Final lift actions associated with topological functors (Q1866043)

If \(U\) is the forgetful functor from the category of topological spaces and continuous functions to the category of sets and functions, then the fiber \(U^{-1}(A)\) of all topologies on the set \(A\) is a poset under reverse inclusion, cocomplete in the sense that it is closed under the taking of arbitrary suprema (intersections in this case). Moreover, if \(f\) is a function from the set \(A\) to itself, and \(\tau\) is a topology on \(A\), then we have the coinduced topology \(f(\tau): =\{u\subseteq A:f^{-1} (u)\in\tau\}\) on \(A\). This assignment defines a left action of the endomorphism monoid of \(A\) on the cocomplete poset \(U^{-1}(A)\), cocontinuous in that it preserves all suprema. The scenario sketched above persists in the more general setting, where \(U:E\to B\) is a topological functor à la \textit{H. Herrlich} [General Topol. Appl. 4, 125-142 (1974; Zbl 0288.54003)]. If \(b\) is a fixed object in \(B\), then for any endomorphism \(f\) on \(b\), and any object \(x\in U^{-1}(b)\), there is a uniquely defined object \(f(x)\in U^{-1}(b)\), the ``codomain for the final lift of \(f\) to \(x\).'' This assignment still gives a cocontinuous left action of the endomorphism monoid of \(b\) on the cocomplete poset \(U^{-1}(b)\), and is called the final lift action. The author's lead result is that if \(b\) is an object in a category \(B\), then every cocontinuous left action of the endomorphism monoid of \(b\) on an arbitrary cocomplete poset may be realized as the final lift action associated to a canonically defined topological functor over \(B\). The author goes on to study the situation where \(b\) is the subobject classifier in a Grothendieck topos \(B\). In this case the endomorphism monoid on \(b\) has a natural cocontinuous poset structure, and hence it may be seen to act cocontinuously on itself. The author gives explicit descriptions of these cocontinuous left actions when \(B\) is the category of sets; further he describes the associated topological functors that realize them.