Subject: categories: Re: Linear--structure or property?

From: "George Janelidze" <[email-masked]>
Date: Fri, 11 Aug 2006 16:53:04 +0200

Dear Steve,

It is true that constructing such examples with more than one object is even
easier in a sense. However your example needs a minor correction:

What is 1+1 in C(x,x) and in C(y,y)?

If it is 0, them M must be a Z/2Z-module, since for every element u in M we
have u+u = 1u+1u = (1+1)u = 0u = 0.

If it is 1, them M must be idempotent (as before: u+u = 1u+1u = (1+1)u = 1u
= u).

If it is 1 in C(x,x) and 0 in C(y,y), then M becomes trivial, which destroys
the example.

And... why did not you and I just take the monoid {0,1}, which becomes a
commutative semiring for both additions?!

More generally, take any non-degenerated Boolean algebra. It has
multiplication=intersection=meet, and at least two additions (symmetric
difference and union=join) both good for that multiplication.

George

----- Original Message -----
From: "Stephen Lack" <[email-masked]>
To: "Categories list" <[email-masked]>
Sent: Friday, August 11, 2006 12:49 PM
Subject: categories: RE: Linear--structure or property?

It's a structure.

Consider the following category C.
Two objects x and y, with hom-categories
C(x,x)=C(y,y)={0,1}
C(y,x)={0}
C(x,y)=M
with composition defined so that each 1 is an
identity morphism and each 0 a zero morphism,
and with M an arbitrary set. Any commutative
monoid structure on M makes C into a linear category.

Steve.

-----Original Message-----
From: [email-masked] on behalf of Michael Barr
Sent: Fri 8/11/2006 6:14 AM
To: Categories list
Subject: categories: Linear--structure or property?

Bill Lawvere uses "linear" for a category enriched over commutative
semigroups.  Obviously, if the category has finite products, this is a
property.  What about in the absence of finite products (or sums)?  Could
you have two (semi)ring structures on the same set with the same
associative multiplication?

Robin Houston's startling (to me, anyway) proof that a compact
*-autonomous category with finite products is linear starts by proving
that 0 = 1.  Suppose the category has only binary products?  Well, I have
an example of one that is not linear:  Lawvere's category that is the
ordered set of real numbers has a compact *-autonomous structure.
Tensor is + and internal hom is -.  Product is inf and sum is sup, but
there are no initial or terminal objects and the category is not linear.