Re: Bi-presheaves

Re: Bi-presheaves

[Ross Street]

Dear Andrew

This is what Lawvere told me about once, long ago.
I think he called it the Isbell envelope; that is what I've called it
ever since. It has nice properties. Lawvere explained that, applied
to finite dimensional vector spaces, it fully contains the category
of banach spaces and bounded linear maps. (I think I've got that
right; it's awhile since I checked it.)

Ross

On 06/03/2009, at 2:34 AM, Andrew Stacey wrote:

> Start with an essentially small category, T, and look at the
> category whose
> objects are triples (P,F,c) where: P is a contravariant functor T -
> > Set, F is
> a covariant functor T -> Set and c is a natural transformation from
> P x F to
> the Hom bi-functor.  Morphisms are pairs of natural transformations
> P_1 -> P_2
> and F_2 -> F_1 that intertwine the natural transformations c_1 and
> c_2.
>
> One could also enrich the whole structure.
>
> Has this cropped up anywhere before?  If so, what is it called and
> where can
> I learn about it?  If not, what shall I call it?
>
> If this is something standard then please pardon my ignorance.  I'm
> fairly new
> to _real_ category theory and am still just learning the basics.
>

Bi-presheaves

[Andrew Stacey]

Many thanks to all those who replied to my question.  The replies were
extremely useful.  I would like to sum up what I've been told to see if I've
understood it correctly.

1. What I described is known as the 'Isbell envelope', and has been known
about for quite some time - if my reading of Lawvere's emails is correct then
the idea dates back to his thesis, after which Isbell worked on the idea and
named it the 'double envelope', consequently Lawvere renamed it the 'Isbell
envelope'.  However, whilst it is known, it is classed as 'folklore' which
I interpret to mean 'everyone knows about it, but no-one has written anything
particular on it' so there's no easy reference to which I could direct someone
(particularly someone like myself not well-versed in the lore of category
theory).

2. It is also a special case of a 'profunctor', which also goes by the names
'distributor', 'commune', and 'bimodule'.  Rather, what I'm describing is
something that can be built out of particular profunctors and natural
transformations - the 'lax factorisation' of Jeff's email.

3. They are also related to Chu spaces - something else completely new to me!

A quick check on MathSciNet provides me with a reasonable reading list.  It
seems, at first glance, easier to find information on Chu spaces than
profunctors, and certainly easier to find it on profunctors than 'Isbell
envelopes'.

What I actually intend doing is fairly simple and I suspect that my audience
(if any) will be more from the non-categorical side of mathematics so I'm
looking more for "here's where this concept has occurred before" rather than
"here's where you can find all the theorems that we need".

For the record, these came up when looking at the various different categories
of "smooth space" that I've encountered.  I'm really a differential topologist
(I hope that that admission doesn't get me expelled from the list) but I like
applying the techniques of differential topology to things that aren't really
smooth manifolds.  This leads to the question of what they actually are and,
as I'm sure everyone here knows, there have been several candidates proposed.
In trying to compare them all, I've been looking for a unified way of
describing them to make it easier to see the differences.  That's where this
notion of two functors and a "composition" came up.  There's an extra part,
which I didn't say originally, in that there are often conditions that these
functors have to satisfy.  That is, I'm really looking at a full subcategories
of the Isbell envelope where some constraints are satisfied.

It probably doesn't class as much of a grand project but it is helping me
learn a little category theory so I hope you all approve of it in that regard!

So many thanks again to all those who replied, and if anyone has any further
words of wisdom to impart then I'm happy to learn more.

Andrew Stacey

Re: Bi-presheaves

[Vaughan Pratt]

Ross Street wrote:
> Given a category A, the Isbell envelope E(A) is the category whose
> objects
> are triplets (F_*, F^*, t) where F_* : A --> Set and F^* : A^{op} -->
> Set are functors
> and  t_{a,b} : F^*(a) x F_*(b) --> A(a,b) is a family natural in a
> and b in A. The
> morphisms are as you described. There is a "double Yoneda" A --> E(A)
> taking
> c to (A(c,-), A(-,c), composition).


(For "the morphisms are as you described" to work, unless I'm missing an
op somewhere, in the move from (P,F,c) to (F_*,F^*,t), P as the
component that "transforms forward" has to be F^*.)

I learned about the Isbell envelope from Ross just this January.  Ross
said he'd learned it from Bill Lawvere, who he said named it that after
learning it from Isbell.

In the course of some follow-up discussion, Rich Wood pointed out to us
the following translation into the language of profunctors of what both
Andrew and Ross wrote just now.  To avoid having to play favourites by
choosing between (P,F,c) and (F_*,F^*,t) I'll pick the neutral (A,X,r),
r: A x X --> ? I've been using myself for this stuff, where the early
letters A,B,... transform forwards and the late letters X,Y,...
transform backwards, just as with Chu spaces.

Organize the presheaves A and X as the profunctors A: C^op x 1 --> Set
and X: 1^op x C --> Set in the category Prof (Australian: Mod) of
profunctors ((bi)modules, distributors) and their natural
transformations.  In notation more suggestive of how they compose, these
become

    A: 1 -|-> C
    X: C -|-> 1
    r: AX --> 1_C

where AX is profunctor composition at 1, and 1_C: C^op x C --> Set is
the identity profunctor 1_C: C -|-> C in Prof, aka the homfunctor of C.

In general, composition of profunctors entails a left Kan extension, but
in this case the composition is taking place at 1, trivializing the
composite AX to A x X: C^op x C --> Set (Andrew's P x F).

The Isbell envelope being strikingly reminiscent of the Chu
construction, it is natural to ask how they're related, which I'll offer
an answer to in the rest of this message.

A little calculation shows that for the Isbell envelope of the
one-morphism category 1 (whose one object I'll denote * in the next
paragraph), E(1) is equivalent to Set x Set^op.  But this is also
Chu(Set,1) where 1 is now the singleton set.

What about Chu(Set,2)?  The trick here is to take C to be the two-object
category 1+1, augmented with two morphisms from what I'll call the
positive copy j of 1 to the negative copy \ell, call this C_2 as the
category naturally associated to the profunctor 2: 1 -|-> 1 defined by
2(*,*) = 2.  I'd been doing this with what I call Dsh(2) below, but Ross
pointed out to me (more generally) that Dsh(K) fully embeds in E(C_K) as
follows.  I'll write it assuming Rich's profunctor typing of A and X as
above so that the second argument of A and the first argument of X is
always the object * of 1.  (In that language Ross's t_{a,b} : F^*(a) x
F_*(b) --> A(a,b) becomes t_{a,b} : \int^c F^*(a,c) x F_*(c,b) -->
A(a,b) which exposes the left Kan nature of the product, albeit trivial
as noted above.)

The objects (A,X,r) of E(C_2) now consist of two pairs of sets: A =
(A(j,*),A(\ell,*)) and X = (X(*,j),X(*,\ell)).  Chu(Set,2) then emerges
from E(C) as the full subcategory of E(C_2) for which A(\ell,*) = X(*,j)
= {}.  In effect A and X behave as though they were single sets A(j,*)
and X(*,\ell), the effect we need for (A,X,r) to be an ordinary Chu space.

The generalization to Chu(Set,K) simply entails setting C_K(j,\ell) = K,
leaving C_K(\ell,j) empty as implicitly assumed above, with |C_K(j,j)| =
|C_K(\ell,\ell)| = 1, and with no other change to the description of the
full subcategory of E(C_K) constituting Chu(Set,K).

In a posting to this list on 9/7/02 with subject line "Presheaves etc.
in a uniform way" I described (even more obscurely than my usual
postings) a common generalization of presheaves and Chu spaces that can
be described far more clearly using the above language.  It amounts to a
generalization of the Isbell envelope (which as I said I only learned
about this January from Ross).  Quite independently of Rich and within a
day of him, Jeff Egger suggested to me exactly the following typing for
A and X in this generalization.

In place of the small C parametrizing the Isbell envelope E(C), take two
small categories J and L (explaining the j and \ell in the above).
Define a *disheaf* (at CT'04 I called these "communes") on a profunctor
K: L -|-> J to be a triple (A,X,r) where A: 1 -|-> J and X: L -|-> 1 are
profunctors and r: AX --> K: L -|-> J is a natural transformation.  By
analogy with Chu(Set,K), write Dsh(Set,K) for the category of disheaves
on the profunctor K, whose morphisms are just as for the Isbell
envelope, which in turn are just as for Chu spaces.

The Isbell envelope E(C) = Dsh(Set,1_C).  This is the sense in which Dsh
generalizes E.

(The "Set" parameter is a bad habit arising out of the notation Chu(V,k)
adopted around 1992, I forget by whom.  It is probably better to write
Chu(K) and Dsh(K) and let V be inferred from context as the ambient V in
which one is working, which I'll do now.)

As pointed out to me by Ross, E(C) generalizes Dsh(K) by taking C to be
what Ross called the *cone* C_K of K: J^op x L --> Set (L -|-> J),
namely J+L augmented with  morphisms from each j to each \ell picked out
of the set K(j,\ell) and composed as prescribed by the functoriality of
K.  (I learned this category representation of profunctors/bimodules the
hard way from Robert Seely at CT'04, who kindly said of my "bipartite
categories" during my talk, "That's a bimodule!")

The generalization is weaker (in some sense) than the other direction,
in that Dsh(K) is not E(C_K) itself but only the full subcategory
obtained by requiring A(j,*) = X(*,\ell) = 0 for all j in \ob(J) and
\ell in \ob(L).

I've recently found Dsh(K) very useful as a foundation for ontology (not
so much Aristotle as XML, which among other things supplies the Web
Ontology Language OWL with its presentation syntax), where it furnishes
an extensional conception of the notion of attribute (which the
Wikipedia article "Property (philosophy)" helpfully points out doesn't
exist), makes sense of C.I. Lewis's highly controversial notion of
qualia from the 1920's (philosophers today divide into qualiaphiles and
qualiaphobes, see the Wikipedia article on "Qualia") as the elements of
K(j,\ell), and offers a mechanism for interaction between the two
components in Cartesian dualism, which failed after a century of
unconvincing candidates for a meaningful such mechanism (Malebranche's
occasionalism, Leibniz's monads, etc.).

 From a more mathematical standpoint disheaves are even more general
than Chu spaces.  I can imagine a few raised eyebrows here---after my
decade of propaganda on Chu spaces as the ultimate universal framework,
the natural question would be, how could anything be more general than a
Chu space?

It depends on whether you're willing to settle for some full subcategory
of some Chu(Set,K) or want your category "on the nose" without the
hassle of having to come up with some ad hoc characterization of the
desired subcategory (cf. the distinction above between Dsh(K) as a
generalization of E(C) and vice versa where only the latter requires the
extra step of specifying a full subcategory).  Instead one can point to
a profunctor K and say that a K-flapdoodle is precisely an object of
Dsh(K).  For example there is a straightforwardly exhibited K such that
Dsh(K) consists essentially of the inelastic acylic graphs, those having
invariant path length.  No presheaf category consists of these, and
Isbell envelopes only contain them as a full subcategory (I don't know
if there's a K large enough that Chu(K) does the same job).

I have a paper on this in the works, focusing mainly on the ontology
application for now and therefore trying (only half successfully I fear)
to minimize the deployment of categorical weapons of math destruction in
order to make it more accessible to those likely to care at all about
ontology, let me know if you're interested.

Vaughan Pratt

Re: Bi-presheaves

[Ross Street]

Given a category A, the Isbell envelope E(A) is the category whose
objects
are triplets (F_*, F^*, t) where F_* : A --> Set and F^* : A^{op} -->
Set are functors
and  t_{a,b} : F^*(a) x F_*(b) --> A(a,b) is a family natural in a
and b in A. The
morphisms are as you described. There is a "double Yoneda" A --> E(A)
taking
c to (A(c,-), A(-,c), composition).

==Ross

On 06/03/2009, at 7:19 PM, Andrew Stacey wrote:

> When you say that you call it 'the Isbell envelope', what do you
> mean?  Is the
> category the 'Isbell envelope of/on the original category' or are
> the objects
> 'Isbell envelopes' and we have the category of Isbell envelopes (in/
> on/of the
> original category)?

Re: Bi-presheaves

[Bill Lawvere]

PS

The Isbell envelope arises from the Isbell conjugate pair
which is the adjoint pair of functors connecting set^(T^op)
and (set^T)^op. It is thus a special case of the general
construction in my thesis (TAC Reprints) of the total
category with two descriptions which objectifies the
notion of adjointness, in order to free it from dependence
on enrichments in small sets. (Of course the example
depends on enrichments in small sets.)


************************************************************
F. William Lawvere, Professor emeritus
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************



On Thu, 5 Mar 2009, Andrew Stacey wrote:

> Dear Categorists,
>
> I'm interested in looking at the following type of thing:
>
> Start with an essentially small category, T, and look at the category whose
> objects are triples (P,F,c) where: P is a contravariant functor T -> Set, F is
> a covariant functor T -> Set and c is a natural transformation from P x F to
> the Hom bi-functor.  Morphisms are pairs of natural transformations P_1 -> P_2
> and F_2 -> F_1 that intertwine the natural transformations c_1 and c_2.
>
> One could also enrich the whole structure.
>
> Has this cropped up anywhere before?  If so, what is it called and where can
> I learn about it?  If not, what shall I call it?
>
> If this is something standard then please pardon my ignorance.  I'm fairly new
> to _real_ category theory and am still just learning the basics.
>
> Thanks,
>
> Andrew Stacey
>
>
>
>

Re: Bi-presheaves

[Bill Lawvere]

Dear Andrew Stacey,

    When John Isbell introduced this construction in the early
1960's, he called it the 'double envelope', so I often call
it the Isbell envelope.
    You just re-discovered it!

Bill Lawvere



************************************************************
F. William Lawvere, Professor emeritus
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************



On Thu, 5 Mar 2009, Andrew Stacey wrote:

> Dear Categorists,
>
> I'm interested in looking at the following type of thing:
>
> Start with an essentially small category, T, and look at the category whose
> objects are triples (P,F,c) where: P is a contravariant functor T -> Set, F is
> a covariant functor T -> Set and c is a natural transformation from P x F to
> the Hom bi-functor.  Morphisms are pairs of natural transformations P_1 -> P_2
> and F_2 -> F_1 that intertwine the natural transformations c_1 and c_2.
>
> One could also enrich the whole structure.
>
> Has this cropped up anywhere before?  If so, what is it called and where can
> I learn about it?  If not, what shall I call it?
>
> If this is something standard then please pardon my ignorance.  I'm fairly new
> to _real_ category theory and am still just learning the basics.
>
> Thanks,
>
> Andrew Stacey
>
>
>
>

Re: Bi-presheaves

[Andrew Stacey]

Ross,

Thanks for the information.  I'm not surprised to hear the name 'Isbell'
connected with this (nor Lawvere) as there are hints of this idea in 'Taking
Categories Seriously' where Lawvere talks about 'Isbell conjugation' (though
in Isbell conjugation one considers the categories of covariant and
contravariant functors as separate).  Looking up 'Isbell envelope' on
MathSciNet came up with nothing whilst 'Isbell conjugation' only came up with
two papers (one reviewed by you, I think!).  One is about 'total categories',
though I've yet to look at it to see if it is relevant.

When you say that you call it 'the Isbell envelope', what do you mean?  Is the
category the 'Isbell envelope of/on the original category' or are the objects
'Isbell envelopes' and we have the category of Isbell envelopes (in/on/of the
original category)?

Thanks for the quick reply,

Andrew

On Fri, Mar 06, 2009 at 04:13:11PM +1100, Ross Street wrote:
> Dear Andrew
>
> This is what Lawvere told me about once, long ago.
> I think he called it the Isbell envelope; that is what I've called it
> ever since. It has nice properties. Lawvere explained that, applied
> to finite dimensional vector spaces, it fully contains the category
> of banach spaces and bounded linear maps. (I think I've got that
> right; it's awhile since I checked it.)
>
> Ross
>
> On 06/03/2009, at 2:34 AM, Andrew Stacey wrote:
>
>> Start with an essentially small category, T, and look at the category
>> whose
>> objects are triples (P,F,c) where: P is a contravariant functor T ->
>> Set, F is
>> a covariant functor T -> Set and c is a natural transformation from P x
>> F to
>> the Hom bi-functor.  Morphisms are pairs of natural transformations
>> P_1 -> P_2
>> and F_2 -> F_1 that intertwine the natural transformations c_1 and
>> c_2.
>>
>> One could also enrich the whole structure.
>>
>> Has this cropped up anywhere before?  If so, what is it called and
>> where can
>> I learn about it?  If not, what shall I call it?
>>
>> If this is something standard then please pardon my ignorance.  I'm
>> fairly new
>> to _real_ category theory and am still just learning the basics.
>>

Bi-presheaves

[Andrew Stacey]

Dear Categorists,

I'm interested in looking at the following type of thing:

Start with an essentially small category, T, and look at the category whose
objects are triples (P,F,c) where: P is a contravariant functor T -> Set, F is
a covariant functor T -> Set and c is a natural transformation from P x F to
the Hom bi-functor.  Morphisms are pairs of natural transformations P_1 -> P_2
and F_2 -> F_1 that intertwine the natural transformations c_1 and c_2.

One could also enrich the whole structure.

Has this cropped up anywhere before?  If so, what is it called and where can
I learn about it?  If not, what shall I call it?

If this is something standard then please pardon my ignorance.  I'm fairly new
to _real_ category theory and am still just learning the basics.

Thanks,

Andrew Stacey