Internal Neighbourhood Structures III: Finite Sum of Subobjects

TL;DR

This paper explores conditions under which certain subobject structures in a category are closed under finite sums, providing new insights into the algebraic and categorical properties of preneighbourhood spaces and related morphisms.

Contribution

It establishes structural conditions on categories that ensure the closure of subobjects and morphisms under finite sums, extending the understanding of internal neighbourhood structures.

Findings

Subobjects of an object are closed under finite sums under certain conditions.

Closed embeddings form a join semilattice that is a biproduct of component semilattices.

Full subcategories of compact or Hausdorff preneighbourhood spaces are closed under finite sums.

Abstract

The notion of an internal preneighbourhood space on a finitely complete category with finite coproducts and a proper $(\mathsf{E}, \mathsf{M})$ system such that for each object $X$ the set of $\mathsf{M}$-subobjects of $X$ is a complete lattice was initiated in \cite{2020}. The notion of a closure operator, closed morphism and its near allies investigated in \cite{2021-clos}. The present paper provides structural conditions on the triplet $(\mathbb{A}, \mathsf{E}, \mathsf{M})$ (with $\mathbb{A}$ lextensive) equivalent to the set of $\mathsf{M}$-subobjects of an object closed under finite sums. Equivalent conditions for the set of closed embeddings (closed morphisms) closed under finite sums is also provided. In case when lattices of admissible subobjects (respectively, closed embeddings) are closed under finite sums, the join semilattice of admissible subobjects (respectively, closed embeddings) of a finite sum is shown to be a biproduct of the component join semilattices. Finally, it is shown whenever the set of closed morphisms is closed under finite sums, the set of proper (respectively, separated) morphisms are also closed under finite sums. This leads to equivalent conditions for the full subcategory of compact (respectively, Hausdorff) preneighbourhood spaces to be closed under finite sums.