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Here is an old discssion box from finite object which hereby I am moving from there to here.
+–{.query}
Toby: I think that I'll move the internal stuff to finite object, to keep each page relatively short.
By the way, do I understand you correctly that ’finite object’ in topos theory by default means ’decidable K-finite object’?
Mike: Okay (to the move). To the question, I’m realizing more and more that I don’t really have the background to be able to say what “topos theorists” say. My only source for this material is the Elephant (and what I’ve been able to deduce on my own, which of course tells us nothing about terminology). The Elephant never says “finite object” unqualified; only “finite cardinal” or “K-finite object” or “decidable K-finite object” or “$\tilde{K}$-finite object.” If “projective” means “externally projective,” and likewise for “choice” and (maybe) “inhabited,” then “finite object” should mean “finite cardinal,” but I wouldn’t use it that way myself out of fear of ambiguity and since “finite cardinal” means the same thing. I don’t see any objection to “internally finite object” meaning “decidable K-finite object,” though.
=–
I am working on finite object.
I have taken the previous material and split it off into a separate Definitions-section and a section Properties – Subobjects of finite objects. Then I started more Properties-subsections, such as Closure properties and Relation to compact objects. I also tried to edit the Definition-section for quick readability, but I am not happy with it yet.
Toby, when you see this here, please check the entry and see if you can live with what I did.
I moved the comment “internally finite = decidable K-finite” up to the Definitions - Internal version section.
At finite object there was (and is) a warning:
Also beware that in category theory the term ’finite object’ is also used in a much more general sense to mean a compact object.
But that’s not the whole story, yet. I have added the sentence:
Similar finitenss meaning may also be attributed to dualizable objects in monoidal categories and to perfect complexes (of abelian sheaves) in geometry.
Since this is kind of important for a global appreciation of the situation, I gave this a table-entry, for inclusion as “Related concepts” in the relevant entries:
This reminds me to ask: is the monoidal category of nuclear TVS (under the projective tensor product, or maybe the completed projective tensor product) compact closed?
I have a dim memory of investigating this once and deciding the answer was “yes”, but if there are experts reading (such as Andrew Stacey or Yemon Choi), maybe they could confirm or deny.
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