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Made a few additions to preimage. Added missing word; added a brief mention of the widely-known general reason for the good preservation-properties of this endofunctor.
The mention of these properties had already been there in preimage, but a reason was still missing. My parenthetical remark should perhaps be expanded and harmonized with existing relevant material on the nLab ($\forall_f$ and $\exists_f$ are already well-documented on some pages), but this requires more care than I can apply to it today. Intend to return to the remark before long.
I changed the wording. The phrases “existential quantifier” and “universal quantifier” are well-established and appropriate here. One could also call $\exists_f$ “direct image along $f$”; I don’t know of a similar analogous phrase for $\forall_f$.
Sometimes I think $\forall_f$ is called the “dual image”. Unfortunately “co-image” seems to be taken for the image with respect to the dual factorization system.
BTW, Peter: it was good to add that reason. I have a feeling that mathematicians at large are not aware of or sensitive to that particular argument (which is constructively valid, e.g., it carries over to general toposes and other general environments). The basic insight of $\exists_f \dashv f^\ast \dashv \forall_f$ is often credited to Lawvere.
I have a feeling that mathematicians at large are not aware of or sensitive to that particular argument
Yes, and instead of
widely-known general reason
I should have written “widely, but by far not widely-enough-known”.
Of course, the links already led to the marvellous nLab page
interactions of images and pre-images with unions and intersections
but personally, I think this can hardly be mentioned too often, and some “foreshadowing” in preimage is useful.
Incidentally, I used “universal image” and “existential image” because Goldblatt uses them on p. 183 of
R. Goldblatt: Varieties of complex algebras. Annals of Pure and Applied Logic 44 (1989) 173–242
a paper I was studying while working on a proof of non-elementarity of a certain class of graphs.
(Not that therefore “universal image” and “existential image” should be used, just to add some context.)
I added some more observations, bringing up for example toposes and also the fact that quantifiers may be viewed as enriched Kan extensions, as famously observed by Lawvere.
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