Cedric Beny
Apr28-04, 01:29 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi,\n\nI am curious about what strict n-category we can generate by\nspecifying a finite set of objects, 1-morphisms, ..., n-morphisms.\n\nBy "generate" I mean completing the category by all possible\ncompositions of k-arrows, k = 1...n. (no two different compositions\nbeing (strictly) equal.)\n\nIt looks like it can give rise to quiet complex structures.\n\nThis has probably been studyied. Any pointer?\n\n\nThe link to physics is as follows: when thinking about fundamental\nphysics one always face methodological questions. Mathematics is the\nmain tool used in physics, Nevertheless physicists rarely study formal\nlogic and have a poor idea of what mathematics are, fundamentally. (I\nfeel this is also true of most mathematicians.) Since we use math so\nmuch it seems crucial to me to be able to justify the usage we make of\nit.\n\nOne can study formal logic, which tells us that all of mathematics can\nbe reduced to algebra (symbols manipulation.) Nevertheless the basic\naxioms, deduction rules, and procedures one needs in order to\nimplement even the simplest mathematical concept (including category\ntheory) seem mysterious and cumbersome and don\'t enlighten much about\nwhy it has to be like that.\n\nIt seems to me that it would be of great interest to have a more\nstraightforward way to formalize mathematics. It would give more\nfreedom and power to the user (the physicist), it would allow to make\nmore efficient computer tools, and it may also help understanding how\nphysics influences the power of mathematical modelling. Indeed\ncomputation is a special case of mathematical reasoning, and from\nquantum computing we know that one cannot think about computation\nwithout making some physical assumptions. The same must be true of\nlogical reasoning and mathematics.\n\nNow John Baez\'s vulgarization papers contagiously spread the feeling\nthat n-categories could be a powerful tool for physical modelling.\n\nI wonder if one could formally define an n-category without having to\ngo through the folklore of formal logic. Indeed one usually define a\n1-category by specifying equations between arrows. Those equations\nhave to be defined in an external language (usually some kind of\nenglish informal first order logic.) But we know that instead we may\nreplace equations by 2-arrows between 1-arrows. So somehow it seems\nthat the 2-category provides part of the language needed to define a\n1-category. Can the rest of the language be found in an n-category?\n\nThe idea would go as follows: in order to define a category (a theory)\none first enumerate a finite (and small) number of k-arrows, k=0...n.\n(the axioms) and eventually promotes some couple of n-arrows to\nisomorphisms (or something like that.) This small set of "objects"\nwould then define a full category by composition of the "axioms". This\ncould be implemented on a computer directly (modulo a few technical\nissues.)\n\nDoes it make sense?\n\nCedric\n\n\n---\nCedric Beny\nGrad Student\nPhysics\nUniversity of Waterloo\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi,
I am curious about what strict n-category we can generate by
specifying a finite set of objects, 1-morphisms, ..., n-morphisms.
By "generate" I mean completing the category by all possible
compositions of k-arrows, k = 1...n. (no two different compositions
being (strictly) equal.)
It looks like it can give rise to quiet complex structures.
This has probably been studyied. Any pointer?
The link to physics is as follows: when thinking about fundamental
physics one always face methodological questions. Mathematics is the
main tool used in physics, Nevertheless physicists rarely study formal
logic and have a poor idea of what mathematics are, fundamentally. (I
feel this is also true of most mathematicians.) Since we use math so
much it seems crucial to me to be able to justify the usage we make of
it.
One can study formal logic, which tells us that all of mathematics can
be reduced to algebra (symbols manipulation.) Nevertheless the basic
axioms, deduction rules, and procedures one needs in order to
implement even the simplest mathematical concept (including category
theory) seem mysterious and cumbersome and don't enlighten much about
why it has to be like that.
It seems to me that it would be of great interest to have a more
straightforward way to formalize mathematics. It would give more
freedom and power to the user (the physicist), it would allow to make
more efficient computer tools, and it may also help understanding how
physics influences the power of mathematical modelling. Indeed
computation is a special case of mathematical reasoning, and from
quantum computing we know that one cannot think about computation
without making some physical assumptions. The same must be true of
logical reasoning and mathematics.
Now John Baez's vulgarization papers contagiously spread the feeling
that n-categories could be a powerful tool for physical modelling.
I wonder if one could formally define an n-category without having to
go through the folklore of formal logic. Indeed one usually define a
1-category by specifying equations between arrows. Those equations
have to be defined in an external language (usually some kind of
english informal first order logic.) But we know that instead we may
replace equations by 2-arrows between 1-arrows. So somehow it seems
that the 2-category provides part of the language needed to define a
1-category. Can the rest of the language be found in an n-category?
The idea would go as follows: in order to define a category (a theory)
one first enumerate a finite (and small) number of k-arrows, k=0...n.
(the axioms) and eventually promotes some couple of n-arrows to
isomorphisms (or something like that.) This small set of "objects"
would then define a full category by composition of the "axioms". This
could be implemented on a computer directly (modulo a few technical
issues.)
Does it make sense?
Cedric
---
Cedric Beny
Grad Student
Physics
University of Waterloo
I am curious about what strict n-category we can generate by
specifying a finite set of objects, 1-morphisms, ..., n-morphisms.
By "generate" I mean completing the category by all possible
compositions of k-arrows, k = 1...n. (no two different compositions
being (strictly) equal.)
It looks like it can give rise to quiet complex structures.
This has probably been studyied. Any pointer?
The link to physics is as follows: when thinking about fundamental
physics one always face methodological questions. Mathematics is the
main tool used in physics, Nevertheless physicists rarely study formal
logic and have a poor idea of what mathematics are, fundamentally. (I
feel this is also true of most mathematicians.) Since we use math so
much it seems crucial to me to be able to justify the usage we make of
it.
One can study formal logic, which tells us that all of mathematics can
be reduced to algebra (symbols manipulation.) Nevertheless the basic
axioms, deduction rules, and procedures one needs in order to
implement even the simplest mathematical concept (including category
theory) seem mysterious and cumbersome and don't enlighten much about
why it has to be like that.
It seems to me that it would be of great interest to have a more
straightforward way to formalize mathematics. It would give more
freedom and power to the user (the physicist), it would allow to make
more efficient computer tools, and it may also help understanding how
physics influences the power of mathematical modelling. Indeed
computation is a special case of mathematical reasoning, and from
quantum computing we know that one cannot think about computation
without making some physical assumptions. The same must be true of
logical reasoning and mathematics.
Now John Baez's vulgarization papers contagiously spread the feeling
that n-categories could be a powerful tool for physical modelling.
I wonder if one could formally define an n-category without having to
go through the folklore of formal logic. Indeed one usually define a
1-category by specifying equations between arrows. Those equations
have to be defined in an external language (usually some kind of
english informal first order logic.) But we know that instead we may
replace equations by 2-arrows between 1-arrows. So somehow it seems
that the 2-category provides part of the language needed to define a
1-category. Can the rest of the language be found in an n-category?
The idea would go as follows: in order to define a category (a theory)
one first enumerate a finite (and small) number of k-arrows, k=0...n.
(the axioms) and eventually promotes some couple of n-arrows to
isomorphisms (or something like that.) This small set of "objects"
would then define a full category by composition of the "axioms". This
could be implemented on a computer directly (modulo a few technical
issues.)
Does it make sense?
Cedric
---
Cedric Beny
Grad Student
Physics
University of Waterloo