From: Marco Grandis <grandis@dima.unige.it>
To: categories@mta.ca
Subject: An autonomous category
Date: Mon, 13 Mar 2006 14:45:25 +0100 [thread overview]
Message-ID: <2DE2CF1F-BF75-4606-990C-19C860157CF4@dima.unige.it> (raw)
The Lawvere category of extended positive real numbers has also an
autonomous structure, with a multiplicative tensor product (instead
of the original additive one). Has this been considered somewhere?
To be more explicit:
The well-known article of Lawvere on "Metric spaces..." (Rend. Milano
1974, republished in TAC Reprints n. 1) introduced the category of
extended positive real numbers, from 0 to oo (infinity included),
with arrows x \geq y, equipped with a strict symmetric monoidal
closed structure: the tensor product is the sum, the internal hom is
truncated difference (with oo - oo = 0).
Now, the same category can be equipped with a multiplicative tensor
product, x.y.
Provided we define 0.oo = oo (so that tensoring by any element
preserves the initial object oo), this is again a strict symmetric
monoidal closed structure, with hom(y, z) = z/y. Now, the
'undetermined forms' 0/0 and oo/oo are defined to be 0.
The new multiplicative structure is even *-autonomous, with
involution x* = 1/x (and 'nearly' compact).
(Note that this choice of values of the undetermined forms comes from
privileging the direction x \geq y, which is necessary if we want
to view metric spaces, normed categories etc. as enriched categories).
Marco Grandis
next reply other threads:[~2006-03-13 13:45 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-03-13 13:45 Marco Grandis [this message]
2006-03-15 0:58 Stephen Lack
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