Abstract
Bourbaki developed the concept of a proper map in topological spaces and proved that a continuous map between topological spaces is proper if and only if it is perfect, known as Bourbaki theorem. Clementino and Tholen extended this concept to lax algebras, formulating a generalized Bourbaki theorem applicable to a special type of category called a $(\S,\Q)$-category. Their theorem states that, under certain conditions, a $(\S,\Q)$-functor is proper if and only if both pullbacks of the functor are closed and a specific transformation is closed. They also provide an equivalent characterization using compactness of fibers. Clementino and Tholen then posed a question: If we slightly modify the conditions in their generalized theorem, do the equivalences still hold? This paper aims to answer this question, investigating the impact of these modifications on the relationship between properness and closure properties.
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