1,270 research outputs found
Totally distributive toposes
A locally small category E is totally distributive (as defined by
Rosebrugh-Wood) if there exists a string of adjoint functors t -| c -| y, where
y : E --> E^ is the Yoneda embedding. Saying that E is lex totally distributive
if, moreover, the left adjoint t preserves finite limits, we show that the lex
totally distributive categories with a small set of generators are exactly the
injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize
the totally distributive categories with a small set of generators as exactly
the essential subtoposes of presheaf toposes, studied by Kelly-Lawvere and
Kennett-Riehl-Roy-Zaks.Comment: Now includes extended result: The lex totally distributive categories
with a small set of generators are exactly the injective Grothendieck
toposes; Made changes to abstract and intro to reflect the enhanced result;
Changed formatting of diagram
Enriched factorization systems
In a paper of 1974, Brian Day employed a notion of factorization system in
the context of enriched category theory, replacing the usual diagonal lifting
property with a corresponding criterion phrased in terms of hom-objects. We set
forth the basic theory of such enriched factorization systems. In particular,
we establish stability properties for enriched prefactorization systems, we
examine the relation of enriched to ordinary factorization systems, and we
provide general results for obtaining enriched factorizations by means of wide
(co)intersections. As a special case, we prove results on the existence of
enriched factorization systems involving enriched strong monomorphisms or
strong epimorphisms
Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory
Given a monad T on a suitable enriched category B equipped with a proper
factorization system (E,M), we define notions of T-completion, T-closure, and
T-density. We show that not only the familiar notions of completion, closure,
and density in normed vector spaces, but also the notions of sheafification,
closure, and density with respect to a Lawvere-Tierney topology, are instances
of the given abstract notions. The process of T-completion is equally the
enriched idempotent monad associated to T (which we call the idempotent core of
T), and we show that it exists as soon as every morphism in B factors as a
T-dense morphism followed by a T-closed M-embedding. The latter hypothesis is
satisfied as soon as B has certain pullbacks as well as wide intersections of
M-embeddings. Hence the resulting theorem on the existence of the idempotent
core of an enriched monad entails Fakir's existence result in the non-enriched
case, as well as adjoint functor factorization results of Applegate-Tierney and
Day
Tangent Categories from the Coalgebras of Differential Categories
Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science
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