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In the doctine page of John’s web, it says
Categorified Gabriel–Ulmer duality is all about recovering the syntactic doctrine from the semantic doctrine.
It struck me that ’recovering’ is the kind of term you hear when people talk about Tannaka reconstruction theorems. Looking at the page - reconstruction theorem - I see it includes both Tannakian examples and Lawvere’s reconstruction theorem, an ancestor of Gabriel-Ulmer duality.
So I wondered if we ever have Gabriel-Ulmer and Tannaka duality treated together, and found Brian Day in Enriched Tannaka reconstruction writing
Our approach provides a synthesis of Tannaka reconstruction and Gabriel-Ulmer duality.
I can see how they’re both reconstructing something by looking at the collection of maps into something else, but I can’t understand Day’s general framework. Can anyone help?
Could you send me the Day’s article ? I do not seem to have access.
Just did so. Let me know if it doesn’t arrive.
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