Closure operators and their middle-interchange law

The paper presents a new definition of closure operator which encompasses the standard Dikranjan–Giuli notion, as well as the Bourn–Gran notion of normal closure operator. As is well known, any two closure operators C, D in a category may be composed in, within order, two different ways. For a subobject M→X one may consider DX(CXM) or DCX(M)(M) as the value at M of a new closure operator D⋅C or D⁎C, respectively. The two binary operations are linked by a lax middle-interchange law. This paper explores situations in which the law holds strictly.