Closure operators with prescribed properties.

*(English)*Zbl 0659.18005
Categorical algebra and its applications, Proc. 1st Conf., Louvain-la- Neuve/Belg. 1987, Lect. Notes Math. 1348, 208-220 (1988).

[For the entire collection see Zbl 0644.00009.]

The basic idea for a closure operator in a category \({\mathcal X}\) is to have for each object X an extensive, increasing and idempotent operation C on the ordered class of its subobjects \(A\to X\); A is said to be closed (resp. dense) if C(A) is isomorphic to A (resp. to X). The author generalizes this notion of closure operator in two ways. Since frequently particular types of subobjects deserve special attention (such as the subspaces in topology), he does not insist on just studying monomorphisms but rather considers subclasses \({\mathcal M}\) of \({\mathcal X}\). Secondly, much of the theory can be developed without requiring idempotency. To convey the paper’s flavor, let me quote prop. 2.02 which has an immediate topological appeal and asserts that, for an object X of \({\mathcal X}\), the following three conditions are equivalent: a) The diagonal \(X\to X\times X\) is closed; b) X is separated; c) the diagonal \(X\to X\times X\) is a sheaf.

The basic idea for a closure operator in a category \({\mathcal X}\) is to have for each object X an extensive, increasing and idempotent operation C on the ordered class of its subobjects \(A\to X\); A is said to be closed (resp. dense) if C(A) is isomorphic to A (resp. to X). The author generalizes this notion of closure operator in two ways. Since frequently particular types of subobjects deserve special attention (such as the subspaces in topology), he does not insist on just studying monomorphisms but rather considers subclasses \({\mathcal M}\) of \({\mathcal X}\). Secondly, much of the theory can be developed without requiring idempotency. To convey the paper’s flavor, let me quote prop. 2.02 which has an immediate topological appeal and asserts that, for an object X of \({\mathcal X}\), the following three conditions are equivalent: a) The diagonal \(X\to X\times X\) is closed; b) X is separated; c) the diagonal \(X\to X\times X\) is a sheaf.

Reviewer: J.Sonner

##### MSC:

18A32 | Factorization systems, substructures, quotient structures, congruences, amalgams |