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added to Isbell envelope the three original reference posted by Richard Garner to the Category Theory mailing list today (or yesterday).
I don’t have time to look into this right now, I just copied those references there for the moment).
I was surprised to see that there’s no comment on the relationship between the Isbell envelope and Cauchy completion of a category (on either page). The connection is really simple: objects in the Isbell envelope are presheaf-copresheaf pairs with a “counit” and the Cauchy completion can be seen as the subcategory where you have a unit making the pair into an adjunction.
This seems like a preferable view to the description at the beginning of Cauchy complete category which defines it as a subcategory of the free cocompletion, because it emphasizes that the cauchy completion isn’t just a limity notion but simultaneously a colimity notion.
It would be fine to add a note on this, of course. Ultimately it seems like another incarnation of the idea of points in the Cauchy completion being adjoint modules, which is there in the very sentence you are critiquing (and of course the simultaneous absolute limit/colimit idea is right there as well).
Right, I was just surprised to see the connection wasn’t noted because objects in the Isbell envelope are so close to being adjoint presheaves!
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