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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Most gratifying! That is just what I was looking for -- thanks! Richard On Mon, May 12, 2014, at 10:16 PM, Michal Przybylek wrote: Dear Richard, What you call "Isbell envelope'', was called by John Isbell "couple category". The construction is defined in the following paper: J. R. Isbell, "Normal completions of categories" Best, Michal On Mon, May 12, 2014 at 6:09 AM, Richard Garner <[1]richard.garner@mq.edu.au> 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Isbell called it the "double envelope". http://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11541-0/S0002-99... On Sun, May 11, 2014 at 10:09 PM, Richard Garner <richard.garner@mq.edu.au> 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Richard, Andrew Stacey asked a related question on the categories list in 2009: http://article.gmane.org/gmane.science.mathematics.categories/124 His summary of responses is here. http://article.gmane.org/gmane.science.mathematics.categories/140 All of the posts can be found at http://search.gmane.org/?query=bi-presheaves&group=gmane.science.mathematics.categories&sort=revdate Cheers, Simon. On 12/05/14 05:09, 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Richard, Two comparison are made between Isbell envelopes and communes in the second last paper listed in my CV at http://boole.stanford.edu/vita.pdf, Pratt, V.R. ?Communes via Yoneda, from an Elementary Perspective?, Fundamenta Informaticae, Vol. 103 Issue 1-4, 203-218, DOI 10.3233/FI-2010-325, IOS Press Amsterdam, 2010. also at http://boole.stanford.edu/pub/CommunesFundInf2010.pdf The two comparisons are on p. 214: "Communes are a generalization of a notion due to Isbell and called by Lawvere the \defn{Isbell envelope} $E(\C)$ of a category $\C$. $E(\C)$ is the special case of a category of communes where the base has the form of a homfunctor $\C\op\times\C\to\Set$, equivalently the identity profunctor $1_C:\C\nrightarrow\C$. An object $D$ of the Isbell envelope can be understood as a commune whose elements are morphisms from objects of $\C$ to $D$ and whose states are morphisms from $D$ to objects of $\C$. Conversely the commune category $\widehat\K$ can be obtained from $E(\check K)$ as the full subcategory of $E(\K)$ consisting of those objects having no elements from $\L$ and no states to $\J$." The acknowledgments section on p.218 gives some background: "Although Bill Lawvere had pointed me at Isbell's papers in connection with left and right adequacy at Category Theory 2004 in Vancouver where I first spoke about communes (during which I was introduced to bimodules by Robert Seely), I first learned about Lawvere's term ``Isbell envelope'' $E(\C)$ for that concept much more recently from Ross Street, which Ross defined for me in terms of left Kan extensions." My CT2011 talk in Vancouver emphasized examples of communes, which is still on my list to write up for publication (currently about halfway down the list). Vaughan On 5/11/2014 9:09 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Simon Willerton has written about the Isbell completion recently, see http://www.tac.mta.ca/tac/volumes/28/22/28-22abs.html. Sam On May 12, 2014 1:10 PM, "Richard Garner" <richard.garner@mq.edu.au> 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Mike Stay -
Richard Garner -
Samuel Dean -
Simon Willerton -
Vaughan Pratt