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# Edit detail for Limits and Colimits revision 1 of 5

 1 2 3 4 5 Editor: Time: 2008/08/14 17:43:50 GMT-7 Note: Products and Sums

changed:
-
**Product** is a limit in the sense of category theory. Given a set of domains
X, Y, ... it constructs a new domain and a function (called 'project') from
the new domain to each domain such that <em>for any other domain A and functions
f:A->X, g:A->Y, ... there exists a unique function called their 'product' from
A into the new domain which commutes with the project functions</em>.

"reference":http://en.wikipedia.org/wiki/Product_%28category_theory%29

\begin{aldor}[limits]
#pile
#include "axiom"
Product(X:Type,Y:Type): with
product: (A:Type, A->X,A->Y) -> (A->%)
project: % -> X
project: % -> Y
Rep == Record(a:X,b:Y)
import from Rep
--
project(x:%):X == rep(x).a
project(x:%):Y == rep(x).b
product(A:Type,f:A->X,g:A->Y):(A->%) ==
(x:A):% +-> per [f(x),g(x)]
--
Product(X:Type,Y:Type,Z:Type): with
product: (A:Type, A->X,A->Y,A->Z) -> (A->%)
project: % -> X
project: % -> Y
project: % -> Z
Rep == Record(a:X,b:Y,c:Z)
import from Rep
--
project(x:%):X == rep(x).a
project(x:%):Y == rep(x).b
project(x:%):Z == rep(x).c
product(A:Type,f:A->X,g:A->Y,h:A->Z):(A->%) ==
(x:A):% +-> per [f(x),g(x),h(x)]
\end{aldor}

**Sum** is a co-limit in the sense of category theory. Given a set of domains
X, Y, ... it constructs a new domain and a function (called 'inject') from
each domain to the new domain such that <em>for any other domain A and functions
f:X->A, g:Y->A, ... there exists a unique function called their 'sum' from
the new domain into A which commutes with the inject functions</em>.

"reference":http://en.wikipedia.org/wiki/Coproduct

\begin{aldor}[colimits]
#pile
#include "axiom"
Sum(X:Type,Y:Type): with
sum: (A:Type, X->A,Y->A) -> (% -> A)
-- Given two functions, each with one of the domains in this Sum
-- and having a common co-domain A: returns the unique function with
-- domain Sum and co-domain A
inject: X -> %
inject: Y -> %
Rep == Union(a:X,b:Y)
import from Rep
--
inject(x:X):% == per [a==x]
inject(y:Y):% == per [b==y]
sum(A:Type,f:X->A,g:Y->A):(%->A) ==
(x:%):A +->
rep(x) case a => f(rep(x).a)
rep(x) case b => g(rep(x).b)
never
--
Sum(X:Type,Y:Type,Z:Type): with
sum: (A:Type, X->A,Y->A,Z->A) -> (% -> A)
-- Given three functions, each with one of the domains in this Sum
-- and having a common co-domain A: returns the unique function with
-- domain Sum and co-domain A
inject: X -> %
inject: Y -> %
inject: Z -> %
Rep == Union(a:X,b:Y,c:Z)
import from Rep
--
inject(x:X):% == per [a==x]
inject(y:Y):% == per [b==y]
inject(z:Z):% == per [c==z]
sum(A:Type,f:X->A,g:Y->A,h:Z->A):(%->A) ==
(x:%):A +->
rep(x) case a => f(rep(x).a)
rep(x) case b => g(rep(x).b)
rep(x) case c => h(rep(x).c)
never
\end{aldor}

Limits and co-limits are dual concepts in category theory.
Notice in particular how the construction of Product and
Sum above implement that duality. I am especially interested
in how the duality between 'rep' and 'per' is involved in
these constructions.



Product is a limit in the sense of category theory. Given a set of domains X, Y, ... it constructs a new domain and a function (called project) from the new domain to each domain such that for any other domain A and functions f:A->X, g:A->Y, ... there exists a unique function called their product from A into the new domain which commutes with the project functions.

reference

aldor
#pile
#include "axiom"
Product(X:Type,Y:Type): with
product: (A:Type, A->X,A->Y) -> (A->%)
project: % -> X
project: % -> Y
Rep == Record(a:X,b:Y)
import from Rep
--
project(x:%):X == rep(x).a
project(x:%):Y == rep(x).b
product(A:Type,f:A->X,g:A->Y):(A->%) ==
(x:A):% +-> per [f(x),g(x)]
--
Product(X:Type,Y:Type,Z:Type): with
product: (A:Type, A->X,A->Y,A->Z) -> (A->%)
project: % -> X
project: % -> Y
project: % -> Z
Rep == Record(a:X,b:Y,c:Z)
import from Rep
--
project(x:%):X == rep(x).a
project(x:%):Y == rep(x).b
project(x:%):Z == rep(x).c
product(A:Type,f:A->X,g:A->Y,h:A->Z):(A->%) ==
(x:A):% +-> per [f(x),g(x),h(x)]
aldor
   Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/limits.as using AXIOM-XL compiler and
options
-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y $AXIOM/algebra Use the system command )set compiler args to change these options. #1 (Warning) Deprecated message prefix: use ALDOR_' instead of _AXL' Compiling Lisp source code from file ./limits.lsp Issuing )library command for limits Reading /var/zope2/var/LatexWiki/limits.asy Product is now explicitly exposed in frame initial Product will be automatically loaded when needed from /var/zope2/var/LatexWiki/limits Sum is a co-limit in the sense of category theory. Given a set of domains X, Y, ... it constructs a new domain and a function (called inject) from each domain to the new domain such that for any other domain A and functions f:X->A, g:Y->A, ... there exists a unique function called their sum from the new domain into A which commutes with the inject functions. reference aldor #pile #include "axiom" Sum(X:Type,Y:Type): with sum: (A:Type, X->A,Y->A) -> (% -> A) -- Given two functions, each with one of the domains in this Sum -- and having a common co-domain A: returns the unique function with -- domain Sum and co-domain A inject: X -> % inject: Y -> % == add Rep == Union(a:X,b:Y) import from Rep -- inject(x:X):% == per [a==x] inject(y:Y):% == per [b==y] sum(A:Type,f:X->A,g:Y->A):(%->A) == (x:%):A +-> rep(x) case a => f(rep(x).a) rep(x) case b => g(rep(x).b) never -- Sum(X:Type,Y:Type,Z:Type): with sum: (A:Type, X->A,Y->A,Z->A) -> (% -> A) -- Given three functions, each with one of the domains in this Sum -- and having a common co-domain A: returns the unique function with -- domain Sum and co-domain A inject: X -> % inject: Y -> % inject: Z -> % == add Rep == Union(a:X,b:Y,c:Z) import from Rep -- inject(x:X):% == per [a==x] inject(y:Y):% == per [b==y] inject(z:Z):% == per [c==z] sum(A:Type,f:X->A,g:Y->A,h:Z->A):(%->A) == (x:%):A +-> rep(x) case a => f(rep(x).a) rep(x) case b => g(rep(x).b) rep(x) case c => h(rep(x).c) never aldor  Compiling FriCAS source code from file /var/zope2/var/LatexWiki/colimits.as using AXIOM-XL compiler and options -O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y$AXIOM/algebra
Use the system command )set compiler args to change these
options.
#1 (Warning) Deprecated message prefix: use ALDOR_' instead of _AXL'
Compiling Lisp source code from file ./colimits.lsp
Issuing )library command for colimits
/var/zope2/var/LatexWiki/colimits
Limits and co-limits are dual concepts in category theory. Notice in particular how the construction of Product and Sum above implement that duality. I am especially interested in how the duality between rep and per is involved in these constructions.