Re: dualities

Re: dualities

[Vaughan Pratt]

John Baez wrote:
> [...]
> So, can we find equally nice examples [of representable dualities] where C and D are instead
> 2-categories?  In particular, can we find examples where C and D
> are 2-categorical generalizations of the 1-categorical examples
> we already know?
>
> In particular, he suggested taking the example where C is the
> category of finite distributive lattices and finding an analogous
> example where C is the 2-category of (maybe finite, in some sense?)
> distributive categories.

Enrico Vitale just sent me the answer for that one:  C = the 2-category
of idempotent-closed categories, D = the 2-category of presheaf
categories.  This categorifies C = Pos, D = StoneDLat by passing from 2
to Set as the enriching autonomous category (so in that sense one could
say we were in 3-CAT all along, though presumably only trivially so by
virtue of only having identity modifications when V = 2, I think).

Although I'd heard the phrase "Morita equivalence" many times over the
years, it meant nothing to me until recently when Bill Lawvere was
talking about graphs as presheaves on the monoid consisting of the three
monotone functions on the ordinal 2 and I finally woke up to the
connection between splitting the two idempotents and ME (the
equivalence, not the condition).  The idempotent closure of that monoid,
meaning the result of splitting the idempotents, is just the initial
segment of Delta of length 2, aka the ordinals 1 and 2 and their
monotone functions.  The impact on the models, here graphs, is that
splitting the idempotents results in giving the self-loops that were
playing the role of vertices their own datatype V, as coded by the
ordinal 1.  This new category of graphs is not the old one as its
objects now have vertices in their own right, but it is equivalent to
the old one.  {2} and {1,2}, each made a category with respectively 3
and 7 monotone functions, are Morita equivalent: they have equivalent
idempotent closures, and homming into Set maps them to equivalent
categories, the iff that makes Morita equivalence important.

ME is the kernel of idempotent closure, which is a categorification,
with Set in place of 2, of the functor Ord --> Pos (Ord the category of
preordered sets, Pos of posets) that collapses the cliques.  The reason
there is no representable duality between Ord and a suitable cousin of
StoneDLat (FinOrd and FinDLat for the Stonaphobes) is that preorders are
equivalent to posets and the Yoneda embedding taking elements of P to
primes in 2^P, while fully faithful, is only good up to equivalence.
(The homfunctor being transposed here is the order <= : P\op x P --> 2.)

The categorification of this, meaning in this case not the passage from
2-CAT to 3-CAT but from enrichment in 2 to enrichment in Set, still has
to deal with equivalence in the same way (though here it goes with the
territory and so is less noticeable than back down at Ord vs. Pos where
we tend to think isomorphism rather than equivalence).  But Hom: C\op x
C --> Set is not itself an equivalence but only a "retract that retracts
retracts", the essence of Morita equivalence (a dual of Freyd's "trivial
for a trivial reason"?).  In order to take the "log to the base Set" we
can't really "retract all the retracts" because we may need to keep some
of them around but then which ones (like picking a dense subset of a
continuum: which subset?).  We can however put them all in, which is to
say, split all the idempotents, so we do that in order to get a normal form.

The rest of this duality is then the triviality that the internal hom of
CAT is contravariant in its first argument.  Morita equivalence is the
only thing to be worried about.

Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category
Theory", namely that Cauchy completion (Kelly's name for the enriched
counterpart of idempotent closure) permits taking logs to any autonomous
base V, then produces a proper class of dualities, one for every
autonomous V.  In particular we can recover Pos\op ~ StoneDLat by taking
V = 2.  (Pos and Ord, preordered sets, while not equivalent any more
than CAT and its subcategory of idempotent-closed categories are
equivalent, have equivalent objects which is all we need ask of a
duality.)  There are two "good" 3-object V's, the non-Heyting one of
which enriches the "prossets" that Haim Gaifman and I wrote about in
LICS'87, so these have their dual objects in the same way, by homming
into 3, a construct I talked about incomprehensibly at the Newton
Institute meeting on geometry in computation some years ago, not
recognizing that it was a duality.  Metric spaces, another duality
there.  And so on.

But then every such duality has its subdualities, for example Set\op ~
CABA as a subduality of Pos\op ~ StoneDLat, so a great many more
dualities there.

Enrico also mentioned the Gabriel-Ulmer duality for locally finitely
presentable categories, and the Adamek-Lawvere-Rosicky duality for
varieties.  Are these in addition to the above or can they be recovered
from them?  Likewise for the duality Peter Johnstone mentioned?

Vaughan Pratt

Re: dualities

[Vaughan Pratt]

Sorry, I should have said "Lawvere's name for..." in

> Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category
> Theory", namely that Cauchy completion (Kelly's name for the enriched
> counterpart of idempotent closure)

Vaughan

Re: dualities

[John Baez]

Hi -

> First, let me say I have avoided contributing to this thread because I
> don't understand what Vaughan is asking.

He asked if dualities categorify.  I guess he meant something
like this:

There are lots of interesting examples of a pair of categories C,D
together with an object c in C and an object d in D such that

hom(-,c): C -> D

and

hom(-,d): D -> C

are part of an equivalence of categories.  In the nicest examples,
c and d are in some sense the same mathematical entity regarded
as living in two different categories - a "schizophrenic object",
in the words of Harold Simmons.

So, can we find equally nice examples where C and D are instead
2-categories?  In particular, can we find examples where C and D
are 2-categorical generalizations of the 1-categorical examples
we already know?

In particular, he suggested taking the example where C is the
category of finite distributive lattices and finding an analogous
example where C is the 2-category of (maybe finite, in some sense?)
distributive categories.

For more on "schizophrenic objects", Peter Johnstone's review
of Clark and Davies' "Natural dualities for the working algebraist"
makes good reading:

http://north.ecc.edu/alsani/ct99-00(8-12)/msg00116.html

Re: dualities

[Vaughan Pratt]

Michael Barr wrote:
> First, let me say I have avoided contributing to this thread because I
> don't understand what Vaughan is asking.  He knows, as well as anyone,
> since he put them on the map, about Chu categories.  He knows about
> *-autonomous categories as well.  So what is the question, really?

Duality is of necessity between categories, and involves associating an
object (say an algebra or space) of one category with its dual in
another, or in the same category in the self-dual case.  Downstairs,
i.e. in 2-CAT.  By "categorifying duality" I meant a duality of
2-categories in which one associates an object (this time a category
rather than an algebra) of one 2-category with its dual in another, with
the functors being reversed (op) as opposed to the natural
transformations (co).  Upstairs, i.e. in 3-CAT.

Regarding Chu, I was going to respond that the Chu construction works
downstairs with categories (from my usual V=Set perspective), or at most
V-categories, categories enriched in V, as objects of the 2-category
V-CAT.  However if the enriched Chu construction can be organized to
allow V to be a 2-category, with Chu(V,k) then being a 3-category, maybe
Mike is on to a promising approach (though it's not clear that's what he
actually meant).  It's an aspect of Chu spaces I know next to nothing
about however.

My first guess would be that it (moving Chu up into 3-CAT) ought to work
fine, with the caveat that the simple notion of Stone topology as a
totally disconnected compact Hausdorff topology would turn into the
proverbial thousand flowers---there's far more room for such stuff in
3-CAT than 2-CAT.  (Actually there's also a lot of unexplored such
territory even just in ordinary Chu(Set,3).)  Along those lines, Peter
Johnstone's mention of quasi-injective toposes dual to continuous
categories in his 1982 paper with Joyal is surely just scratching the
surface of the possible permutations and combinations up there in 3-CAT.

   Peter's
> example is one of the very few *-autonomous categories I cannot relate to
> Chu.  Complete (say inf) semi-lattices is another.

Oh, but complete inf semilattices are one of the most elegant
self-dualities of chupology.  They embed in Chu(Set,2) as those
biextensional Chu spaces (biextensional = no repeated rows or columns),
of any cardinality, such that the set of rows is closed under arbitrary
AND (think of them as bit vectors) and likewise for the set of columns.
  No other conditions.  Using OR instead of AND for both rows and
columns gives sup semilattices.  But that was in my previous post, where
I also mentioned that XOR in place of OR, at least for finite Chu
spaces, embeds FinVct_{GF(2)}.  In either case the symmetry of the
conditions makes the self-duality immediate.  All that is in the 1999
Coimbra notes I cited.

Vaughan

Re: dualities

[Ronnie Brown]

Peter Freyd writes on Pontrjagin duality. I would like to mention the
generalisation given in

 (with P.J. HIGGINS and S.A. MORRIS), ``Countable products of lines  and
circles: their closed subgroups, quotients and duality properties'',  {\em
Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32.

One point made is that a duality is not necessarily inherited by closed
subgroups and Hausdorff quotients. If it is, it is called a strong duality.
Then strong duality is inherited by closed subgroups and Hausdorff
quotients!

I have taught the classification of closed subgroups of R^n in an analysis
course. It is a nice result, and the sums you can set use duality in a nice
way - treatment borrowed from Bourbaki.

Ronnie Brown

> Next is Pontryagin's: the category of locally compact groups.  The
> original Pontryagin duality easily generalizes: the category of
> locally compact modules over a given commutative ring is self-dual.
> (In the non-commutative case one also obtains a duality but not a
> self-duality -- unless, of course, the ring is self-dual.) A corollary
> is that the category of discrete left  R-modules is dual to the
> category of compact right  R-modules

Re: dualities

[Michael Barr]

First, let me say I have avoided contributing to this thread because I
don't understand what Vaughan is asking.  He knows, as well as anyone,
since he put them on the map, about Chu categories.  He knows about
*-autonomous categories as well.  So what is the question, really?

The simplest answer to Mamuka's question is the duality between vector
spaces (over any field, including the 2 element field) and linearly
compact vector spaces.  In the case of a finite field, linear compactness
is the same as the ordinary topological kind.  One proof of this fact is
that the category of finite dimensional spaces is self-dual and if two
categories are dual, the inductive completion of one is dual to the
projective completion of the other.  For finite dimensional vector spaces,
the inductive completion is vector spaces and the projective completion is
linearly compact ones.  Another example is the obvious duality between
finite sets and finite boolean algebras that gives Stone duality on one
hand and the duality between Set and CABA on the other, depending which
one you complete which way.

Most examples I am aware of of self-dualities are Chu categories (or chu
categories).  And if V_k is the category of k-vector spaces, then
Chu(V_k,k) (an object is a pair of spaces and a bilinear pairing into k)
is *-autonomous, as is chu(V_k,k) of separated extensional pairs.  Peter's
example is one of the very few *-autonomous categories I cannot relate to
Chu.  Complete (say inf) semi-lattices is another.

Michael

Re: dualities

[K C H Mackenzie]

Quoting Peter Freyd <pjf@saul.cis.upenn.edu>:

> On the subject of favorite dualities:
>
> Surely the most important are the self-dualities and the most
> important of these (so important we stop noticing it as we age) is the
> category of finite-dimensional vector spaces over a given field.

Something on this has been done.

Duality for vector bundle objects in the category of Lie groupoids
was done by Jean Pradines in 1988, and is part of the fundamental
work on symplectic groupoids. The cotangent bundle $T^*G$ of any
Lie groupoid $G$ has a groupoid structure with base the dual of
$AG$, the Lie algebroid of $G$, and Pradines' construction
realizes this as the dual of the tangent prolongation $TG$ of $G$.

A double vector bundle (in the sense of Ehresmann) is a particular
instance of a vector bundle in the category of Lie groupoids.
Pradines' duality can be applied to such a structure in two ways,
and these do not commute. If $D$ is a double vector bundle over
vector bundles $A$ and $B$, each of which is a vector bundle over
a manifold $M$, then $D$ can be dualized over $A$ and over $B$.
These dualization operations generate the dihedral group of order 6.
See `Duality and triple structures', pp455--481 of `The breadth of
symplectic and Poisson geometry', (Weinstein Festschrift), Progr.
Math., Birkh\"auser Boston, 2005.

Alfonso Gracia-Saz and I are preparing a paper on the duality
of $n$-fold vector bundles.

Details and references for the double case can be found in my
`General Theory of Lie groupoids and Lie algebroids', Cambridge,
2005, Chapter 9.

Whether categlorification would add anything to this I do not know.

Kirill Mackenzie

Re: dualities

[Vaughan Pratt]

In addition to Peter's nice collection of self-dualities there are the
Kleisli and Eilenberg-Moore categories of "the" covariant power-set monad
(there are really two such monads but either will do), respectively Rel
and complete semilattices, both self-dual.  One that Mike Barr introduced
me to is the subcategory of Rel whose morphisms are the partial
injections, those binary relations such that if (x,y) and (x,z) are both
present then y = z and likewise for their converses.  My personal
favorites are finite chains with bottom (showing that \Delta, as the base
category of the presheaf category of simplicial sets, comes very close to
being self-dual; had \Delta itself been self-dual, Set^{\Delta\op} and
Set^\Delta would have been the same thing), and semilattices with a top
and all nonempty sups (my candidate for a self-dual system of event/state
structures before I replaced it with Chu spaces).

One should also mention the topological vector spaces in Barr's book on
*-autonomous categories, whose self-duality does for the
finite-dimensional vector spaces mentioned by Peter what Pontryagin
duality does for finite abelian groups.

A feature of Chu spaces I particularly like is that each of the above,
as well as finite-dimensional vector spaces and finite abelian groups,
can be described as that full subcategory of Chu(Set,K) (K = 2 except
for vector spaces and abelian groups) consisting of biextensional Chu
spaces whose rows and columns satisfy the same closure conditions.  For
example the category of finite-dimensional vector spaces over GF(2) (a
sneaky way to stick to Chu(Set,2)) embeds in Chu(Set,2) as precisely
those finite biextensional Chu spaces whose rows and columns, viewed as
bit vectors in the sense a machine-language programmer understands the
concept, are both closed under bitwise XOR.  This example is given as an
exercise at the end of Chapter 2 of
http://boole.stanford.edu/pub/coimbra.pdf, my notes for the July 1999
Coimbra School cotaught with John Baez and Cristina Pedicchio.
Proposition 2.2 in the same chapter obtains complete semilattices (of
any cardinality) as those Chu spaces whose rows and columns are closed
under bitwise OR, with the self-duality of CSLat as the immediate
Corollary 2.3.  This shows that the self-dualities CSLat and
Vct_{GF(2)}, at least for finite objects, arise identically except for
how they combine their bit-vectors, namely with respectively OR and XOR.

Vaughan


Peter Freyd wrote:
> On the subject of favorite dualities:
>
> Surely the most important are the self-dualities and the most
> important of these (so important we stop noticing it as we age) is the
> category of finite-dimensional vector spaces over a given field.
>
> Next is Pontryagin's: the category of locally compact groups.  The
> original Pontryagin duality easily generalizes: the category of
> locally compact modules over a given commutative ring is self-dual.
> (In the non-commutative case one also obtains a duality but not a
> self-duality -- unless, of course, the ring is self-dual.) A corollary
> is that the category of discrete left  R-modules is dual to the
> category of compact right  R-modules. (For 50 years I've been trying to
> turn this into an exercise in abelian categories. There's a nice
> reduction down to the proposition that  R/Z  is a cogenerator for the
> category of compact abelian groups, but that fact seems to require
> some non-trivial functional analysis.) Strange that two of the most
> important "dualities" are both Pontryagin's. The other is in algebraic
> topology theory.
>
> Then, of course there's my present favorite: the category of finitely
> presented group-valued functors from the category of finitely
> presented modules over a commutative ring.
>
>

Re: dualities

[Mamuka Jibladze]

> Next is Pontryagin's: the category of locally compact groups.  The
> original Pontryagin duality easily generalizes: the category of
> locally compact modules over a given commutative ring is self-dual.
> (In the non-commutative case one also obtains a duality but not a
> self-duality -- unless, of course, the ring is self-dual.) A corollary
> is that the category of discrete left  R-modules is dual to the
> category of compact right  R-modules. (For 50 years I've been trying to
> turn this into an exercise in abelian categories. There's a nice
> reduction down to the proposition that  R/Z  is a cogenerator for the
> category of compact abelian groups, but that fact seems to require
> some non-trivial functional analysis.)

This reminded me of a long-standing torture: does anybody know an elementary
proof at least of the particular case when the base ring - thus the dualizer
too - has only two elements? (Demanded by Guram Bezhanishvili; I agreed to
try thinking on this one as it seemed somehow close to Boolean algebras,
but...)

> Then, of course there's my present favorite: the category of finitely
> presented group-valued functors from the category of finitely
> presented modules over a commutative ring.

Yes, yes?

dualities

[Peter Freyd]

On the subject of favorite dualities:

Surely the most important are the self-dualities and the most
important of these (so important we stop noticing it as we age) is the
category of finite-dimensional vector spaces over a given field.

Next is Pontryagin's: the category of locally compact groups.  The
original Pontryagin duality easily generalizes: the category of
locally compact modules over a given commutative ring is self-dual.
(In the non-commutative case one also obtains a duality but not a
self-duality -- unless, of course, the ring is self-dual.) A corollary
is that the category of discrete left  R-modules is dual to the
category of compact right  R-modules. (For 50 years I've been trying to
turn this into an exercise in abelian categories. There's a nice
reduction down to the proposition that  R/Z  is a cogenerator for the
category of compact abelian groups, but that fact seems to require
some non-trivial functional analysis.) Strange that two of the most
important "dualities" are both Pontryagin's. The other is in algebraic
topology theory.

Then, of course there's my present favorite: the category of finitely
presented group-valued functors from the category of finitely
presented modules over a commutative ring.