A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

Abstract

In this paper we show that each factorization structure M on a small category X , satisfying certain conditions, yields a presheaf M on X and a morphism of presheaves m :. −→ M. We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on X (b) subobjects of M (c) morphisms from M to (d) weak Lawvere-Tierney topologies (e) weak Grothendieck topologies (f) closure operators on Sets X op .

Weak Topologies on Toposes

“…In this section, we present the basic properties of weak LT-topologies on toposes. To begin, we recall the following definition from [15].…”

“…Considering LT-topologies in the framework of [2], a weak topology is exactly an LT-topology without idempotency. The term 'weak Lawvere-Tierney topology' was coined by Hosseini and Mousavi in [15]. On the other hand, modal closure operators on a category and its types are of interest to some mathematicians, e.g.…”

Morphism classes producing (weak) Grothendieck topologies, (weak) Lawvere--Tierney topologies, and universal closure operations

“…(a) Let (X, ≤) be any partially ordered set such that every nonempty subset of X has a maximum (≤ op is then indeed a total order and (X, ≤ op ) is well-ordered). Obviously every sieve on an object x ∈ X is principal and weak Grothendieck topology and j is a weak Lawvere-Tierney topology, see [3].…”

“…As a special case one can take X = {· · · , −3, −2, −1} in the usual order and to the total sieve and for m, weak Grothendieck topology and j is a weak Lawvere-Tierney topology, see [3]. …”

Quasi Right Factorization Structures as Presheaves

“…Let P : def = P X1 , see [5], where X 1 is the class of all morphisms in X . Calling a 2-reflection in Lax(PrOrd X op ) a lax 2-reflection, we have: …”