3 Aug 2008 02:01

### place-value notation

Hi,

Category theoretic notions such as sum and colimit can be thought of
as generalizations of the elementary arithmetic operation of adding
numbers. Given that place-value notation is the syntax (in the sense
of underlying structure or format) of addition, is there a category
theoretic account of place-value notation?

thank you,
posina


6 Aug 2008 17:00

### Re: Paper on slice stability in Locale Theory

On Thu, 31 Jul 2008, Townsend, Christopher wrote:

> Feedback is welcome. For example, I have always attributed the result
> Loc/Y =3D Loc_Sh(Y) to Joyal and Tierney. Am I right?=20
>
I think it predates the Joyal--Tierney work by a couple of years.
It's (more or less) present in the long Fourman--Scott paper on
sheaves and logic in SLN 753 (the Proceedings of the 1977 Durham
symposium), but they don't claim originality for it.

Peter Johnstone


7 Aug 2008 13:27

### Category <b>2</b>

Folk,

The title is revised and notations "top" and "bottom"
have been added.  The latest scan is here.

http://carnot.yi.org/Category2.jpg

Regards,       ... Peter E.

--

--
http://members.shaw.ca/peasthope/
http://carnot.yi.org/  = http://carnot.pathology.ubc.ca/


8 Aug 2008 22:40

### Re: Paper on slice stability in Locale Theory

The result was in the air - and I'm sure that Joyal in particular saw
it clearly.

For me it (or at least the logical version of it given in Fourman-
Scott) came as an answer to Dana's request for a description of an
internal cHa in a category of "Omega sets". I don't think I had
already heard it from Joyal - but I'm sure he already knew it. It was
easy to see that an object of Loc/Omega represented an internal cHa,
by generalisation of the representation of examples such as the
internal cHa O(R) in Sh(X) as O(RxX) -> O(X), or relaxation of the
representation of sheaves by local homeomorphisms.

Our paper was deliberately (and perhaps mistakenly) concrete rather
than abstract, so we didn't think to phrase the representation
explicitly as an equivalence of categories.

A related (but even more logical and obstinately pointed) result is
given in
Michael P. Fourman. T1 spaces over topological sites. J. Pure and
Applied Algebra, 27(3):223-224, March 1983.

On 6 Aug 2008, at 16:00, Prof. Peter Johnstone wrote:

> On Thu, 31 Jul 2008, Townsend, Christopher wrote:
>
>> Feedback is welcome. For example, I have always attributed the result
>> Loc/Y =3D Loc_Sh(Y) to Joyal and Tierney. Am I right?=20
>>
> I think it predates the Joyal--Tierney work by a couple of years.
> It's (more or less) present in the long Fourman--Scott paper on


13 Aug 2008 02:23

### Set as a monoidal category

Dear Categoreans,

I know three monoidal structures on the category of sets, all of them
symmetric. Two are the product and coproduct, and I'll leave it to
your imagination to figure out the third one.

My question is: are these the only three? Proofs, counterexamples, or
references appreciated.

Thanks, -- Peter


13 Aug 2008 05:45

### Co-categories

I've been thinking idly about a concept dual to categories
in much the same way that co-algebras are dual to algebras,
and I've decided that I'd like to more about it.
To be precise, if V is a monoidal category,
then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C],
while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C].
(You can fill in the rest of the definition for yourself.)

Searching Google, this concept appears to be known (under this name)
in the case where V is Abelian, but I'm not so interested in that.
I'm more interested in the case where V is a pretopos (like Set)
equipped with the coproduct (disjoint union) as the monoidal structure (x).
My motivation is that this concept is important in constructive analysis
when V is a Heyting algebra equipped with disjunction as (x).
(This defines a V-valued apartness relation on the set of objects;
but I'm stating even this fact in more generality than I've ever seen.)

So if anyone has heard of this concept where V is not assumed abelian,
or even knows of a good introduction where V is assumed abelian,
then I would be interested in references.

--Toby


13 Aug 2008 11:36

### Re: Set as a monoidal category

Dear Categoreans,

It is standard practice to answer a different question! But I can't resist referring to
R. Brown   Ten topologies for $X\times Y$'', {\em Quart. J.Math.}
(2) 14 (1963),  303-319.
and asking if one can modify these topologies or underlying sets by some process to work sensibly for the
category of sets? (a compact subset of a discrete space is of course finite). Maybe it is not possible.

Ronnie

----- Original Message ----
From: Peter Selinger <selinger <at> mathstat.dal.ca>
To: Categories List <categories <at> mta.ca>
Sent: Wednesday, 13 August, 2008 1:23:59 AM
Subject: categories: Set as a monoidal category

Dear Categoreans,

I know three monoidal structures on the category of sets, all of them
symmetric. Two are the product and coproduct, and I'll leave it to
your imagination to figure out the third one.

My question is: are these the only three? Proofs, counterexamples, or
references appreciated.

Thanks, -- Peter


13 Aug 2008 03:36

### RE: Set as a monoidal category

Dear Peter,

There's a paper

Algebraic categories with few monoidal biclosed structures or none

of Foltz, Lair, and Kelly which studies monoidal closed structures on various
categories, and shows that the cartesian closed one is the only possibility for Set.
More generally, it shows that for many categories we know well, the only possible
monoidal closed structures are the ones we know well.

But this depends heavily on the closedness. Without that, as you say, one can use
the cocartesian monoidal structure (the coproduct).

Here's a further infinite family of monoidal structures on Set. Let A be any set. Then
define the tensor product * by X*Y=AXY+X+Y.

Steve.

-----Original Message-----
From: cat-dist <at> mta.ca on behalf of Peter Selinger
Sent: Wed 8/13/2008 10:23 AM
To: Categories List
Subject: categories: Set as a monoidal category

Dear Categoreans,

I know three monoidal structures on the category of sets, all of them
symmetric. Two are the product and coproduct, and I'll leave it to
your imagination to figure out the third one.


13 Aug 2008 16:37

My self-published advanced undergraduate algebra  text
"Introductory Algebra, Topology, and Category Theory"
has been  available by mail order since mid-2006.  Recently I have made
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Martin Dowd


13 Aug 2008 18:29

### Re: Co-categories


Dear Toby,

What you call a cocategory enriched over V can also be
described as a category enriched over V^op. These have been
studied by Paddy McCrudden in his thesis under the name
"coalgebroids" (= many-object coalgebras). The main results
are on a generalised notion of Tannakian duality; and on
transfer of extra structure across this duality. See
respectively:

[1] Paddy McCrudden, Categories of Representations of Coalgebroids,
Advances in Mathematics Volume 154, Issue 2, Pages 299-332

Theory and Applications of Categories, Vol. 7, pp 71-147.

Richard

--On 12 August 2008 20:45 Toby Bartels wrote:

> I've been thinking idly about a concept dual to categories
> in much the same way that co-algebras are dual to algebras,
> and I've decided that I'd like to more about it.
> To be precise, if V is a monoidal category,
> then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C],
> while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C].
> (You can fill in the rest of the definition for yourself.)
>
> Searching Google, this concept appears to be known (under this name)