Eraldo Giuli

Closure operators I

Abstract Closure operators in an ( E , M )-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M . Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X . Every closure operators admits an idempotent hull and a weakly hereditary core. Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotent, weakly hereditary, standard closure operators correspond to idempotent radicals (= torsion theories).

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.

Onm-separated projection spaces

Closure operators in the category of projection spaces are investigated. It is shown that completeness, absolutes-closure ands-injectivity coincide in the subcategory of separated projection spaces and that there compactness with respect to projections implies completeness.

Factorizations, injectivity and compactness in categories of modules

A notion of closure operator for modules is used to characterize factorization structures in categories of modules. Moreover compactness, injectivity and absolute closedness are studied with respect to such closure operators. A criterion for compactness of modules is obtained in terms of injectivity or absolute closedness of the quotients extending recent results of Temple Fay.

Openness with Respect to a Closure Operator

We study the notions of closed, open, initial and final morphism with respect to a closure operator and show that they have a perfectly symmetric pullback behaviour. We also investigate their interaction with closed and with open subobjects and the impact of the existence of suitably defined complements of subobjects.

A Topologist’s View of Chu Spaces

For a symmetric monoidal-closed category

What is a Quotient Map with Respect to a Closure Operator

\mathcal{X}

Objects with closed diagonals

and any object K, the category of K-Chu spaces is small-topological over

Closure Operators with Respect to a Functor

\mathcal{X}

Bases of topological epi-reflections

and small cotopological over