70 research outputs found
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
C-system of a module over a monad on sets
This is the second paper in a series that aims to provide mathematical
descriptions of objects and constructions related to the first few steps of the
semantical theory of dependent type systems.
We construct for any pair , where is a monad on sets and is
a left module over , a C-system (contextual category) and
describe a class of sub-quotients of in terms of objects directly
constructed from and . In the special case of the monads of expressions
associated with nominal signatures this construction gives the C-systems of
general dependent type theories when they are specified by collections of
judgements of the four standard kinds
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