273,864 research outputs found
Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets
Bishop's informal set theory is briefly discussed and compared to Lawvere's
Elementary Theory of the Category of Sets (ETCS). We then present a
constructive and predicative version of ETCS, whose standard model is based on
the constructive type theory of Martin-L\"of. The theory, CETCS, provides a
structuralist foundation for constructive mathematics in the style of Bishop.Comment: 28 page
Category Theory in Coq 8.5
We report on our experience implementing category theory in Coq 8.5. The
repository of this development can be found at
https://bitbucket.org/amintimany/categories/. This implementation most notably
makes use of features, primitive projections for records and universe
polymorphism that are new to Coq 8.5.Comment: This is the abstract for a talk accepted for a presentation at the
7th Coq Workshop, Sophia Antipolis, France on June 26, 201
Quotient completion for the foundation of constructive mathematics
We apply some tools developed in categorical logic to give an abstract
description of constructions used to formalize constructive mathematics in
foundations based on intensional type theory. The key concept we employ is that
of a Lawvere hyperdoctrine for which we describe a notion of quotient
completion. That notion includes the exact completion on a category with weak
finite limits as an instance as well as examples from type theory that fall
apart from this.Comment: 32 page
A C-system defined by a universe category
This is a major update of the previous version. The methods of the paper are
now fully constructive and the style is "formalization ready" with the emphasis
on the possibility of formalization both in type theory and in constructive set
theory without the axiom of choice.
This is the third paper in a series started in 1406.7413. In it we construct
a C-system starting from a category together with a
morphism , a choice of pull-back squares based on
for all morphisms to and a choice of a final object of . Such a
quadruple is called a universe category. We then define universe category
functors and construct homomorphisms of C-systems defined by
universe category functors. As a corollary of this construction and its
properties we show that the C-systems corresponding to different choices of
pull-backs and final objects are constructively isomorphic.
In the second part of the paper we provide for any C-system CC three
constructions of pairs where is a universe
category and is an isomorphism.
In the third part we define, using the constructions of the previous parts,
for any category with a final object and fiber products a C-system
and an equivalence
Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory
In Riesz space theory it is good practice to avoid representation theorems
which depend on the axiom of choice. Here we present a general methodology to
do this using pointfree topology. To illustrate the technique we show that
almost f-algebras are commutative. The proof is obtained relatively
straightforward from the proof by Buskes and van Rooij by using the pointfree
Stone-Yosida representation theorem by Coquand and Spitters
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