2,459 research outputs found
A Cartesian Closed Extension of the Category of Locales
We present a Cartesian closed category ELOC of equilocales,
which contains the category LOC of locales as a reflective full subcategory.
The embedding of LOC into ELOC preserves products and all exponentials of exponentiable locales
Ionads
The notion of Grothendieck topos may be considered as a generalisation of
that of topological space, one in which the points of the space may have
non-trivial automorphisms. However, the analogy is not precise, since in a
topological space, it is the points which have conceptual priority over the
open sets, whereas in a topos it is the other way around. Hence a topos is more
correctly regarded as a generalised locale, than as a generalised space. In
this article we introduce the notion of ionad, which stands in the same
relationship to a topological space as a (Grothendieck) topos does to a locale.
We develop basic aspects of their theory and discuss their relationship with
toposes.Comment: 24 pages; v2: diverse revisions; v3: chopped about in face of
trenchant and insightful referee feedbac
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Formal concept analysis has grown from a new branch of the mathematical field
of lattice theory to a widely recognized tool in Computer Science and
elsewhere. In order to fully benefit from this theory, we believe that it can
be enriched with notions such as approximation by computation or
representability. The latter are commonly studied in denotational semantics and
domain theory and captured most prominently by the notion of algebraicity, e.g.
of lattices. In this paper, we explore the notion of algebraicity in formal
concept analysis from a category-theoretical perspective. To this end, we build
on the the notion of approximable concept with a suitable category and show
that the latter is equivalent to the category of algebraic lattices. At the
same time, the paper provides a relatively comprehensive account of the
representation theory of algebraic lattices in the framework of Stone duality,
relating well-known structures such as Scott information systems with further
formalisms from logic, topology, domains and lattice theory.Comment: 36 page
Predicative toposes
We explain the motivation for looking for a predicative analogue of the
notion of a topos and propose two definitions. For both notions of a
predicative topos we will present the basic results, providing the groundwork
for future work in this area
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