Dear all, Thanks for the useful responses. My reading of the---admittedly extensive---Isbell corpus was clearly not (left or right) adequate. To summarise, the original published references where things are done, by Isbell, with Isbell envelopes (under the name "couple categories") appear to be: [1] John R. Isbell, Structure of categories, Bulletin of the American Mathematical Society 72 (1966), 619– 655. [2] John R. Isbell, Normal completions of categories, Reports of the Midwest Category Seminar, vol. 47, Springer, 1967, 110–155. while Vaughan Pratt also draws attention to their appearance in: [3] Vaughan Pratt, Communes via Yoneda, from an elementary perspective, Fundamenta Informaticae 103 (2010), 203–218. There are, of course, related constructions in linear logic, and in fact the Isbell envelope turns up, after a fashion, in the Chu appendix to Mike Barr's "Star-autonomous categories" (there called, again, the "double envelope"). Richard On Mon, May 12, 2014, at 02:09 PM, Richard Garner wrote:
Dear categorists,
One of the more folklorish constructions in category theory is that of the Isbell envelope. The folklorishness, in this case, seems to be so severe that I cannot find mention made of it in any published article at all (though there are several to the related notion of Isbell conjugacy). I am writing, therefore, in the hope that this is only due to my own poor knowledge of the literature, and that some other reader of this list may be able to put me to rights.
Richard
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