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Construction of closure operations in a category of presheaves. (English) Zbl 07582433

Summary: We construct some types of universal closure operations induced by certain collection of morphisms. For this purpose, we use Lawvere-Tierney topologies and universal closure operations that correspond to each other to establish the equivalent conditions over the collection of morphisms. In this way we use multiple sieves instead of principal sieves for constructing results. Examples are also given to illustrate the established results.

MSC:

06A05 Total orders
06A15 Galois correspondences, closure operators (in relation to ordered sets)
18F10 Grothendieck topologies and Grothendieck topoi
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[1] monomorphisms (epi-morphisms), then the n-identity property as well as the n-quasi-meet property hold. Thus, under the above conditions, the induced functor M n , map j n , and universal operation c n satisfy (1), (2), (3), and (5) of theorems 1.11, 2.4 and 2.10, respectively. As a special case let X be the full subcategory of T op (category of topological spaces and continuous functions) consisting of finite ordinal topological spaces and consider X as all monomorphisms (epimorphisms).
[2] Example 2.13. Let (X, ≤) be a preordered set and X = C(X, ≤) be the category it induces (see [1]). We know in case x ≤ y, Hom(x, y) has a unique morphism, which we denote by (x, y). It is not hard to see that ⟨(a, x)⟩ • (b, x) = {(c, b) : c ≤ b and c ≤ a} and that ((b, x) ⇒
[3] ⟨(a, x)⟩) ̸ = ∅ if and only if a meet a ∧ b exists, in which case (a ∧ b, b) ∈ ((b, x) ⇒ ⟨(a, x)⟩) or equivalently ⟨(a, x)⟩ • (b, x) = ⟨(a ∧ b, b)⟩. Let M be a class of morphisms of X . M satisfies the principality property if and only if for each (a, x) ∈ M/x and (b, x) ∈ X 1 /x a meet a ∧ b exists and (a ∧ b, b) ∈ M/b; M has enough retractions (almost enough retractions) if and only if for each x, 1 x ∈ M/x (1 x ∈ M/x or M/x ̸ = ∅);
[4] M has the n-identity property if and only if for all x and for all sieves S on x, if the set M S has at most n maximal elements in M, then it contains 1 x ; M has the n-maximal property if and only if for all x, every nonempty subset of (M/x, ≤ op ) has at most n maximal elements and also for all x and (a, x) ∈ X 1 /x, either there is (b, x) in
[5] M/x such that b ∼ = a or for all (b, x) ∈ M/x, b ≥ a; and finally M has the n-quasi-meet property if and only if M has local binary meet (i.e. for all objects x, M/x has binary meet). In case (X, ≤) is a partially ordered set, every maximum or meet that exists is unique and if (X, ≤) is a lattice then every binary meet exists and is unique. 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. Now suppose for all x, M/x ̸ = ∅ and for a ≤ b ≤ x, (a, x) ∈ M if and only if (a, b) ∈ M and (b, x) ∈ M. One can then verify that M satisfies the principality as well as all the properties listed in Definition 1.5. References
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