Re: Isbell envelope

[Vaughan Pratt <pratt@cs.stanford.edu>]

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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