Re: Query

Re: Query

[categories]

Date: Sun, 29 Jun 1997 11:58:50 -0700 (PDT)
From: Dr. Farogh Dovlatshahi <frank@primenet.com>

 This is not an answer to your question. But your posting prompts me to
ask a question in the semantics underlying applications of cateogories.

 What is the underlying metaphysics of perception (event, processes,
relations ?). How do you capture 'time'. It seems to me the dynamism is in
the arrows -- and there perhaps is where time comes in. 

 It has always seemed to me that the applications of Cat. The. (for
example in the study of Dynamical Systems) has served as a language to
talk about mathematical, computatianal and procedural complexities ONCE
THESE ARE ALREADY IN PALCE. Cat. is not to replace them as an alternative.

Dr. Farogh Dovlatshahi

> 
> 
> 
> Dear Colleagues,
> 
> I am applying tools of category theory to research in artificial
> perception and cognition. A basic category is proposed where every
> perception is an object and morphisms capture the flow between
> perceptions. Natural transformations capture paths to more cognitive
> perceptions.
> An Anonymous referee remarked that my category is `very closely related
> to comma categories'.
> 
> Can anybody refer me to written material that introduces comma categories? 
> Also, please let me know if you are aware of other research that applies
> categorical tools to research in artificial perception and cognition.
> 
> Thanks
> 
> Zippie
> 
> e-mail zippie@actcom.co.il
> 
> Dr. Zippora Arzi-Gonczarowski
> Typographics, Ltd.
> 46 Hehalutz St.
> Jerusalem 96222, Israel
> 
> 

Re: query

[Tom Leinster]

Dear Jim,

On Wed, 24 Jun 2009, jim stasheff wrote:
> Mac Lane coherence can be deduced from the simple connectivity of the
> associahedron

Surely that's not true, assuming that by "Mac Lane coherence" you mean Mac
Lane's coherence theorem for monoidal categories.  The associahedra (and
in particular the pentagon) say nothing about the unit coherence
isomorphisms,

X \otimes I ----> X <---- I \otimes X.

To make it true, surely you need to weaken Mac Lane's theorem to a
statement about "semigroupal" categories, i.e. monoidal categories without
unit...?

Best wishes,
Tom



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RE: query

[Noson S. Yanofsky]

> -----Original Message-----
> From: categories@mta.ca [mailto:categories@mta.ca] On Behalf Of jim
> stasheff
> Sent: Wednesday, June 24, 2009 12:18 PM
> To: Categories list
> Subject: categories: query
>
> Mac Lane coherence can be deduced from the simple connectivity of the
> associahedron
> Is it written that way anywhere?
>
>  jim

Hi,
Yes. My thesis. "Obstructions to Coherence: Natural Noncoherent
Associativity and Tensor Functors", City University of New York, 1996.

The part about the associahedron was published in
Obstructions to Coherence: Noncoherent Associativity
The Journal of Pure & Applied Algebra. 147 no. 2, Pgs 175 - 213. (2000).
or http://xxx.lanl.gov/abs/math.QA/9804106

The second part about the tensor functors was never published.

I look at the fundamental group of the associahedra thought of
as groupoids (called the "Catalan groupoids"). They are all trivial.
But then I ask, what if the pentagons do not commute? The fundamental group
of the Mac Lane non-commuting pentagon is Z. And I get generators and
relations for all the higher non-commuting associahedra. They are not
free groups from n=7 on.

I do a similar thing for non-coherent tensor functors (monoidal functors).

All the best,
Noson




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query

[jim stasheff]

Mac Lane coherence can be deduced from the simple connectivity of the
associahedron
Is it written that way anywhere?

 jim



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Query

[Johannes Huebschmann]

Dear All

Given a Lie group G and a G-representation V, according
to Hochschild-Mostow, the
ordinary Eilenberg-Mac Lane construction, suitably interpreted
in terms of smooth functions, yields a differentiably injective
resolution of V over G. More precisely, the Eilenberg-Mac Lane
construction (dual bar construction) arises here as the differentiable
cosimplicial G-module having, in degree p,
the space of smooth V-valued maps on a product of p+1 copies
of G, with the ordinary coface and codegeneracy operators.

Suppose now that G is connected and finite-dimensional and let
K be a maximal compact subgroup. Hochschild-Mostow have also shown that the
V-valued differential forms on G/K then yield an injective
resolution of V over G as well. This kind of construction actually
goes back to van Est.

The standard procedure yields comparison maps between the two resolutions.
In degree zero the comparison is, of course, achieved by the obvious map
from C^{\infty}(G/K,V) to C^{\infty}(G,V) induced by the projection
from G to G/K and by the obvious map
from C^{\infty}(G,V) to C^{\infty}(G/K,V) induced by integration over K.

Does anybody know whether, in the literature, the constituents of
a comparison map in higher degrees have been spelled out explicitly
somewhere?

Many thanks in advance

Best regards

Johannes



HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

TEL. (33) 3 20 43 41 97
      (33) 3 20 43 42 33 (secretariat)
      (33) 3 20 43 48 50 (secretariat)
Fax  (33) 3 20 43 43 02

e-mail Johannes.Huebschmann@math.univ-lille1.fr

query

[jim stasheff]

Apparently Graeme's approach to infinite sloop spaces or rather
to one fold loop spaces
has be further abstracted to produce what is known as a `Segal category'

at a quick glance, it seems to me these are related to Fukaya's A_\infty
cats
as my approach to \Omega X is related to Graeme's

anyone seen this worked out or even commented on?

jim

Query

[Oswald Wyler]

For every set Z, there is a self-adjoint contravariant functor Q=ENS(--,Z),
with unit/counit h:Id-->Q^op Q given by (h_X)(x)(f)=f(x).  Let Q-alg denote
the category of algebras for the monad induced by this self-adjunction.
If Z is not empty or a singleton, then the comparison functor ENS^op-->Q-alg
is an equivalence by results of M. Sobral.  If Z has two members, then Q-alg
is isomorphic to CaBool, the category of complete atomic Boolean algebras.
What is known about Q-alg if Z has more than two members (beyond the fact
that Q-alg and CaBool are equivalent)?

Re: query

[Michael Batanin]

This is a summary of my correspondence with J.Stasheff.


James Stasheff wrote:

> As monoids can be described as categories with one object,
> one can consider \Aoo structures on categories with a notion of
> homotopy, e.g. topological categories or differential graded categories.
> To be more precise, the set of objects and the set of morphisms
> carry a notion of homotopy. As usual, one deals with composable
> morphisms and then weakens the axiom of associativity up to homotopy
> in the strong sense in order to
> define \Aoo - categories.  This was first done by Smirnov by 1987
> \cite{smirnov:baku} to handle functorial homology operations and
> their dependence on choices (cf. indeterminacy). More recently, Fukaya
> \cite{fukaya:1} reinvented \Aoo -categories with remarkable applications to
> Morse theory and Floer homology.
> Inspired by this work, Nest and Tsygan have proposed an \Aoo -category
> with automorphisms of an associative algebra as objects and for
> the space of morphisms, a twisted version of the Hochschild complex
> of the corresponding endomorphism algebras.

Michael Batanin:

One can generalize "ordinary" category theory in the different ways. One
can consider internal category theory, enriched category theory. We can
also consider a category as a special sort of simplicial set. All this 
points of view have their own A_{\infty}-analogues.

I realize, that the approach of Smirnov, Fukaya and others is a
generalization of "internal" category theory. In my paper "Monoidal
globular categories as a natural environment ..."(Adv.Math. 136, 39-103
(1998)) I also consider a Cat-internal version of
A_{\infty}-\omega-category (so it involves a weak form of interchange
law)that I call monoidal globular category. A surprising coherence
theorem sais that a general monoidal globular category is equivalent to
a strict one (the internal category structure on objects aloows to
strictify interchange low). 

In another my paper "Homotopy coherent category theory and
A_{\infty}-structures in monoidal categories" (JPAA 123(1998),67-103) I
defined an enriched version of A_{\infty}-category. So 
we have a honnest set of objects but morphisms are objects of a monoidal
simplicial categories with a Quillen model structure. I also can define
what A_{\infty} functor is and prove an appropriate coherence and
homotopy invariance theorems. 
Another nice theorem sais that A_{\infty}-categories and their
A_{-infty}-functors form an A_{infty}-category in a natural way. 
  
The simpliocial point of view on A_{\infty}-categories goes back to 
Boardman and Vogt book. The corresponding notion is a simplicial set
satisfying some weak Kan conditions. This approach was extensively used
by T.Porter and J.-M.Cordier (see T.Porter's answer on Jim's query). 
 


James Stasheff:
> Since in a category we are concerned with $n$-tuples of morphisms
> only when they are composable, it is appropriate to similarly relax
> the composition operations for in defining an operad.  the result is
> known as a partial operad and appears in two different contexts:
> in the mathematical physics of vertex operator algebras (VOAs)
> \cite{yizhi} and

Mivhael Batanin:
 In my work "Globular monoidal
categories ..." I introduced the n-dimensional operads over trees. A
1-dimensional operad in this sense is not exatly the same as usual
non-symmetric operad as every operation may have source and target and
we can multiply just composable chains of operations. A one object
version of this may be identify with a usual nonsymmetric operad. (In my
paper I use \omega-operads). I wonder if a partial operad is the same as
my 1-operad?  

Michael.

Re: query

[Marco Grandis]

I have not seen Fukaya's paper.

If the problem is to formalise higher homotopies for, say, topological
spaces or chain complexes,  together with their operations, there is a
simple solution based on the path endofunctor (or, dually, the cylinder
endofunctor; or their adjunction) and its powers.

The path endofunctor  P  is constructed so that a homotopy  a: f -> g: X ->
Y  amounts to a map  a: X -> PY;  the maps  f, g  are recovered by means of
the two faces  d-, d+: PY -> Y.
(For  Top,  PY  is obviously the space  Y^[0, 1]  of paths in  Y,  with the
compact-open topology; for chain complexes  (PY)_n  =  Y_n + Y_(n+1) + Y_n,
with suitable differential.)

P  comes equipped with various natural transformations (faces, degeneracy,
connections, symmetries, concatenation...), satisfying "algebraic"
coherence axioms (a sort of "cubical comonad" with additional structure).
These transformations represent the basic structure of lower order
homotopies.

But now you have, practically for free:

- n-tuple homotopies, represented by the power endofunctor  P^n,
- n-homotopies, represented by a subfunctor  P_n  of  P^n,
- their operations, via the usual algebra of natural transformations,
- the deduced coherence relations of the latter.

Eg, a double homotopy, represented by a map  X -> P^2(Y),  has four faces
given by the four natural transformations  d-P, d+P, Pd-, Pd+: P^2 -> P;
if  k  is the concatenation of homotopies,  kP  and  Pk  are the vertical
and horizontal concatenation of double homotopies, and so on; if  k  is
associative, as is the case for chain complexes but not for spaces, so are
all higher concatenations. A double homotopy is said to be a 2-homotopy
when its "vertical" faces (say) are trivial, i.e. factor through the
degeneracy  1 -> P.
(For  Top,  P^2(Y)  is the space of maps from the standard square  [0, 1]^2
to Y;  P_2(Y)  is the subspace of those maps which are constant on the
vertical faces of the square.)

This way of deducing higher homotopies and their operations from the lower
ones can be found in:

M. Grandis, Categorically algebraic foundations for homotopical algebra,
Appl. Categ. Structures 5 (1997), 363-413.

M. Grandis, On the homotopy structure of strongly homotopy associative
algebras, J. Pure Appl. Algebra, 134 / 1 (1999 ?), 15-81.

Regards,   Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it

http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/

Re: query

[Ross Street]

>The language of higher category theory in more than analogous
>to homotopy theory, especially in the cellular version.  What
>is the appropriate reference for a non-categorical reader?
>Have the A_\infty categories of Smirnov or Fukaya been treated
>in the categorical literature??

Fortunately, since the A_\infty categories (described sketchily in the
preprint I have of Fukaya) use chain complexes rather than topological
spaces (so that I believe a 1-object A_\infty category is an algebra for
the A_\infty non-permutative operad on chain complexes - an A_\infty
DG-algebra), we do not need to realise the whole homotopy types dream (as
described by John Baez) to give a categorical description of A_\infty
categories. Several years ago I remember discussing this by email with Jim
Stasheff.  Dominic Verity was here at the time.  I cannot remember all the
details but the two basic ingredients were some free strict n-categories
made from the set (whose elements are to be the objects of the A_\infty
category) in much the way the orientals are constructed, and the
construction (below) mentioned in my Oberwolfach Descent Theory notes of
September 1995.

        There is a connection between the Gray tensor product and ordinary
chain complexes.  Each chain complex R gives rise to a (strict)
omega-category J(R) whose 0-cells are 0-cycles  a  in  R, whose 1-cells  b
: a --> a'  are elements  b  in  R_1  with  d(b) = a'- a,  whose 2-cells  c
: b --> b'  are elements  c  in  R_2  with  d(c) = b'- b,  and so on.  All
compositions are addition.  This gives a functor  J :  DG --> omega-Cat
from the category  DG  of chain complexes and chain maps.  In fact,  J :
DG --> omega-Cat  is a monoidal functor where  DG  has the usual tensor
product of chain complexes and  omega-Cat  has the Gray tensor product.  By
applying  J  on homs, we obtain a (2-) functor  J_* :  DG-Cat --> V_2-Cat,
where  V_2  is  omega-Cat  with the Gray-like tensor product (extending the
natural tensor product of oriented cubes as described in Sjoerd Crans
thesis).  In particular, since  DG  is closed, it is a DG-category and we
can apply  J_*  to it.  The V_2-category  J*(DG)  has chain complexes as
0-cells and chain maps as 1-cells; the 2-cells are chain homotopies and the
higher cells are higher analogues of chain homotopies.

Best regards,
Ross

Re: query

[john baez]

Jim Stasheff writes:

> The language of higher category theory in more than analogous
> to homotopy theory, especially in the cellular version.  What
> is the appropriate reference for a non-categorical reader?
> Have the A_\infty categories of Smirnov or Fukaya been treated
> in the categorical literature??

Unfortunately the truly appropriate reference has not yet been
written, because the equivalence between weak infinity-groupoids
and homotopy types has not yet worked out in full detail, at least
not in the cellular version.   The *dream* of translating all of 
homotopy theory into higher category theory is outlined in:

John Baez and James Dolan, Categorification, to appear in Proceedings
Workshop on Higher Category Theory and Mathematical Physics at 
Northwestern University, Evanston, Illinois, March 1997, eds. Ezra
Getzler and Mikhail Kapranov, preprint available as math.QA/9802029.

This also has lots of references to different places where various
bits of the dream have been realized.  

Basically the dream consists of working out the following correspondence:

HIGHER CATEGORY THEORY             HOMOTOPY THEORY

omega-groupoids                    homotopy types
n-groupoids                        homotopy n-types
k-tuply groupal omega-groupoids    homotopy types of k-fold loop spaces
k-tuply groupal n-groupoids        homotopy n-types of k-fold loop spaces
k-tuply monoidal omega-groupoids   homotopy types of E_k spaces
k-tuply monoidal n-groupoids       homotopy n-types of E_k spaces
stable omega-groupoids             homotopy types of infinite loop spaces
stable n-groupoids                 homotopy n-types of infinite loop spaces
Z-groupoids                        homotopy types of spectra
 
How do A_infinity categories fit in?  As far as I can tell they should
correspond to omega-categories where all j-morphisms are invertible for
j > 1.  

query

[James Stasheff]

The language of higher category theory in more than analogous
to homotopy theory, especially in the cellular version.  What
is the appropriate reference for a non-categorical reader?
Have the A_\infty categories of Smirnov or Fukaya been treated
in the categorical literature??
thanks

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds

Re: query

[categories]

Date: Mon, 6 Oct 1997 15:48:08 -0400 (EDT)
From: John R Isbell <ji2@acsu.buffalo.edu>

Dear Warren,
   With Schaum's Outline on set theory, I would recommend
the new (new as proper book; it had a long development)
book "Conceptual Mathematics" by F. W. Lawvere and S. H.
Schanuel. They developed the book as a text for a
'mathematics for poets' course.
    John Isbell

query

[categories]

Date: Wed, 1 Oct 1997 10:17:10 -0400
From: wglick@CapAccess.org


My name is Warren Glick.  Until illness forced
me into a disability retirement, I was a reference
librarian in the Philosophy Division of the
main branch of the DC Public Library.  Currently,
I am working my way through the Schaum's Outline
Book on Set Theory.  I am a beginner and would
enjoy receiving pointers and guidance on the
relationship between set theory and category theory.


On the Web I have read brief summaries on 
the Bourbaki School's view of set theory and
structuralism in the philosophy of mathematics,
as it relates to set theory.  Please e-mail
and/or mail me any information that might 
assist me.

Thanks in advance for your support.


Warren Glick
4850 Conn. Ave., NW #619
Washington, DC 20008
wglick@capaccess.org

query

[categories]

Date: Wed, 1 Oct 97 04:53:29 +1000
From: burns burns@latcs1.lat.oz.au



I've quite a lot to ask, but for this post I'll explain what I've
been about. It's PhD research, but I came to category theory only
in the last year, so let me lay out my background.


1. Context.

The goal is to formulate coordinate systems as first-class entities
in programming languages. But what is a coordinate system, in
computational terms? Some more or less equivalent answers:

Operationally:

* The initial node, or any derived node, in a directed graph where the
edges are coordinate transformation functions.

Denotationally:

* A chart mapping from an open set into a vector space. (Manifold  
theory.)

* A signature or ADT (where it appears as type F):

	Types:
		X ; coordinate tuples
		T ; coordinate transformations
		F ; coordinate systems
		P ; points
	Operations:
		compose: T x T -> T
		invert: T -> T
		apply: T x X -> X
		plot: F x X -> P
		measure : F x P -> X
		step: T x F -> F
	Identities:
		plot(step(t,f),apply(t,x)) = plot(f,x)
		measure(f,plot(f,x)) = x
		plot(f,measure(f,p)) = p
		+ group identities for T

* A functor on a Cartesian closed category with additive structure,
preserving biproducts (vector bases); or more ambitiously, preserving
a set of invariant relations axiomatizing Euclidean geometry.

It's the simple linear case of this last, which I want to develop here.


2. Background.

I got this stuff originally from Arbib & Manes, Freyd, Barr & Wells,
and Pierce; while I bounced off Eilenberg and Asperti & Longo. What
I found I was stubbornly keeping out of the library was MacLane,
_Categories for the Working Mathematican_, and the 1985 _Category
Theory and Computer Programming_, with its nice essays by Rydehard,
Poigne, Pitt et al.


3. Essentials.

The foundations are Cartesian closed categories and abelian categories.

Briefly, CCCs have values, products and powers; while ACs have
abelian group structure on Hom-sets, a common zero, and biproducts;
as well as kernels and stuff which I'm not up with yet.

(I'm also still lost with limits and adjunctions. It's frustrating that
I can work through examples, and still miss the principle. I imagine
I really understood adjoints, a whole lot would snap into focus.)


4. Sketch Synopsis.

(Terminology:
	domain = type = object
	arrow = morphism
	power domain = exponential = Hom-set (in CCCs)
)


4.1. CCC Isomorphisms.

CCC's are held together by domain constructors: 1, x, and [->],
with corresponding isomorphisms:

	A =~= [1->A]			; domains have values
	[M->A] x [M->B] =~= [M -> AxB]	; there are finite products
	[MxA -> B] =~= [M -> [A->B]]	 ; Hom-sets are power domains

[NB! I get the impression that writers use "power domain" in a different
sense. "Hom domain", "arrow domain", "function domain" might all be
better.)

Also

	A x B =~= B x A

for products in general. Products are associative, powers aren't.

4.2. The Field Domain.

To put it to work, assume primitive domains:

	I	 stands for a finite index set
	R	stands for the reals, with the ring structure:

	(+) : R x R -> R
	(.) : R x R -> R
	0 : 1 -> R
	1 : 1 -> R
	(-) : R -> R

(To turn R into a field, we have to break it up as a coproduct:

	R = {0} + R-{0}

so that we can define

	(^-1) : R-{0} -> R-{0}

but this complication can be left until kernels are opened up.)


R is a module, with a one-element basis {1}, and it's also its
own linear automorphism set,

	R =~= L[R->R]
	r <-> (lambda(.)) r

The vectorial properties, addition and scalar multiplication,
carry over to any domain [A->R]. For [A->R] to be a module,
there must be a finite basis for it. As far as I can tell,
being "finite" only requires that operations (e.g. sum) over
all elements are defined.

4.3. Biproduct from CCC.

Now bring in domain I. Equip it with an equality mapping, the
Kronecker delta:

	(=) : I x I -> R : (i,j) |-> if (i = j) 1, else 0

With the CCC isomorphisms, plus R =~= L[R->R], this can be
expressed as:

	delta : I x R x I -> R : (i,r,j) |-> if (i = j) r , else 0

and we use the usual exponential curry-eval diagram, and a bit of
extra currying, to resolve it to:

	eta : I -> [R -> [I->R]] : i |-> (r |-> (j |-> if(i=j) r,else 0))
	pi :  I -> [[I->R] -> R] : i |-> ((j |-> x[j]) |-> x[i])

This expresses the biproduct (sketch diagram here)

	 eta i     pi i
	R -> [I->R] -> R

with (eta i) and (pi i) as injection and projection sets. The eta's
produce vectors, the pi's take components. In effect they are a
vector basis and a covector basis for [I->R].

We want a name for this category; I don't know what the standard is,
so I'll refer to the domain [I->R] as X within the category, and
X(I) for the category itself.

4.4. Duality Functor

The contravariant hom-functor X(I)(-,R) gives the dual category,
which we could call X*(I) = Xop(I). The hom-functor action is:

	X(I)(-,R):

	A |=> [A->R]
	f: A ->B |=> f#: [B->R] -> [A->R]

In particular, this gives:

	(pi* i) = (eta i)# : [X->R] -> R : x* |-> x* eta i
	(eta* i) = (pi i)# : R -> [X->R] : r |-> r pi i

and

	(eta** i) = (eta i)##
	(pi** i) = (pi i)##

after some fiddling around with [1-> ]'s.

4.5. Chart and Transformation Functors.

Next, suppose we have another category, say P, with its own
copies of I and R as well as a domain E. And suppose there is
an isomorphism:

	f : E -> X
	~f : X -> E

then (as far as I can tell), it defines an invertible functor:

	Chart(f): P -> X(I)

with the action:

	R |=> R
	I |=> I
	E |=> X
	eta i |=> ~f eta i : R -> E
	pi i  |=>  (pi i) f : E -> R

and Chart(f), at last, is what we mean by a coordinate system!


Thirdly, if we have a matrix:

	m : I x I -> R

and require its adjoint to be defined (i.e. it's non-singular),
then

	t(m) = Sum(i,j) ( t<i,j> (eta i) (pi j) ) : X -> X

defines a functor as above:

	Trans(t(m)) : X(I) -> X(I)

The adjoint matrix has to be defined, to preserve the biproduct.
If it isn't, the dual functor won't define a surjection.


4.6 Category: XTFP(I)?

Now let me put together what I think I've got. (If I'm qualifying
assertions a lot, that is because the concept is clear enough for
making assertions, but the hard work is only half done.)

Overall, we have a collection of categories which we can denote

	{Trans(t)(X), I, Trans(t) eta(X), Trans(t) pi(X)}

including the structure

	X(I) = {[I->R], I, eta, pi}

which we first made up (and which remains somewhat special, because
it's the only one for which the eta's and pi's are the columns and
rows of the identity matrix, i.e. Kronecker delta for I).

Make a category of this collection. Its domains will be the categories
in the collection, and its arrows will be the transformation functors
Trans(t(m)). Call this XT(I); perhaps a nicer name would be
Bases(I).

But more, we can include XT(I) in a larger category, of images of
the whole thing. This will be a category of domains P, with associated
chart functors Chart(f): P -> X.

It would be reasonable to call the the category containing XT(I)
and the image of it defined by a single P and F, XTF(I,P) or
Altas(I,P). And the whole gazoo, comprising as many of these as
we care to make up P's, we can call XTFP(I), or LinearGeometry(I).


4.7 Back to ADT's.

One test of whether the construction has gelled, is to match it
against the ADT signature at the beginning. That signature is
what I was originally calling "XTFP", before I started using
categories, and the reason I've gone in for categories was to
impose on it a discipline of function domains and ranges.

The core of the signature is:

	Operations:
		compose: T x T -> T
		invert: T -> T
		apply: T x X -> X
		plot: F x X -> P
		measure : F x P -> X
		step: T x F -> F

With the categorial development, write this as:

	compose : (t1,t2) |-> t1 t2
	invert : t |-> ~t
	apply : (t,x) |-> t x
	plot: (f, x) |-> ~f x
	measure: (f,p) |-> f p
	step: (t,f) |-> t f

But also requiring:

	invert: f -> ~f

The ADT is thus included in XTFP(I); but the latter now has a good
deal more structure, because it also contains R, with addition and
scalar multiplication defined on I x I x ... x I -> R, and all arrow
domains which are isomorphic to it via the CCC and R <-> L[R->R]
isomorphisms.

What's more, if we extend still further, to a category containing
XTFP(I) for all indices I, this larger category (XTFP) is exactly
Vect.

4.8  In Practice

The categorial version of the ADT is nice in some basic ways.
The identities, such as

	plot(step(t,f),apply(t,x)) = plot(f,x)

come out like

	~(t f) (t x) = ~f ~t t x = ~f x

with the cancellation being obvious, where the original is
rather ad hoc.

Considering this as an expression syntax, it's pretty readable.
In fact, this form is what I have, in an implementation of the
original ADT in Mathematica 2.2. I extend the types to lists
or arrays in the types, and the expression syntax then works
for regular hierarchic figures. I build small models in this
XTFP implementation: 3D stick-figure bicycles, fractals, etc.
I think of it as a more elaborate version of LOGO.

Part of what I'm wrestling with, is a way to use Mathematica's
substitution rules to make the t's and f's work _as_ Trans
and Chart functors. This is application of categories (to the
definition of functional programming languages) with a vengeance.

I haven't seen it done quite like this in the literature, but
then I may be missing something in denotational semantics or
typed-lambda, which states it all but states it obscurely.


4.9  Geometric Semantics

A big concern of mine, is that programming languages for graphics
and geometry lack geometric meaning. Because the conventions of
numerical representation are settled _during_ the analysis of a
modelling problem, and _before_ programming begins, the actual
code does not support any shifting between alternate representations;
that is, there's no such thing as change of variable, re-solving
equations for a different variable set, etc.

Bringing coordinate systems into a programming as first-class entities
is supposed to be a first step towards improving the state of affairs.
Now we can see coordinate systems and transformations as functors,
the problem can be seen as, how to keep the language in a form to
which functors can be applied.

The immediate successes in geometric semantics with XTFP(I), are

1. Points and their coordinates are distinguished.
2. Covariant and contravariant vectors are distinguished.
3. Active and passive coordinate transformations are distinguished.


4.10  The Further Prospect.

The basic operation in XTFP(I) turns out to be: substitute for the
eta i and pi i, any equivalent set of basis vectors.

Looking back, it impresses me that we can get all this from:

1. Field structure on R
2. Equality on I

The next thing to go for, is the remainder of tensor analysis.
The compounds

	eta i pi j
	pi i eta j

are arrow-successive; but there is no tensor product defined,
to make

	eta i eta j

meaningful, for instance.


Further, up til now we have made minimal use of structure on I.
But there are some obvious morphisms between XTFP(I)'s, for
different I's.

In particular, if we have I, then we have the lattice of subsets
of I; and these subsets index related spaces, with related vector
bases. E.g. if

	I = (1,2,3,4)

indexes vectors in R^4 = [I->R], then

	I<I = ((1,2),(1,3),(1,4),(2,3),(2,4),(3,4))

indexes bivectors. The combinatorics of the I's implies the
form of the cross-product function, that is, how to arrange the
2x2 minors of a matrix {x1,x2} with x1,x2 in X(I). So to speak.

I'm working on a smooth way to formalize this in general. The
essential concept is that there must be some _functor_ from the
category of I's and combinatorial operations, to the category
of the graded subspaces (scalar, vector, bivector,..,(n-1)-vector,
determinant) of X(I). It's one of those cases where one is sure
that someone must have done such a thing already, but one can't
find it in the literature.


4.11 Thanks for Listening.

That's the state of play.

I guess my conclusion on categorial development, is that you can
spend a lot of effort getting to the elements of the area you want
to model; but when you've done it, you have a formalism that's
pretty well purged of ad-hoc assumptions. It's done with the
absolute minimum.

Now, everyone wants feedback on their ideas, and I'm no exception;
but I'm not expecting anyone to review this in detail. What would
help me a lot is any indication that others have gone up the same
route, of combining CCCs or lambda-calculus with arithmetic, to
define vector analysis in the same sort of way.

That's all. Now I'll be hoping to gain some enlightenment from
listening in to your discussions.

-------------------------------------------------------------------
Jonathan Burns        |
burns@latcs1.lat.oz.au|   I don't care what the Dark Matter is
Dept.Computer Science |   as long as it's not spiders.
LaTrobe University    |                  - The Classic .sigs
-------------------------------------------------------------------

Re: Query

[categories]

Date: Tue, 1 Jul 1997 09:49:21 -0700
From: Michael J. Healy 206-865-3123 <mjhealy@redwood.rt.cs.boeing.com>


Since there have been some replies to the query about artificial perception, 
I suppose it's OK to mention my own work in progress.  I have been doing 
research in the formal semantics of neural networks.  I am working on a 
mathematical model in which concepts (formulas) are formed in memory as 
colimits.  The diagrams involve neural structures representing other 
concepts, going all the way back to simple percepts.  A concept is stored 
in memory as a neuron or neuron pool together with its attendant synaptic 
connections.  Logically closed portions of memory are theories.  Functors 
and natural transformations enter in in the usual fashion of categorical 
model theory.  

There is still a lot of work to do on this, and I am still learning the 
mathematics.  I do have a proposed neural implementation of it, and am 
working on a paper.  Previous work along these lines has involved geometric 
logic, so that I could understand some of the basics of learning, which for 
me involves working with an observational logic.  I have a paper on this, 
too.  Finding reviewers for this kind of material in the neural network 
community has been difficult.  If any of this sounds interesting enough 
to discuss, I certainly wouldn't mind getting some feedback from category 
theorists.

Sincerely, 
Mike Healy
--

===========================================================================
                                         e	     
Michael J. Healy                          A
                                  FA ----------> GA
(425)865-3123                     |              |
FAX(425)865-2964                  |              |
                               Ff |              | Gf
c/o The Boeing Company            |              |   
PO Box 3707  MS 7L-66            \|/            \|/
Seattle, WA 98124-2207            '              '
USA                               FB ----------> GB
                                         e            "I'm a natural man."
michael.j.healy@boeing.com                B
-or-  mjhealy@u.washington.edu

============================================================================

Re: Query

[categories]

Date: Sun, 29 Jun 1997 20:49:19 -0400
From: Michael Barr <barr@triples.math.mcgill.ca>

Not that I am blowing my own horn (but why not), you will find something
about comma categories in Barr & Wells, Category Theory for Computing
Science.  But no, I know of no other research along those lines.

Michael Barr

Query

[categories]

---------- Forwarded message ----------
Date: Thu, 26 Jun 1997 13:57:18 +0300 (IDT)
From: ZIPPIE Gonczarowski <zippie@actcom.co.il>
To: cat-dist@mta.ca


Dear Colleagues,

I am applying tools of category theory to research in artificial
perception and cognition. A basic category is proposed where every
perception is an object and morphisms capture the flow between
perceptions. Natural transformations capture paths to more cognitive
perceptions.
An Anonymous referee remarked that my category is `very closely related
to comma categories'.

Can anybody refer me to written material that introduces comma categories?

Also, please let me know if you are aware of other research that applies
categorical tools to research in artificial perception and cognition.

Thanks

Zippie

e-mail zippie@actcom.co.il

Dr. Zippora Arzi-Gonczarowski
Typographics, Ltd.
46 Hehalutz St.
Jerusalem 96222, Israel

Re: query

[categories]

Date: Mon, 10 Feb 1997 07:50:48 -0500 (EST)
From: Peter Freyd <pjf@saul.cis.upenn.edu>

I've been talking about paracategories for several years now.
But they require some more axioms. And they don't require 
inverses.

query

[categories]

Date: Sat, 8 Feb 1997 15:19:56 -0500 (EST)
From: James Stasheff <jds@math.unc.edu>

is there a name for a gadget with stict unit
strict inverses but multiplcation not always defined
and associativity holds when both a(bc) and (ab)c
are defined
BUT neither bracketing existing implies the other exists
references?
thanks

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
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