An elementary question

An elementary question

[Dana Scott]

The category of posets (= partially ordered sets) and monotone
maps is often used as an easy example -- different from the category
of sets -- that has products, coproducts, and is cartesian closed
but not a topos.

Let P and Q be two posets.  Define (P (<) Q) as the modified
coproduct where all the elements of P are made less than all the
elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
definition as a functor in the category of posets?



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Re: An elementary question

[alex]

I hope so much that I do not say something monumentally stupid. I am
just a student and do not want to attract negative attention, but is
there not, for any P, Q a whole partial order of "generalized
coproducts", in the sense that these consist of coproducts plus some
relations of the form (p \leq q) or vice versa, where these coproducts
are ordered by saying that one "generalized coproduct" is "smaller" than
the other iff from (p \leq q) in the second it follows that this
relation also holds in the first one? So the modified coproduct you
describe would be the least element in this partial order of generalized
coproducts. Furthermore, this modified coproduct, its dual and the
normal coproduct would be the only ones that could actually made into a
functor from Pos*Pos to Pos, since these are the only ones describable
through of the generalized coproducts describable through a universal
property: mapping each two partial orders to the lowest element in the
partial order of the lattice in the first case, to the highest one in
the second, and to the coproduct in the third. Of course it all depends
on what you consider nice, but this is the nicest I can see, at least.

I hope this is at least halfway correct (although I think it should be,
since I spent some time recently contemplating a similar structure I
found in a pet project of mine) and comprehensibly formulated. English
is not my first language and describing mathematical structures in
writing without the use of latex is a bit hard for me. I can work it out
a bit more formal in a latex file if you want.

Hope I could help,

Alex

On 13.08.2017 21:55, Dana Scott wrote:
> The category of posets (= partially ordered sets) and monotone
> maps is often used as an easy example -- different from the category
> of sets -- that has products, coproducts, and is cartesian closed
> but not a topos.
>
> Let P and Q be two posets.  Define (P (<) Q) as the modified
> coproduct where all the elements of P are made less than all the
> elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
> definition as a functor in the category of posets?
>



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Re: An elementary question

[Patrik Eklund]

What would be the practical applications of that construction?

If P and Q are two-pointed, truth in one is falser than false in the
other. If P and Q are powersets, the full set in one is more empty than
the empty in the other.

Even the mathematical justfication of that construction is a bit far
fetched, isn't it?

However, we might have those posets as economists and mathematicians.
The best economist is worse than the worst mathematician, or the best
mathematician is worse than the worst economist. Such attitudes do exist
but they are not very practical, are they?

The general intuition, however, if my intuition about what the general
intuition is in this situation is correct or at least ordered, is
interesting when going towards ordering categorical objects based on
structure the objects respectively embrace. Monoidal categories
involving that tensor are interesting, and we already have a fair
understanding about what they can do for us in many application areas,
so introducing non-commutativity via modified coproducts sounds like
something that might be already in the making. I wouldn't be surprised
at all if that indeed is the case.

Note also that the objects themselves are not the whole story but that
"orderal" or "posetal" category may turn out to be a most interesting
underlying category for many applications involving applications. We
indeed already see that in the case of monoidal categories.

Looking forward to formulations of monoidal-posetal categories!

Best,

Patrik



On 2017-08-13 22:55, Dana Scott wrote:
> The category of posets (= partially ordered sets) and monotone
> maps is often used as an easy example -- different from the category
> of sets -- that has products, coproducts, and is cartesian closed
> but not a topos.
>
> Let P and Q be two posets.  Define (P (<) Q) as the modified
> coproduct where all the elements of P are made less than all the
> elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
> definition as a functor in the category of posets?
>
>

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Re: An elementary question

[Paul Blain Levy]

Dear Dana,

Let (P_i | i in I) be a family of posets and < a well-ordering of I.

The <-lexicographic sum of (P_i | i in I) is given by Sum_{i in I} P_i
with (i,x) <= (j,y) if i=j and x<=y or i<j.

It may be viewed as a representing object as follows.

A cocone (f_i | i in I) in Poset from (P_i | i in I) to a poset V is
"<-lexicographic" when

for all i < j and  x in P_i and y in P_j we have f_i(x) <= f_i(y).    (*)

So we obtain a right Poset-module, i.e. functor Poset --> Set

sending V to the set of <-lexicographic cocones from (P_i | i in I) to V.

The <-lexicographic sum is a representing object for this functor.

Therefore, for fixed (I,<), it extends uniquely to a functor Poset^I -->
Poset making the representation natural.

However, property (*) is not "categorical" in the sense of making sense
in an arbitrary category.  So this probably doesn't answer your question.

Paul




On 13/08/17 20:55, Dana Scott wrote:
> The category of posets (= partially ordered sets) and monotone
> maps is often used as an easy example -- different from the category
> of sets -- that has products, coproducts, and is cartesian closed
> but not a topos.
>
> Let P and Q be two posets.  Define (P (<) Q) as the modified
> coproduct where all the elements of P are made less than all the
> elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
> definition as a functor in the category of posets?
>

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Re: An elementary question

[Mike Stay]

On Sun, Aug 13, 2017 at 10:42 PM, Patrik Eklund <peklund@cs.umu.se> wrote:
> What would be the practical applications of that construction?

Scan lines on your monitor screen: the rightmost point on the first
line has an address or index that is less than the leftmost point on
the second.
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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Re: An elementary question

[Robin Cockett]

A great question ... and  I do not have an answer for it.  However,  regarding P (<) Q as a (posetal) module between the posets it does have the striking property that it is the final module!


-robin

________________________________
From: Dana Scott <scott@cs.cmu.edu>
Sent: Sunday, August 13, 2017 1:55:11 PM
To: categories@mta.ca
Subject: categories: An elementary question

The category of posets (= partially ordered sets) and monotone
maps is often used as an easy example -- different from the category
of sets -- that has products, coproducts, and is cartesian closed
but not a topos.

Let P and Q be two posets.  Define (P (<) Q) as the modified
coproduct where all the elements of P are made less than all the
elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
definition as a functor in the category of posets?


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Re: An elementary question

[Patrik Eklund]

Right.

So a bit like the carriage return at the of a line in written text. This
would be an application for the structure of the whole text.

Maybe so, but if that text (maybe shown on a monitor) is about drug
treatment of hypertension, we have a poset of events leading to blood
vessels being too stiff because angiotensin receptors are too many.
Receptor blockers is one treatment. Another is diuretics affecting
unfavourable water retention, which makes the blood volume unnecesary
large. Such water retention disturbance is also the result of a poset of
events leading to that disturbance.

Now, and unfortunately, medical mathematics does not embrace these
things, and in fact, when medication with both angiotensin receptor
blockers and diuretics, it is not known with which to start and wait
before starting with the other. Decades ago it was all about diuretics,
but now it's different. Why? Certainly not because of a better
understanding of the intertwining of those posets.

So in that text, readable on the monitor, there are indeed partial
orders, and if you, Mike, want to focus on the carriage return, fine.

Patrik

PS We've written a paper on the order of interventions, based on using
modules, non-commutativity and over monoidal closed categories. If
anybody is interested in a copy, just let me know.



On 2017-08-14 21:43, Mike Stay wrote:
> On Sun, Aug 13, 2017 at 10:42 PM, Patrik Eklund <peklund@cs.umu.se>
> wrote:
>> What would be the practical applications of that construction?
>
> Scan lines on your monitor screen: the rightmost point on the first
> line has an address or index that is less than the leftmost point on
> the second.
> --
> Mike Stay - metaweta@gmail.com
> http://www.cs.auckland.ac.nz/~mike
> http://reperiendi.wordpress.com


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Re: Re: An elementary question

[Vaughan Pratt]

> On 2017-08-13 22:55, Dana Scott wrote:
>>  Does (P (<) Q) have a nice categorical
>> definition as a functor in the category of posets?

Yes [3].

On 08/13/17 9:42 PM, Patrik Eklund wrote:
> What would be the practical applications of that construction?

Sequential composition, aka concatenation, aka ordinal addition [1,2].
Second diagram of Figure 1, /et seq,/ [3] answers Dana's question more
generally for V-categories, with the category of preordered sets as the
case V = 2.  The simplified proof of Theorem 9 notwithstanding, our
paper could benefit today from a more pedagogically sensitive treatment.

Vaughan Pratt

[1] G. Birkhoff. An extended arithmetic. Duke Mathematical Journal,
3(2), June 1937.

[2] G. Birkhoff. Generalized arithmetic. Duke Mathematical Journal,
9(2), June 1942.

[3]  Casley, R.T., Crew, R.F., Meseguer, J., and Pratt, V.R., ``Temporal
Structures'', Proc. Category Theory and Computer Science 1989, ed. D.
Pitt et al, LNCS 389, 21-51, Springer-Verlag, 1989. Revised journal
version in Mathematical Structures in Computer Science, Volume 1:2,
179-213, July 1991.  A version missing some figures is downloadable as
http://boole.stanford.edu/pub/man.pdf, the missing figures should be in
the older version http://boole.stanford.edu/pub/man90.pdf.


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Re: An elementary question

[Mike Stay]

On Mon, Aug 14, 2017 at 10:50 PM, Patrik Eklund <peklund@cs.umu.se> wrote:
> So in that text, readable on the monitor, there are indeed partial orders,
> and if you, Mike, want to focus on the carriage return, fine.

The rendering of text or graphics on a screen made of scan lines is
just one easy example of serialization, a nontrivial practical task on
which millions of man hours have been spent.  The lexicographic
ordering is well-defined and well-motivated, and it's reasonable to
ask for a nice characterization.
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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Re: An elementary question

[Joachim Kock]

> Let P and Q be two posets.  Define (P (<) Q) as the modified
> coproduct where all the elements of P are made less than all the
> elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
> definition as a functor in the category of posets?

Hi Dana,

unless I misunderstand the question, (<) is the join operation,
which makes sense more generally for categories, and more
generally for simplicial sets, or augmented simplicial sets.
Here it is simply the cocontinuous extension (in each variable)
of ordinal sum (i.e. the Day convolution tensor product of
ordinal sum).

(It plays an crucial role in the development of higher category
theory, thanks to the discovery by Andr?? Joyal that slice and
coslice can be defined as right adjoints to join with a fixed
object.  (These are generalised slices and coslices, with the
classical notions corresponding to the cases of join with a point.)
This is the construction that allows for the definition of limits
and colimits in infinity-categories, and hence the starting point
for generalising category theory from categories to infinity-
categories.)

[A. Joyal: Quasi-categories and Kan complexes, JPAA 2002]

Cheers,
Joachim.



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Re: An elementary question

[Branko Nikolić]

Dear Dana,

I'm not sure if the following construction is the one you are looking
for, but it's the only categorical (in fact 2-categorical) description
I could think of, and it is related to Robin Cockett's answer.

If you view posets as categories (ncatlab.org/nlab/show/partial+order)
then P and Q can be seen as objects of the 2-category Cat of
categories, functors and natural transformations. Furthermore, instead
of functors we can look at modules (aka profunctors or distributors,
ncatlab.org/nlab/show/profunctor) and their morphisms, to get the
bicategory Mod.
The situation you described corresponds to the terminal module between
Q and P (1-cell in Mod which is a terminal object in the hom-category
Mod(Q,P)). The poset you obtain by taking the "modified coproduct" has
the universal property of being the lax colimit of that 1-cell...
A more general construction is explained here
http://maths.mq.edu.au/~street/Pow.fun.pdf

Best regards,
Branko

On 16 Aug 2017 11:50 pm, "Joachim Kock" <kock@mat.uab.cat> wrote:
>
> Let P and Q be two posets.  Define (P (<) Q) as the modified
> coproduct where all the elements of P are made less than all the
> elements of Q.  QUESTION. Does (P (<) Q) have a nice categorical
> definition as a functor in the category of posets?


Hi Dana,

unless I misunderstand the question, (<) is the join operation,
which makes sense more generally for categories, and more
generally for simplicial sets, or augmented simplicial sets.
Here it is simply the cocontinuous extension (in each variable)
of ordinal sum (i.e. the Day convolution tensor product of
ordinal sum).

(It plays an crucial role in the development of higher category
theory, thanks to the discovery by Andr?? Joyal that slice and
coslice can be defined as right adjoints to join with a fixed
object.  (These are generalised slices and coslices, with the
classical notions corresponding to the cases of join with a point.)
This is the construction that allows for the definition of limits
and colimits in infinity-categories, and hence the starting point
for generalising category theory from categories to infinity-
categories.)

[A. Joyal: Quasi-categories and Kan complexes, JPAA 2002]

Cheers,
Joachim.


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