ON GENERALIZED BOURBAKI THEOREM IN (T,V)-CATEGORIES
JUANDA KELANA PUTRA

Masaryk University

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 (T,V)-
category. Their theorem states that, under certain conditions, a (T,V)-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 con-
ditions 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.

Keywords: Proper map, Perfect map, (T,V)-category, (T,V)-functor

Topic: Minisymposia Geometry and Topology