Date: Tue, 6 Aug 1996 19:02:15 0300 (ADT)
Subject: extensive stuff
Date: Tue, 6 Aug 1996 14:24:15 0400
From: Peter Freyd
The phrase DISTRIBUTIVE CATEGORY is established as referring to a
category with finite products and coproducts wherein coproducts
distribute with products. The phrase EXTENSIVE CATEGORY refers to
a cartesian category (that is, one with finite limits) with finite
coproducts wherein coproducts are preserved by pullbacks.
(Bill, they tell me you gave it this name because measurements tend
to be valued in such. Right?)
An equivalent definition of an extensive category is a cartesian
category that's _locally_ distributive, that is, every slice category
is distributive.
I've recently finally been able to find the right expansion of what
I've called "cartesian logic" (the syntax of cartesian categories) to
what I guess I will have to call "extensive logic" (the syntax of
extensive categories). Cartesian logic can be sloganized as the logic
of _unique_ existentials. For extensive logic add _exclusive_
disjunctions. (Yes, all you purists, "sloganized" is in the OED.
Actually, the American Heritage Electronic Dictionary would seem to
define it as the result of turning something into a Scottish war cry.)
With cartesian logic we first obtain a completeness theorem with
respect to the semantics of cartesian categories (by constructing the
free cartesian category from a given theory and noting what rules of
inference are needed to make the construction work) and then we obtain
a completeness theorem with respect to the "elemental semantics" by
using the fact that setvalued representations for any small cartesian
category are collectively faithful. By the time we're done we need
that the representations reflect not just equality and isomorphisms
but nonsplitepis into nonepis. (The Cayley representations, of
course, do all this.)
Similarly for extensive logic. Here we need the fact that the
setvalued representations of any small extensive category are
collectively faithful, indeed, collectively reflect (not just equality
and isomorphisms but) split epis. By "representation" I mean a functor
that preserves finite limits and finite coproducts. It would not be
enough to preserve just finite products and coproducts for the
semantics. That is, I must work in the context of extensive categories
not distributive categories. (Cayley, of course, no longer suffices.)
Do distributive categories arise in nature that aren't extensive?
The quickest artificial example is the full subcategory, *A*, of
(*S*)x(*S*), where *S* is the category of sets. A pair is
in *A* if either both X and Y are nonempty or both are empty.
*A* is coreflective, hence cartesian. It's closed under the formation
of products and coproducts, hence distributive (indeed, it's closed
under the formation of exponentials, hence an exponential category).
*A* is not an extensive category. <{a,b},{a,b}> is the coproduct of
<{a},{a}> and <{b},{b}>. The intersection of each of these
subobjects with <{a},{b}> is <{},{}> hence pulling back along the
inclusion map of <{a},{b}> does not preserve coproducts.
But I seem to have stumbled across a more natural example. In Cats and
Alligators a pair of idempotents e, e' are said to be "neighbors"
if ee'e = e and e'ee' = e'. So, let's understand a SEMIGROUP OF
NEIGHBORING IDEMPOTENTS to mean a semigroup satisfying the further
equations:
xx = x,
xyx = x.
Note that as a consequence: xyz = (xzx)yz = x(z(xy)z) = xz.
The category of semigroups of neighboring idempotents is a
distributive category because, even better, it's an exponential
category. Construct A => B in the naive way, that is, as the set of
homomorphisms from A to B. The semigroup structure on A => B is
given pointwise: for homomorphisms f,g:A > B, define
fg = \a.(fa)(ga). The equation xyz = xz forces fg to be a
homomorphism.
A homomorphism h:X*A > B curries to a homomorphism h':X >(A => B)
where the naive definition works for h', to wit, h' = \x.(\a.f).
The equation xx = x is just what's needed to see that h'x is a
homomorphism for each x, and then to see that h'(xy) = (h'x)(h'y).
To see that the category of semigroups of neighboring idempotents is
not an extensive category consider the fourelement semigroup with
multiplication given by:
a b c d
________
a a c c a
b d b b d
c a c c a
d d b b d
It is a coproduct (in the category of semigroups of neighboring
idempotents) of its oneelement subsemigroups {a}, {b}. (In other
words, it's the free semigroup of neighboring idempotents generated
by a and b.) Now intersect these generating subsemigroups with the
oneelement subsemigroup {c} to see that pulling back along the
inclusion map of {c} does not preserve coproducts.
Still here? Guess what. The two categories are equivalent and the
equivalence carries one example to the other.
Given an object, , in *A* turn XxY into a semigroup by
defining = . This is how the equivlence gets from
*A* to the category of neighboring idempotents. Getting back is left
to the reader.
Now the real question: how much of all this is already in Johnstone?
Date: Wed, 7 Aug 1996 12:55:55 0300 (ADT)
Subject: Peter Freyd's letter of 6 Aug
Date: Wed, 7 Aug 1996 17:44:00 +1000 (EST)
From: Max Kelly
I refer to Peter Freyd's interesting letter of 6 Aug concerning
distributive and extensive categories.
Peter, you have not got quite right the current nomenclature for
these: the definitive account of their interconnexions is in
[Carboni, Lack, Walters, Introduction to extensive and distributive
categories, JPAA 84 (1983), 145158]. Their name for your "extensive"
is "lextensive", which I think is due to Bill Lawvere; it means "lex
and extensive", where "lex' is used to mean "having all finite limits".
(By the way, I absolutely detest this usage of "lex"; a CATEGORY cannot
be left exact!) Their "extensive" categories have only finite COPRODUCTS
as part of the structure; but these are to be such that the
canonical A/a x A/b > A/(a+b) is to be an equivalence of categories.
The point is important because a MORPHISM of extensive categories
need preserve only finite coproducts, not finite limits (as a morphism
of lextensive categories must). So the 2categories involved are
quite different, and this affects the notion of free categorywith
structure.
One other thing: those semigroups you discuss are what I called
"middleignoring semigroups" in my paper with Pultr [ On algebraic
recognition of directproduct decompositions, JPAA 12(1978), 207224],
where I showed them equivalent to pairs of sets with neither empty
or both  only as the simplest and most trivial example of our
extension of Michael Barr's result on algebras for the "nth
power monad" sending A to A^n in any category. The funny thing is
that I spoke on this as your guest in the colloquium at Philadelphia
in 1977. Anyway, the more general situation of that paper may give
more examples of distributive categories  I haven't yet had time
to think about it. The category of middleignoring semigroups, by
the way, and more generally the category of algebras for the nth
power monad P_n on the category of sets, is symmetric monoidal
closed by Fred Linton's old result, since this monad is
commutative  at least I think it must be so, without stopping
now to check it.
About the nomenclature: do people agree that "lex" is really
terrible? Peter Johnstone in Sussex recently called it
something like a twicedead metaphor  but now I forget what
he wanted in its place  was it Peter Freyd's "cartesian" ?
Max Kelly.
Date: Wed, 7 Aug 1996 12:57:05 0300 (ADT)
Subject: Re: extensive stuff
Date: Wed, 7 Aug 96 10:17 BST
From: Dr. P.T. Johnstone
>Now the real question: how much of all this is already in Johnstone?
Not much of it, if you mean what is in Johnstone's published work,
rather than in Johnstone's mind. But my paper "A syntactic approach
to Diers' localizable categories" in Springer LNM 753 (the 1977 Durham
Symposium proceedings) is relevant: in it I introduced what I then
called "disjunctive logic", which is exactly what Peter now wants to
call "extensive logic" (and I guess that's a better name). What I was
doing there was to describe a class of theories whose model categories
(in Sets) were just the "localizable categories" (aka multiply
presentable categories) introduced by Yves Diers in his thesis: there
is nothing about extensive categories in my paper, because I didn't
know the concept at that time. However, I've known for some years now
that extensive categories are exactly the class of categories in which
this fragment of logic should be modelled  but I haven't found a
suitable opportunity to set this down in print.
Incidentally, the corresponding class of sketches (those with arbitrary
finite cones, but only discrete finite cocones) has been studied by
many people  see for example Barr & Wells (TTT), page 292.
Peter Johnstone
Date: Wed, 7 Aug 1996 12:57:56 0300 (ADT)
Subject: Extensive stuff
Date: Wed, 7 Aug 96 10:41 BST
From: Dr. P.T. Johnstone
A quick PS: since "semigroups of neighbouring idempotents" satisfy the
identity xyz = xz, they already have a name: they are what the
semigrouptheorists call "rectangular bands". As such, they appear in
my paper "Collapsed toposes and cartesian closed varieties" (J. Algebra
129 (1990); see top of p. 462), as the simplest nontrivial example of
what I called a "commutative hyperaffine theory". The fact that they
form a category equivalent to the twovalued collapse of Set x Set is
in my paper (though it was known long before), as is the fact that this
category is cartesian closed but not locally cartesian closed  and from
the proof that it's not lcc you can easily extract a proof that it's not
extensive.
Peter Johnstone
Date: Wed, 7 Aug 1996 12:59:18 0300 (ADT)
Subject: Re: extensive stuff
Date: Wed, 7 Aug 1996 09:40:48 0300
From: RJ Wood
Dear Peter
Everybody keeps rediscovering your *A*. It probably belongs in Insights.
Mike Barr's ``The Point of the Empty Set'' shows that the Ifold product
functor, restricted to the full subcategory of *S*^I determined by the
``pure functors'', is monadic, in fact VTT. For I=2 the monad in question
is TX=X^2 and a Talgebra is thus a set with a binary operation
satisfying
xx=x
(xy)(zw)=xw
(These are obviously equivalent to associativity and the equations you
gave.) Anyway, this explains the equivalence of categories you mentioned.
I certainly wasn't aware that *A* is distributive but not extensive. I'd
just like to point out that *A* and its generalizations is also useful for
showing that certain apparently multisorted algebraic categories are actually
single sorted. For example, the obvious category of all modules over
all rings is monadic over *A* and thus by the above also monadic over *S*.
Others will have more to say on this. On a personal note, I first learned of
*A* from my friend Kip Howlett who picked it up at some conference around
`71. He realized that it was what I needed for my MSc thesis that involved
Msets with variable M and applications to automata theory. Anyway, sorting
out *A* got me into category theory.
Best regards
RJ
> The quickest artificial example is the full subcategory, *A*, of
> (*S*)x(*S*), where *S* is the category of sets. A pair is
> in *A* if either both X and Y are nonempty or both are empty.
> *A* is coreflective, hence cartesian. It's closed under the formation
> of products and coproducts, hence distributive (indeed, it's closed
> under the formation of exponentials, hence an exponential category).
>
> *A* is not an extensive category. <{a,b},{a,b}> is the coproduct of
> <{a},{a}> and <{b},{b}>. The intersection of each of these
> subobjects with <{a},{b}> is <{},{}> hence pulling back along the
> inclusion map of <{a},{b}> does not preserve coproducts.
>
> But I seem to have stumbled across a more natural example. In Cats and
> Alligators a pair of idempotents e, e' are said to be "neighbors"
> if ee'e = e and e'ee' = e'. So, let's understand a SEMIGROUP OF
> NEIGHBORING IDEMPOTENTS to mean a semigroup satisfying the further
> equations:
> xx = x,
> xyx = x.
>
> Note that as a consequence: xyz = (xzx)yz = x(z(xy)z) = xz.
>
> The category of semigroups of neighboring idempotents is a
> distributive category because, even better, it's an exponential
Date: Wed, 7 Aug 1996 13:01:16 0300 (ADT)
Subject: Re: extensive stuff
Date: Wed, 7 Aug 1996 11:07:35 0400
From: Peter Freyd
Yikes! Paul Taylor has pointed out to me that I left out one of the
conditions for distributive and extensive categories, to wit, the
disjointness of coproducts. (Please note, however, that all the
categories that I said were distributive are distributive.)
He also points out that finite limits are not needed to say that
coproducts are stable. Of course. But they're certainly needed for
extensive logic. So: should the standard definition of extensive
categories include finite limits? Or should I  help!  go around
talking about "locally distributive logic"? (I included finite limits
in the definition of locally distributive, but needn't have done so:
the condition that the category and that every slice of the category
be distributive easily implies finite limits.)
Date: Wed, 7 Aug 1996 13:20:36 0300 (ADT)
Subject: Re: extensive stuff
Date: Wed, 7 Aug 1996 09:13:45 0700
From: james dolan
Still here? Guess what. The two categories are equivalent and the
equivalence carries one example to the other.

Given an object, , in *A* turn XxY into a semigroup by
defining = . This is how the equivlence gets from
*A* to the category of neighboring idempotents. Getting back is left
to the reader.
yes, the algebraic theory of semigroups retracts onto the initial
theory in two ways (left projection and right projection), so we get a
theory morphism to the square of the initial theory that's surjective
for some reason or other. the square of the initial theory is also
the natural structure theory of the right adjoint but very slightly
nonmonadic functor "binary cartesian product of sets".
i think these funny semigroups are some variety of "bands".
unfortunately i can't remember what a "band" is exactly.
Now the real question: how much of all this is already in Johnstone?
don't remember seeing it discussed there, but at least the part about
the funny semigroups being the algebras for the monad associated to
the slightly nonmonadic adjunction between sets and pairs of sets is
probably wellknown, i'd guess.
Date: Sat, 10 Aug 1996 23:52:22 0300 (ADT)
Subject: Re: extensive stuff
Date: Thu, 8 Aug 1996 11:43:51 +1000 (EST)
From: stevel@maths.su.oz.au
Dear Peter,
I too ``rediscovered'' your *A*, precisely in looking
for an example of a distributive category which failed
to be locally distributive, but never realized that it
had such an illustrious history.
As for distributive categories which fail to be extensive,
there is another important and natural class of examples.
Any distributive lattice, thought of as a preorder, is a
distributive category, indeed a locally distributive category
but has coproducts which are very far from being disjoint
and so fails to be extensive.
In a distributive category admits a subdirect decomposition
into a distributive and extensive category, and a distributive
preorder, in the following way. Given a distributive category D, one
can form the preorder reflection D_pr, and this is a distributive
preorder and the projection D > D_pr a distributive functor.
On the other hand, one can form the ``extensive reflection'' D_ext
of D (this is the image of D under the left biadjoint to the
inclusion of the 2category of distributive and extensive cats
in the 2category of distributive cats), and the projection
D > D_ext is also a distributive functor, and moreover the
induced functor D > D_ext x D_pr is fully faithful. This is
similar to a result of Cockett which fully embeds a _locally_
distributive category in the product of a lextensive category
and a distributive preorder.
The category D_ext has a very simple construction.
An object of D_ext is an arrow a:A>1+1 in D.
An arrow from a:A>1+1 to b:B>1+1 is an arrow f:A>B+1 in D
satisfying the condition
f
A > B+1
 
a   b+1
 
v v
1+1> 1+1+1
inj_13
Steve.
Date: Sat, 10 Aug 1996 23:53:19 0300 (ADT)
Subject: Re: extensive stuff
Date: Thu, 8 Aug 1996 11:59:20 +1000 (EST)
From: stevel@maths.su.oz.au
>
> He also points out that finite limits are not needed to say that
> coproducts are stable. Of course. But they're certainly needed for
> extensive logic. So: should the standard definition of extensive
> categories include finite limits? Or should I  help!  go around
> talking about "locally distributive logic"? (I included finite limits
> in the definition of locally distributive, but needn't have done so:
> the condition that the category and that every slice of the category
> be distributive easily implies finite limits.)
>
>
As Max Kelly pointed out, an extensive category with finite limits
has been called a lextensive category.
The problem still remains what one should call an extensive category
with finite products (such a category being necessarily distributive).
``Extensive and distributive category'' is a bit of a mouthful, and
prextensive is obviously unacceptable. Other possibilities that have
been suggested include ``2rig'' and ``arithmetic category''.
Steve.
Date: Sat, 10 Aug 1996 23:54:20 0300 (ADT)
Subject: Re: extensive stuff
Date: Thu, 8 Aug 96 09:55 BST
From: Dr. P.T. Johnstone
Max asks:
>About the nomenclature: do people agree that "lex" is really
terrible? Peter Johnstone in Sussex recently called it
something like a twicedead metaphor  but now I forget what
he wanted in its place  was it Peter Freyd's "cartesian" ?
and Peter asks:
should the standard definition of extensive
categories include finite limits? Or should I  help!  go around
talking about "locally distributive logic"?
Yes, I've been a convert for some time now to the Freydian use of
"cartesian" for categories having (or functors preserving) all finite
limits. I know this annoys the computerscience people who want to
use it for (finite products but not equalizers), but I can't think of
a better term. Actually, as applied to categories, "left exact" is a
thricedead metaphor (twicedead as applied to functors, since "exact
sequence" is a dead metaphor for "exact differential", and "left exact"
as applied to functors is a dead metaphor for "preserving the left
hand ends of exact sequences"). What status that gives to the term
"lextensive" for "extensive plus all finite limits", I shudder to
think.
Should the standard definition of extensive categories include
finite limits? Obviously Max (and Bob Walters, and Steve Lack, and
probably Bill Lawvere) would say "no". But if you want to develop
a syntax for these categories, you're not going to get very far
without the finite limits; so we do need a name for "extensive plus
finite limits". Perhaps, after all, we should revive the term
"disjunctive" from my SLNM 753 paper, and call them "disjunctive
categories". What do other people think?
Peter Johnstone
P.S.  If anyone was confused by the two messages I contributed to
this discussion yesterday, please note that they were sent out in
the reverse of the order in which I sent them in. Not that it
matters very much.
Date: Sat, 10 Aug 1996 23:55:28 0300 (ADT)
Subject: Re: Peter Freyd's letter of 6 Aug
Date: Thu, 8 Aug 1996 11:47:50 +0200
From: Dr. Reinhard B/rger (Prof. Dr. Pumpl^nn)
Just for completeness, I think that the category of middleignoring
semigroups is equivalent to the category of pairs of sets which are
either both empty or both nonempty. Another example of a category
which is distributive but not extensive is the dual of the category
of unital rings; note that the dual of the category of unital
commutative rings is even extensive.
Greetings
Reinhard
Date: Wed, 14 Aug 1996 23:04:48 0300 (ADT)
Subject: disjunctive stuff
Date: Sun, 11 Aug 1996 16:11:09 0400
From: Peter Freyd
Regarding some comments of Peter Johnstone:
I haven't succeeded in interpreting disjunction in arbitrary wcartesian
extensive categories and would therefore hesitate in calling them
"disjunctive categories."
Computer scientists used to use the phrase "concrete" to mean "well
pointed" but seemed to have stoped as they became aware of the clash
with existing category terminology. (By the way, I can't think at the
moment of many people in CS, other than Robin C and friends, who use
"cartesian" just to mean products.)
Date: Wed, 14 Aug 1996 23:06:33 0300 (ADT)
Subject: alternating stuff
Date: Sun, 11 Aug 1996 17:15:58 0400
From: Peter Freyd
How about "alternating categories"? Alternatives, in ordinary
language, are usually understood to be mutually exclusive. So an
extensive cartesian category (i.e. a locally distributive category
with terminator) would be called an "alternating category" and the
corresponding syntax, "alternating logic".
A major problem: it would be hard to keep others  since I find it
hard to keep myself  from corrupting this to "alternative logic".
(On the other hand, whenever one is an environment where "linear
logic" is sure to be totally misinterpreted, one could claim also to
be studying "alternative logic".)
Let me go on record here for the syntax. As in cartesian logic
conjunction is the only connective and the only terms are conjunctions
of primitive predicates each followed by an appropriate sequence of
variables. Cartesian logic has just one primitive assertion, written
A ue> B, where A and B are terms. Given an elemental interpretation
of the primitive predicates, A ue> B is satisfied in the elemental
cartesian semantics if for every instantiation of the variables of A
such that A holds there is a unique instantiation of the remaining
variables of B such that B holds. In alternating logic the
primitive assertions are written
A ue> B1B2...Bn
where A, B1, B2,...,Bn are terms. Such an assertion is said to be
satisfied in the elemental alternating semantics if for every
instantiation of the variables of A such that A holds there is a
unique index, i, such that the remaining variables of Bi can be
instantiated so that Bi holds and, further, there is just one such
instantiation of the remaining variables of Bi.
E.G.: for (decidable) fields, add to the cartesian theory of unital
rings the alternating axiom
x=x ue> (x=0)(xy=1).
As for the categorical semantics, given an alternative category in
which each primitive predicate has been interpreted, extend the
interpretation to terms  just as for cartesian logic  using finite
limits.
(For example: if A has variables x and y, and B has variables
y and z then, using brackets to designate the interpretations,
the interpretation of A^B is characterized by
[A^B]
l/ \r
[A] [B]
/ \ / \
[x] [y] [z]
where the rhombus is a pullback.)
Note that the interpretation of a conjunction comes equipped with
the two maps, l and r.
In cartesian logic the key definition:
A ue> B is satisfied iff l:[A^B] > [A] is an isomorphism.
In alternating logic:
A ue> B1B2...Bn is satisfied iff the l's combine to give
an isomorphism [A^B1] + [A^B2] +...+ [A^Bn] > [A].
te: Sun, 18 Aug 1996 11:29:00 0300 (ADT)
Subject: alternating stuff
Date: Thu, 15 Aug 1996 00:22:45 0400
From: James Otto
Date: Sun, 11 Aug 1996 17:15:58 0400
From: Peter Freyd
How about "alternating categories"? Alternatives, in ordinary
language, are usually understood to be mutually exclusive. So an
Alternating (Turing) machines are fundamental to complexity and logic
programming. The alternation is of bounded quantifiers.
The exclusive, inclusive distinction is captured by the orthogonal,
injective distinction, which is a strong model, weak model
distinction.
extensive cartesian category (i.e. a locally distributive category
with terminator) would be called an "alternating category" and the
corresponding syntax, "alternating logic".
...
Let me go on record here for the syntax. As in cartesian logic
....
In cartesian logic the key definition:
A ue> B is satisfied iff l:[A^B] > [A] is an isomorphism.
On the models side of dualities, this is precisely being orthogonal to
a map.
In alternating logic:
A ue> B1B2...Bn is satisfied iff the l's combine to give
an isomorphism [A^B1] + [A^B2] +...+ [A^Bn] > [A].
On the models side of dualities, this is precisely being orthogonal to
a small cone with discrete base. E.g. see Ad\'amek and Rosick\'y's
book including its historical remarks e.g. on Y. Diers.
I like P. Johnstones's `multi' for the map, cone distinction.
On the theory side of dualities, P. Johnstones's remarks may help me.
Just out of curiousity, if one drops the `closed' (which I do not)
from `cartesian closed' or `locally cartesian closed', what would one
expect to have?
Regards, Jim Otto
Date: Sun, 18 Aug 1996 11:29:59 0300 (ADT)
Subject: Re: alternating stuff
Date: Thu, 15 Aug 1996 17:01:33 +1000 (EST)
From: stevel@maths.su.oz.au
> Date: Wed, 14 Aug 1996 23:06:24 0300 (ADT)
> From: categories
>
> Date: Sun, 11 Aug 1996 17:15:58 0400
> From: Peter Freyd
>
> How about "alternating categories"? Alternatives, in ordinary
> language, are usually understood to be mutually exclusive. So an
> extensive cartesian category (i.e. a locally distributive category
> with terminator) would be called an "alternating category" and the
> corresponding syntax, "alternating logic".
>
An extensive cartesian category is _not_ the same as a locally
distributive category with terminator: as has already been pointed
out, extensive categories are also required to have disjoint
coproducts, which locally distributive categories need not, as the
example of distributive lattices shows.
Although I entirely agree that calling a category ``lex'' if it has
finite limits is a bad thing, and am not thrilled about the name
lextensive for an extensive category with finite limits, I do think
that the name does have some advantages. In particular it makes clear
that the two notions are closely linked. As various people have
pointed out, for many uses the extensive categories won't be much use
unless you have the finite limits as well; the reason, at least from
my point of view, for isolating the extensive categories is to
emphasize the fact that they are in fact categories with finite
coproducts satisfying a certain _property_ which does not depend on
the _structure_ of having all finite limits.
As an ``excuse'' for the name lextensive, one can think of it as
standing for (finite )l(imits +)extensive.
Steve.
Date: Sun, 18 Aug 1996 11:31:13 0300 (ADT)
Subject: Re: disjunctive stuff
Date: Thu, 15 Aug 96 10:27 BST
From: Dr. P.T. Johnstone
I'm not sure about `alternating logic' and `alternating categories';
perhaps I just need time to get used to them. I think the trouble is
that in everyday speech `alternating' means something dynamic
(switching back and forth between two states), and the logic doesn't
have any such feature. `Alternative' doesn't have the same connotation,
but I agree that `alternative logic' is impossible.
The original reason for `disjunctive' was a pun: it stands for both
`disjunction' and `disjoint'. The way I formulated the syntax
(essentially, an extension of Michel Coste's version of cartesian logic),
you are allowed to write down a disjunction of formulae if you can
prove that it's disjoint (just as you can use an existential quantifier
if you can prove that the thing being quantified is unique).
Peter Johnstone
Date: Sun, 18 Aug 1996 11:33:04 0300 (ADT)
Subject: Re: extensive stuff
Date: Thu, 15 Aug 1996 13:31:16 +0000
From: Steve Vickers
If "lex" fills a need in our vocabulary  "having finite limits" when
applied to structures, "preserving finite limits" when applied to
transformations , can't we forgive it the deadness of its metaphorical
origins? I suspect there are other metaphorically dead words in mathematics
 what about ring or field?
The fundamental problem with "Cartesian" now is that it is ambiguous.
Whenever it's used we have to investigate whether nonproduct limits are
required.
Steve Vickers.
Date: Sun, 18 Aug 1996 11:34:04 0300 (ADT)
Subject: Re: disjunctive stuff
Date: Thu, 15 Aug 1996 15:09:14 0400
From: Michael Barr
A propos the discussion of "cartesian", I might add that I think
Descartes also invented the idea of the graph of a function
and that is an equalizer. FWIW.
Michael
Date: Sun, 18 Aug 1996 11:37:23 0300 (ADT)
Subject: strongly (or exclusive) disjunctive logic: Hu, Tholen
Date: Sat, 17 Aug 1996 11:22:07 0400
From: James Otto
Dear People,
Perhaps I already said more about strongly (or exclusive) disjunctive
logic than I wished. (So this is 2 of 2.) But I should note
H. Hu, W. Tholen, Limits in free coproduct completions, JPAA 105
('95)
Rather than a duality as in
P. Gabriel, F. Ulmer, Springer LNM 221 ('71)
M. Makkai, A. Pitts, TAMS 299 ('87)
but somewhat as in
P. Johnstone, In Springer LNM 753 ('79)
they construct a dual and a double dual:
small with multilimits of finite diagrams
 flat functors to set
v
finitely accessible with connected limits
 functors to set preserving filtered colimits and connected limits
v
having finite limits and stable disjoint small coproducts
with net image the coprimes.
By the way, for accessible categories one could see
J. Ad\'amek, J. Rosick\'y, Cambridge ('94)
F. Borceux, Volumes 12, Cambridge ('94)
for alternating Turing machines
D. Bovet, P. Crescenzi, Prentice Hall ('94)
and for quantifiers, games, and interactive proofs (a book which I am
less familiar with than the previous 4)
J. K\"obler, U. Sch\"oning, J. Tor\'an, Birkh\"auser ('93)
Regards, Jim Otto
Date: Thu, 22 Aug 1996 09:27:03 0300 (ADT)
Subject: Re: cartesian/(L)extensive/... stuff
Date: Tue, 20 Aug 1996 15:38:11 0600 (MDT)
From: Robin Cockett
(1) First, in response to Peter's earlier comments about the term "cartesian":
I would certainly like to consider Peter as a friend! Some names for
mathematical concepts are like old jeans: the more threadbare and holes
they have the more comfortable they feel. I suspect the term cartesian
might be one such.
I tend to think of a "cartesian tensor" as being a product  as
saying a bit of structure is cartesian usually means it arises
from limits  and have abused terminology by calling what I should have
probably called a "cartesian tensor category" (or a "cartesian monoidal
category") a "cartesian category"  yes this is a category with products.
I am usually careful, however, to make this usage explicit as I am aware
that equalizers are often assumed.
(2) Another article of comfy clothing is the term extensive! I agree
with Steve Lack that it is unfortunate that extensive categories are not
assumed to have a final object (as these creatures are the more common)
I merrily suggest abusing terminology to avoid that hiccup. I would be less
happy, however, to let the LEXness  or should I say cartesianness  be
carried by the context. My reason for this hesitation is as follows:
If you start with a distributive category and form its extensive completion,
as explained by Lack, then all datatypes (natural numbers, lists, etc.) lift
into that completion. These datatypes are shapely in the sense of
Barry Jay (naturallity square are cartesian etc.). However, if you,
taking the construction from another angle, freely add equalizers to a
distributive category with datatypes (i.e. finitely complete it) all bets
are off: certainly datatypes which do lift need not be shapely but I conjecture
that there is, in fact, no guarantee that they lift at all ...
Conjecture: There is a distributive category X with (strong) NNO such that
========== E(X) is its equalizer completion does not have a (strong) NNO.
I do not have a proof that this is so ...or not!
Coproducts appear to lift as the equalizer completion 2functor
preserves products. However, clearly (consider a distributive lattice) the
resulting coproducts need not be extensive  although, of course, the category
E(X) is distributive (another source of nonextensive distributive categories).
If the distributive category is separated (or decidable) in the appropriate
sense then certainly datatypes lift (as the extensive completion and
equalizer completion then coincide). This is the reason why the initial
distributive category with (strong) datatypes may be finitely completed to
preserve all datatypes.
This makes me sensitive about the passage to Lextensive even from extensive.
To obtain a completion in a RIGHT SENSE may be a little more delicate (or
brutish ... depending on your approach). The point is there are some
outstanding issues here ..
(3) Lastly a remark on the connection between distributive categories
and categorical proof theory:
One motivation for developing the theory of weakly distributive categories
(wdc) was to provide a unification of the proof theory of "classical" and
linear logic. In particular, we supposed that the "andor" fragment of
classical logic has as a proof theory the free distributive category on
its propositions. Accordingly, in that original paper, we sketched a proof
that distributive categories are cartesian wdc's (i.e. wdc's in which the
tensor is a product and the cotensor a coproduct). Subsequently we never
revisited the result.
Over the summer while studying the nucleus of these categories we realized
something was amiss. Reexaming the proof we realized that one of the
"obvious" coherence conditions was obviously false. In fact, so
badly does it fail for distributive categories that the revision of the
result states:
Prop. A cartesian wdc which is simultaneously
a distributive category is necessarily a preorder.
It should be mentioned that cartesian wdc's abound: pointed sets, vector
spaces, semilattices, ... are examples. Thus cartesian wdc's definitely do
not collapse.
(Back to names!!!!
This does remove one of our motivations for the name "weakly distributive
categories": Barr suggested the term "linearly distributive". However,
to us the original name is now one of those comfy bits of clothing
(even if somewhat frayed) ... In fact, if we had NOT made this
oversight it is likely we would have been altogether more hesitant
in the development of weakly distributive categories! Mathematics
moves in mysterious ways.)
There is philosophical significance to the correct result: classical
semantic settings separate at an earlier stage than we had suspected from
(categorical) proof theoretic settings. In particular, distributive
categories (a core fragment of classical setting) do not permit the
process of cut elimination (unless they are preorders).
robin
(Robin Cockett)
(p.s. Revised papers are available under Seely's home page:
ftp://triples.math.mcgill.ca/pub/rags/ragstriples.html)
Date: Thu, 22 Aug 1996 22:46:00 0300 (ADT)
Subject: Re: cartesian/(L)extensive/... stuff
Date: Fri, 23 Aug 1996 11:02:53 +1000 (EST)
From: Steve Lack
>
> (2) Another article of comfy clothing is the term extensive! I agree
> with Steve Lack that it is unfortunate that extensive categories are not
> assumed to have a final object (as these creatures are the more common)
I can't quite imagine in what context I might have said that; it is
certainly true that life becomes easier when the extensive category
in question has a terminal object, but the whole point is that the
notion of extensivity is a property of finite coproducts. (So in
particular an extensive functor is one that preserves finite
coproducts only, although of course it then follows that pullbacks
along coproduct injections are also preserved.)
>
> If you start with a distributive category and form its extensive completion,
> as explained by Lack, then all datatypes (natural numbers, lists, etc.) lift
> into that completion. These datatypes are shapely in the sense of
> Barry Jay (naturallity square are cartesian etc.). However, if you,
> taking the construction from another angle, freely add equalizers to a
> distributive category with datatypes (i.e. finitely complete it) all bets
> are off: certainly datatypes which do lift need not be shapely but I conjecture
> that there is, in fact, no guarantee that they lift at all ...
>
> Conjecture: There is a distributive category X with (strong) NNO such that
> ========== E(X) is its equalizer completion does not have a (strong) NNO.
>
> I do not have a proof that this is so ...or not!
>
> Coproducts appear to lift as the equalizer completion 2functor
> preserves products. However, clearly (consider a distributive lattice) the
> resulting coproducts need not be extensive  although, of course, the category
> E(X) is distributive (another source of nonextensive distributive categories).
In fact,
(1) If X is distributive then E(X) is locally distributive
but
(2) If X is distributive and E(X) is extensive then X is
equivalent to the trivial category 1.
To (freely) pass from a distributive category to a lextensive one, you
first form E(X) and then the slice category p/E(X) where p is the equalizer
i
>
p > 1 > 1+1
j
of the coproduct injections i,j:1>1+1.
The passage from extensive categories to lextensive categories, is, as
Robin says, more delicate.
Steve Lack.
Date: Sat, 24 Aug 1996 11:44:46 0300 (ADT)
Subject: Re: cartesian/(L)extensive/... stuff
Date: Fri, 23 Aug 1996 11:53:12 0600 (MDT)
From: Robin Cockett
I did mean extensive + products and mistyped "final object." The point
being that these are common beasties which deserve a snappy name!
Further, Steve Lack is correct about the fact that if E(X) is extensive and
X is distributive then X = 1. So I should correct my comments about when
the equalizer completion and extensive completion coincide! ... and I
have to chuckle at this point as I have tripped on my own snag ...
The problem is the initial object.
In a distributive category this object is only one way connected to the rest
of the category so that the poset collapse of a distributive category is always
nontrivial while the category itself is. This means the added equalizer
i
>
p > 1 > 1+1
j
of the coproduct injections i,j:1>1+1 in E(X) is never isomorphic to the
initial object. Hence Steve Lack's comment.
Some time ago (when I was in Oz, in fact) I sensitized the community
there to exactly these issues. In fact, I became a bit of a heretic by
considering distributive categories without an initial object (I called these
predistributive). It is the initial of these gadgets (with "nonempty"
inductive datatypes) which has E(X) extensive ... and the initial object is
provided precisely by the added equalizer p, above.
However, it really is infinitely better to talk about LEXT(X), the
lextensive completion, not the raw E(X) where some preinitial baggage
may still be present.
The conjecture still stands but is better expressed in the form:
Conjecture: There is a distributive category X with (strong) NNO such that
========== LEXT(X) is its lextensive completion does not have a (strong) NNO.
robin
Date: Tue, 27 Aug 1996 10:19:53 0300 (ADT)
Subject: terminology
Date: Mon, 26 Aug 1996 11:07:26 +1000 (EST)
From: Steve Lack
Regarding the debate on cartesian/extensive, we'd like to suggest
the name "arithmetic category" for an extensive category with products.
Certainly such categories seem well structured enough to do some
arithmetic, and free such are closely enough related to the free
analogies in the algebraic context (ie rigs) to be thought of as
behaving "arithmetically" in some sense.
Robbie Gates
Steve Lack


robbie gates 
apprentice algebraist  http://cat.maths.usyd.edu.au/~robbie
pgp key available 
Date: Wed, 28 Aug 1996 10:42:44 0300 (ADT)
Subject: Re: terminology
Date: Tue, 27 Aug 1996 14:45:24 +0000
From: Steve Vickers
>Regarding the debate on cartesian/extensive, we'd like to suggest
>the name "arithmetic category" for an extensive category with products.
>
>Certainly such categories seem well structured enough to do some
>arithmetic, and free such are closely enough related to the free
>analogies in the algebraic context (ie rigs) to be thought of as
>behaving "arithmetically" in some sense.
>
>Robbie Gates
>Steve Lack
The phrase "Arithmetic Category" with this sense doesn't sit comfortably
next to Joyal's "Arithmetic Universes", in which the word "arithmetic" also
conveys recursive structure.
Steve Vickers.
Date: Thu, 29 Aug 1996 13:19:42 0300 (ADT)
Subject: Re: alternating stuff
Date: Thu, 29 Aug 1996 14:59:54 +0200 (MET DST)
From: Jiri Rosicky
In a recent paper (An algebraic description of locally multipresentable
categories, TAC 2 (1996), 4054), we have introduced
essentially multialgebraic theories and showed that they correspond
to locally finitely multipresentable categories in the same way as
essentially algebraic theories correspond to locally finitely
presentable categories.
J.Adamek, J.Rosicky
Date: Thu, 29 Aug 1996 13:19:00 0300 (ADT)
Subject: Re: terminology
Date: Thu, 29 Aug 1996 03:00:41 0400 (EDT)
From: F William Lawvere
The term EXTENSIVE was applied to certain categories in 1991
by Carboni, Lawvere, and Walters on the basis of the following
considerations.
For centuries, mathematical philosophy has distinguished between
extensive quantities and intensive quantities, for example in
thermodynamics of inhomogeneous bodies, volume, mass, energy, and
entropy on the one hand are distinguished from pressure, density,
temperature on the other. Lawvere in 1982 (SLNM 1174) and in
1990 (Categories of Space and of Quantity) had proposed to
explain these as modes of variation of quantity. Quantities vary
over a domain of variation in both cases. ( A domain of variation
is a "space" , which in turn is an object in a category of space,
which will be rather more determinate than "category" in general).
An extensive quantity type is a covariant functor from a category
of space, preserving finite coproducts, to a linear category (=
one in which coproducts and products coincide). A distribution,
a measure, a current, a homology class on a sum domain is
given by a tuple of such , one on each summand ; thus all these
are elements of extensive quantity types. An intensive quantity
type is a contravariant functor from a category of space which
also takes coproducts to products ; but more: intensive quantities
usually act linearly on extensive quantities, lending them (not only
a linear but also a) multiplicative structure which is also preserved
by the contravariant functorality. For example, functions act as
densities on distributions and measures, similarly differential
forms act on currents, and cohomology classes act on homology classes.
Often intensive quantity types are representable and related extensive
types are definable as linear duals ( Riesz, Pontrjagin, Eilenberg Mac
Lane etc),but these are not the only possibilities.
A fundamental example of a linear "category" is the 2category of
all categories with coproducts; it has an obvious abstraction functor
to the linear category of commutative monoids, by taking isomorphism
classes. Part of the idea of Ktheory and of Khomology and of
the "nonlinear Ktheory" which I with Schanuel and others have been
pursuing under the name of "objective number theory" is that it is
useful to "objectify" quantities by lifting their type across this
abstraction functor.
The most fundamental measure of a thing is the thing itself. But measures
can be pushedforward ( a common colloquial expression for the covariant
functorality of extensive quantity).However pullback is more familiar
(already in 1844 Grassmann complained that intensives were more familiar
than Ausdehnungen) : On any category with pullbacks, there is the
contravariant functor to cat which takes the "slice" categories
of each object.This functor is commonly viewed as consisting if
"functions", namely functions whose values are the fibers. Indeed if the
category satisfies suitable conditions , this will be an intensive
quantity type. But what extensives will it act on ?
On any category at all the slice categories constitute an even simpler
covariant catvalued functor, simply composing the transforming map
following the structural map to define the new structural map. It is
often appropriate to consider that the structural map distributes
the total in the base ( though distributions usually do not have values
at points, they do often have totals ) and that the mentioned composing
pushes the distribution forward. Indeed if the category has coproducts,
this naive pushing forward is automatically linear.
Thus a category with coproducts defines an extensive quantity type on
itself if and only if it is an extensive category.
If we call lextensive any extensive category with (finite) limits, then
also by pullback, the intensives act on the extensives in a way that
satisfies all reasonable functoralities, including the crucial CCR
or "projection formula". More exactly, note that there will typically be
lots of subcategories of such a "category of space" which are extensive
but do not themselves have products or even a terminal object;
it suffices to be closed under sums and summands. Namely the empty
space together with all spaces of dimension exactly 7. Any such
subcategory A defines the extensive quantity type "distributions
of Adimensional spaces " in each base space X, namely the subslice
category. Given A and B two extensive subcategories, there is an
objective intensive quantity type which acts roughly as " BA dimensional
cohomology" namely for any X consider the category of all those spaces
over X such that for any space over X whose total is in A, the pullback
has total in B ; this pulls back along any change of X.
Of course many extensive subcategories of a (lextensive) category of space
will have products, for example (as Joyal pointed out to me in 1984)
if A=B is the "compact" objects , thus defining the extensive notion
of distributing a compact object in a base space, the corresponding
intensive quantities which act on these are the proper fiberings
.
Date: Fri, 30 Aug 1996 13:57:50 0300 (ADT)
Subject: Re: terminology
Date: Fri, 30 Aug 1996 12:08:55 0400 (EDT)
From: F William Lawvere
The following is an improved version of the text sent Thurs. Any
suggestions for further improvements will be welcome . Note that
the distinction between the objective extensive and the objective
intensive is essentially "sampling/sorting" distinction discussed in
Conceptual Mathematics. Bill
The term EXTENSIVE was applied to certain categories in 1991
by Carboni, Lawvere, and Walters on the basis of the following
considerations.
For centuries, mathematical philosophy has distinguished between
extensive quantities and intensive quantities, for example in
thermodynamics of inhomogeneous bodies, volume, mass, energy, and
entropy on the one hand are distinguished from pressure, density,
temperature on the other. Lawvere in 1982 (SLNM 1174) and in
1990 (Categories of Space and of Quantity) had proposed to
explain these as modes of variation of quantity. Quantities vary
over a domain of variation in both cases. ( A domain of variation is a
"space" ,
which in turn is an object in a category of space,
which will be rather more determinate than "category" in general).
An extensive quantity type is a covariant functor from a category
of space, preserving finite coproducts, to a linear category (=
one in which coproducts and products coincide). A distribution,
a measure, a current, a homology class on a sum domain is
given by a tuple of such , one on each summand ; thus all these
are elements of extensive quantity types. An intensive quantity
type is a contravariant functor from a category of space which
also takes coproducts to products ; but more: intensive quantities
usually act linearly on extensive quantities, lending them (not only
a linear but also a) multiplicative structure which is also preserved
by the contravariant functorality. For example, functions act as
densities on distributions and measures, similarly differential
forms act on currents, and cohomology classes act on homology classes.
Often intensive quantity types are representable and related extensive
types are definable as linear duals ( Riesz, Pontrjagin, Eilenberg Mac
Lane etc),but these are not the only possibilities.
A fundamental example of a linear "category" is the 2category of
all categories with coproducts; it has an obvious abstraction functor
to the linear category of commutative monoids, by taking isomorphism
classes. Part of the idea of Ktheory and of Khomology and of
the "nonlinear Ktheory" which I with Schanuel and others have been
pursuing under the name of "objective number theory" is that it is
useful to "objectify" quantities by lifting their type across this
abstraction functor.
The most fundamental measure of a thing is the thing itself. But measures
can be pushedforward ( a common colloquial expression for the covariant
functorality of extensive quantity).However pullback is more familiar
(already in 1844 Grassmann complained that intensives were more familiar
than Ausdehnungen) : On any category with pullbacks, there is the
contravariant functor to cat which takes the "slice" categories
of each object.This functor is commonly viewed as consisting of
"functions", namely functions whose values are the fibers. Indeed if the
category satisfies suitable conditions , this will be an intensive
quantity type. But what extensives will it act on ?
On any category at all the slice categories constitute an even simpler
COVARIANT catvalued functor, simply composing the transforming map
following the structural map to define the new structural map. It is
often appropriate to consider that the structural map distributes
the total in the base ( though distributions usually do not have values
at points, they do often have totals ) and that the mentioned composing
pushes the distribution forward. Indeed if the category has coproducts,
this naive pushing forward is automatically linear.( But even though the
slice categories have terminal objects, these are not preservedby
the relevant extensive functorality.)
THUS A CATEGORY WITH COPRODUCTS DEFINES
AN EXTENSIVE QUANTITY TYPE ON ITSELF
IF AND ONLY IF IT IS AN EXTENSIVE CATEGORY.
If we call lextensive any extensive category with (finite) limits, then
also by pullback, the intensives act on the extensives in a way that
satisfies all reasonable functoralities, including the crucial CCR
or "projection formula". More exactly, note that there will typically be
lots of subcategories of such a "category of space" which are extensive
but do not themselves have products or even a terminal object ;
it suffices to be closed under sums and summands. Namely the empty
space together with all spaces of dimension exactly 7. Any such
subcategory A defines the extensive quantity type "distributions
of Adimensional spaces " in each base space X, namely the subslice
category. Given A and B two extensive subcategories, there is an
objective intensive quantity type which acts roughly as " BA dimensional
cohomology" namely for any X consider the category of all those spaces
over X such that for any space over X whose total is in A, the pullback
has total in B ; this pulls back along any change of X.
Of course many extensive subcategories of a (lextensive) category of space
will have products, for example (as Joyal pointed out to me in 1984)
if A=B is the "compact" objects , thus defining the extensive notion
of distributing a compact object in a base space, the corresponding
intensive quantities which act on these are the PROPER fiberings
.
Date: Thu, 5 Sep 1996 14:11:33 0300 (ADT)
Subject: Correction to paper  distributive is not weakly distributive
Date: Tue, 3 Sep 1996 16:13:35 0400
From: Robert A. G. Seely
The following notice and discussion amplifies some recent remarks made
by Robin Cockett on the CATEGORIES list.
We wish to announce a correction to a statement in the paper
Weakly distributive categories
by J.R.B. Cockett and R.A.G. Seely
An error in Proposition 3.1, where we claimed that distributive
categories are weakly distributive, was found in proof. The result is
totally incorrect: a distributive category is a cartesian weakly
distributive category if and only if it a preorder. (Note: a weakly
distributive category may be cartesian  by which we just mean the
tensor and cotensor ("par") are cartesian product and coproduct
respectively  without being a preorder; it is the distributivity that
causes the collapse.)
In particular, any distributive category which satisfies equation (13):
\delta^R_R
(A+B)x(C+D) > A+(Bx(C+D))
 
\delta^L_L  
v 
((A+B)xC)+D  1 + \delta^L_L
 
\delta^R_R + 1  
v a v
(A+(BxC))+D > A+((BxC)+D)
(where we write x for the tensor, + for the cotensor (par),
and 1 for identity)
for the choice of weak distributions described in the paper is
immediately a preorder. This because in that diagram if A=D=1 and B=C=0
then, up to equivalence, we obtain for the two sides of diagram the
coproduct embeddings of 1 into 1+1. This suffices to cause collapse.
The argument can be modified to show that in any distributive category
which is simultaneously weakly distributive (no matter how the weak
distributions are defined), Boolean negation must have a fixed point.
This also suffices to cause collapse.
A consequence of this observation is that the categorical proof theory
of notnecessarilyintuitionist AND/OR logic is somewhat subtle. In the
absence of any connective for implication, there is no apparent a priori
reason not to have multipleconclusion sequents; let's see what this
yields. We start with the premise that a good semantics for AND/OR
logic ought to be a polycategory; in particular, that the morphisms
interpreting the following two derivations must be equal. (That these
are equal is a consequence of the polycategory definition, but you can
judge them on their own merits if you like. This type of permutation of
cuts is pretty standard, and categorical cut elimination then would
demand that they be equal.)
(Notation: I use > for the sequent turnstile, and x and + for AND and
OR. The interpretation of the commas is, as is usual in such logics,
AND on the left and OR on the right, so there are evident identity maps
representing A,B > AxB and A+B > A,B. All deduction steps are cuts.
The cut rule is XX,A > YY and WW > A,UU entail XX,WW > YY,UU
and variants via exchange.)
B,C > BxC A+B > A,B

A+B,C > A,BxC C+D > C,D

A+B,C+D > A,BxC,D
B,C > BxC C+D > C,D

= B,C+D > BxC,D A+B > A,B

A+B,C+D > A,BxC,D
But here's the catch  with the obvious interpretation, these come out
different in SETS: think of the image of a pair in (A+B)x(C+D),
where a \in A and d \in D. For the top map, this is mapped to a,
whereas for the bottom map it is mapped to d. This is just our equation
(13) again, so the point of our initial comment is that in any
distributive category, with any interpretation, these two maps are equal
iff the category is a preorder.
This is a pretty "stripped down" example  it seems that categorical cut
elimination is inconsistent with using distributive categories for
AND/OR logic and general sequents. This problem is averted of course if
one restricts oneself to "intuitionist" sequents (with the right of the
turnstile restricted to single formulas), but then this result may be
seen as indicating how the folkloric result concerning the collapse of
categorical proof theory for classical logic (Joyal) doesn't really
depend on very much structure  note that we have assumed no structure
rules beyond cut, and the linear versions of the AND/OR sequent rules;
the collapse just needs multipleconclusion sequents and distributivity.
It is interesting to note, however, that by carefully choosing the weak
distributions one can construct a cartesian weakly distributive category
from an elementary distributive category by simply passing to the
Kleisli category of the ``exception monad'' E(X) = X+1. So, for example,
although SETS is not weakly distributive itself, POINTED_SETS is.
The error means, of course, that all discussion in the paper of
nonposetal distributive categories as examples of weakly distributive
categories must be discounted. This mainly affects the Introduction and
Section 3, where Proposition 3.1 must be restated as indicated above,
and the surrounding text must take this restatement into account. In
particular, Theorem 3.3, although still correct, ought to be stengthened
to state that a cartesian weakly distributive category is a preorder if
and only if it has a strict initial object.
A version of this paper which contains a rewritten Introduction and
Section 3 may be found on rags' WWW home page at this URL:
.
These comments will also appear in the published version of the paper
(to appear in JPAA).
Finally, the inevitable controversy about terminology: we have decided
to continue calling these categories "weakly distributive", since we
have done so for so long and in so many places. Besides, Hyland and
dePaiva had arrived at the same name for the "weak distributivities",
independently, and at the same time. But we keep an open mind about
these matters: if another name seems to have nearuniversal approval, we
will adopt it too. The most promising seems to be Barr's suggestion of
"linearly distributive". Indeed, had that suggestion been made in 1991,
we might have adopted it then (it certainly beats "dissociative
categories"!)
Robin Cockett
Robert Seely
(for ftp help: rags@math.mcgill.ca)