RE: Linear--structure or property?

[F W Lawvere <wlawvere@buffalo.edu>]

Sorry Mike, I believe you misunderstood my definition of "Linear
category". It is in my paper

"Categories of Space and of Quantity"
International Symposium on Structures in Mathematical Theories, San
Sebastian (1990), published in the Book "The Space of Mathamtics.
Philosophical, Epistemological and Historical Explorations. DeGruyter,
Berlin (1992), 14-30.

Namely, because "additive" was already established standard usage that
included negatives, and because the importance in algebraic geometry etc.
of rigs other than rings and distributive lattices had been historically
underestimated, I chose the name "linear" because it would at least
have an intuitive resonance with physicists and computer scientists and
others who routinely apply linear algebra to positive and other contexts.
Thus, the definition is simply

a category with finite products and coproducts which agree (i.e. there is
a zero object which permits the definition of identity matrices, which are
required to be invertible).

Of course this implies enrichment in additive monoids.

However, the question of whether a multiplicative monoid has a unique
addition is of interest. I believe it was resolved in the negative by the
Tarski school of universal algebra, although I cannot recall the
reference, nor whether their examples were as straightforward as Steve
Lack's.

In ambient categories other than abstract sets, that question has been of
interest to me in particular in connection with the foundations of smooth
geometry and calculus. Euler's definition of real numbers as ratios of
infinitesimals leads immediately to a definition of multiplication as
composition of pointed endomorphisms of an infinitesimal object T. This
endomorphism monoid R of course has a zero element and hence there are two
canonical injections from it into R x R and the requirement on an addition
is that it be R-homogeneous and restrict to the identity along both those
injections. In classical algebraic geometry, where T is coordinatized as
the spectrum of the dual numbers, this addition is unique. In a
forthcoming paper I try to approach that result from a conceptual
standpoint, by invoking expected functorial properties of integration or
differentiation. The matter still needs to be clarified.

Concerning Mike's example of the real numbers as a *-autonomous category
under addition, I found it useful to note, in my 1983 Minnesota Report on
the existence of semi-continuous entropy functions, that the
system of extended real numbers, including both positive and negative
infinity, is also a closed monoidal category (even though not all objects
are reflexive). There are actually two such structures, depending on which
sense of the ordering one takes as the arrows of the category; the
definition of addition (which is to be the tensor product) must preserve
colimits in each variable (where colimits has the two possible
meanings). Besides its utility in freely performing certain operations in
analysis, this structure strikingly illustrates a point often made to
young students: subtraction is just addition of negatives, provided one is
in a group like the real numbers; however, in general the binary operation
of subtraction can be merely adjoint to addition and, in fact, the
condition that A is a "compact" object in a SMC

   for all B, A* @ B --> hom(A,B)is invertible

precisely characterizes the finite real numbers.

Best wishes,

Bill

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On Fri, 11 Aug 2006, Stephen Lack wrote:

> 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: cat-dist@mta.ca 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.
>
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