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? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (9)
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alex -
Branko Nikolić -
Dana Scott -
Joachim Kock -
Mike Stay -
Patrik Eklund -
Paul Blain Levy -
Robin Cockett -
Vaughan Pratt