31,474 research outputs found
A co-free construction for elementary doctrines
We provide a co-free construction which adds elementary structure to a
primary doctrine. We show that the construction preserves comprehensions and
all the logical operations which are in the starting doctrine, in the sense
that it maps a first order many-sorted theory into a the same theory formulated
with equality. As a corollary it forces an implicational doctrine to have an
extentional entailment
Elementary quotient completion
We extend the notion of exact completion on a weakly lex category to
elementary doctrines. We show how any such doctrine admits an elementary
quotient completion, which freely adds effective quotients and extensional
equality. We note that the elementary quotient completion can be obtained as
the composite of two free constructions: one adds effective quotients, and the
other forces extensionality of maps. We also prove that each construction
preserves comprehensions
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
Old School Catalog 1912-13, The Department of Law
https://scholar.valpo.edu/oldschoolcatalogs/1011/thumbnail.jp
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