The preservation of meets and joins, and hence whether a monotone map sustainsgenerative effects, is tightly related to the concept of a Galois connection, or moregenerally an adjunction.

In his work on generative effects, Adam restricts his attention to maps that preservemeets, even while they do not preserve joins. The preservation of meets implies that themapbehaves well when restricting to a subsystem, even if it can throw up surpriseswhen joining systems

n [Ada17], Adam thinks of monotone maps as observations. A monotone map:P!Qis a phenomenon ofPas observed byQ. He defines generative effects of such a mapto be its failure to preserve joins (or more generally, for categories, its failure topreserve colimits)