Isbell envelope

Isbell envelope

[Richard Garner]

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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Re: Isbell envelope

[Mike Stay]

Isbell called it the "double envelope".

http://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11541-0/S0002-9904-1966-11541-0.pdf

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
>

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Re: Isbell envelope

[Simon Willerton]

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
>


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Re: Isbell envelope

[Vaughan Pratt]

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

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Re: Isbell envelope

[Samuel Dean]

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

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Re: Isbell envelope

[Richard Garner]

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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Re: Isbell envelope

[Richard Garner]

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




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