30 research outputs found
Basic Subtoposes of the Effective Topos
We employ a new tool (sights) to investigate local operators in the Effective
Topos. A number of new such local operators is analyzed using this machinery.
Moreover, we investigate a local operator defined in the thesis of A. Pitts,
and establish that its corresponding subtopos satisfies true arithmetic.Comment: 26 page
A 2-Categorical Analysis of the Tripos-to-Topos Construction
We characterize the tripos-to-topos construction of Hyland, Johnstone and
Pitts as a biadjunction in a bicategory enriched category of equipment-like
structures. These abstract concepts are necessary to handle the presence of
oplax constructs --- the construction is only oplax functorial on certain
classes of cartesian functors between triposes. A by-product of our analysis is
the decomposition of the tripos-to-topos construction into two steps, the
intermediate step being a weakened version of quasitoposes
Some Properties of Internal Locale Morphisms Externalised
We study morphisms of internal locales of Grothendieck toposes externally:
treating internal locales and their morphisms as sheaves and natural
transformations. We characterise those morphisms of internal locales that
induce surjective geometric morphisms and geometric embeddings, demonstrating
that both can be computed `pointwise'. We also show that the co-frame
operations on the co-frame of internal sublocales can also be computed
`pointwise' too.Comment: 46 pages. Updated version for submission. New sections on examples
and applications, in addition to general improvement
An informal introduction to topos theory
This short expository text is for readers who are confident in basic category
theory but know little or nothing about toposes. It is based on some impromptu
talks given to a small group of category theorists.Comment: 27 pages. Version 3: expanded section on set theory; other minor
edit
The Modified Realizability Topos
The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, Hyland-Ong-Ritter) and strong normalization. In this paper this topos is investigated from a general logical and topostheoretic point of view. It is shown that Mod (as we call the topos) is the closed complement of the effective topos inside another one; this turns out to have some logical consequences. Some important subcategories of Mod are described, and a general logical principle is derived, which holds in the larger topos and implies the well-known Independence of Premiss principle
More on Geometric Morphisms between Realizability Toposes
Geometric morphisms between realizability toposes are studied in terms of
morphisms between partial combinatory algebras (pcas). The morphisms inducing
geometric morphisms (the {\em computationally dense\/} ones) are seen to be the
ones whose `lifts' to a kind of completion have right adjoints. We characterize
topos inclusions corresponding to a general form of relative computability. We
characterize pcas whose realizability topos admits a geometric morphism to the
effective topos.Comment: 20 page
Grothendieck quasitoposes
A full reflective subcategory E of a presheaf category [C*,Set] is the
category of sheaves for a topology j on C if and only if the reflection
preserves finite limits. Such an E is called a Grothendieck topos. More
generally, one can consider two topologies, j contained in k, and the category
of sheaves for j which are separated for k. The categories E of this form, for
some C, j, and k, are the Grothendieck quasitoposes of the title, previously
studied by Borceux and Pedicchio, and include many examples of categories of
spaces. They also include the category of concrete sheaves for a concrete site.
We show that a full reflective subcategory E of [C*,Set] arises in this way for
some j and k if and only if the reflection preserves monomorphisms as well as
pullbacks over elements of E.Comment: v2: 24 pages, several revisions based on suggestions of referee,
especially the new theorem 5.2; to appear in the Journal of Algebr