free setoids. By using dependent W-types we can also
avoid elimination into a type universe. The results have been verified in Coq
and a formalisation is available on the author's GitHub page
Dependently-Typed Formalisation of Typed Term Graphs
We employ the dependently-typed programming language Agda2 to explore
formalisation of untyped and typed term graphs directly as set-based graph
structures, via the gs-monoidal categories of Corradini and Gadducci, and as
nested let-expressions using Pouillard and Pottier's NotSoFresh library of
variable-binding abstractions.Comment: In Proceedings TERMGRAPH 2011, arXiv:1102.226