Re: A small cartesian closed concrete category

[peasthope@shaw.ca]

Folk,

At Thu, 14 Feb 2008 15:06:49 -0500 I wrote,
"Is there a cartesian closed concrete category which 
is small enough to write out explicitly?"
 
At Fri, 15 Feb 2008 08:47:57 +0000 Philip Wadler srote,
"... please summarize the replies ... and send ... to the ... list?
... interested to see if you receive a positive reply."

I've counted 16 respondents!  The question is 
answered well.  With my limited knowledge, the 
summary probably fails to credit some of the 
responses adequately but this is not intentional.
Thanks to everyone who replied!

5 messages mentioned Hyting-algebras.
Never heard of them.  Lawvere & Schanuel 
do not mention them in the 1997 book.  
Will store the terms for future reference.

Fred Linton wrote,
"... skeletal version of the full category
... having as only objects the ordinal numbers 0 and 1.

Here 0 x A = 0, 1 x A = A, 0^1 = 0, 0^0 = 1, 1^A = 1.
In other words, B x A = min(A, B), B^A = max(1-A, B)."

My product diagrams are at 
  http://carnot.yi.org/category01.jpg
.

Now I can try to illustrate the uniqueness 
of map objects according to L&S, page page 314, 
Exercise 1.  Does this category have a name?   
Is Boolean Category sensible?

Two messages mentioned lambda calculus.
Another topic for future reference.

Stephen Lack asked "How small is small? 
How explicit is explicit?"  Probably 
several other readers wondered the same.
Fred's reply is small enough and explicit 
enough to write out in detail.

One message addressed the term "concrete".  
I referred to Concrete Categories in the 
Wikipedia.

Matt Hellige mentioned categories a little 
bigger than that described by Fred.  
For instance, objects 0, 1, 2, 3.
Map A -> B exists iff A < B.

B x A =? min(A, B)  
I should sketch the details of some of these 
examples beyond the 0, 1 case above.

Andrej Bauer described Fred's category in the context 
of Heyting algebra and noted a proof by 
Peter Freyd.

Thorsten Altenkirch mentioned an equational 
inconsistency which is beyond my present 
grasp.

Apologies to anyone who's reply is not  
addressed adequately.  If someone requests, 
I can revise the summary and resubmit it.

Thanks,         ... Peter E.

Desktops.OpenDoc  http://carnot.yi.org/