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# Edit detail for Limits and Colimits revision 5 of 5

 1 2 3 4 5 Editor: Bill Page Time: 2008/12/17 08:57:05 GMT-8 Note: Equalizer

changed:
-**Product** is a limit in the sense of category theory. Given a set of domains
Limits

**Product** is a limit in the sense of category theory. Given a set of domains

changed:
-**Sum** is a co-limit in the sense of category theory. Given a set of domains
**Equalizer** is a limit. Given two domains X, Y and two functions
f:X->Y, g:X->Y, the Equalizer is a new domain E and a function
eq:E->X for which f eq = g eq such that <em> for any other domain M
and function m:M->X for which f m = g m, there exists a unique
function u:M->E such that u eq = m</em>.

"reference":http://en.wikipedia.org/wiki/Equaliser_(mathematics)#In_category_theory

In Spad Equalizer can be defined as a 'SubDomain' as follows:
)abbrev domain EQLZR Equalizer
Equalizer(X:SetCategory,Y:SetCategory,f:X->Y, g:X->Y): SetCategory with
eq: % -> X
== SubDomain(X,f #1 = g #1) add
--
eq(x:%):X == x pretend X

)abbrev package EQLZF EqualizerFunctions
EqualizerFunctions(X:SetCategory,Y:SetCategory,f:X->Y,g:X->Y, M:SetCategory,m:M->X): with
u: M->Equalizer(X,Y,f,g)
--
u(x:M):Equalizer(X,Y,f,g) == m(x) pretend Equalizer(X,Y,f,g)

\begin{axiom}
)show Equalizer(Float,Float,sin,cos)
\end{axiom}

Co-limits

**Sum** is a co-limit in the sense of category theory. Given a set of domains



## Limits

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"
Product2(X:SetCategory,Y:SetCategory): with
construct: (X,Y) -> %
coerce: % -> OutputForm
product: (A:Type, A->X,A->Y) -> (A->%)
project1: (%) -> X
project2: (%) -> Y
Rep == Record(a:X,b:Y)
import from Rep
--
construct(x:X,y:Y):% == per [x,y]
coerce(x:%):OutputForm ==
bracket([coerce(rep(x).a)$X, coerce(rep(x).b)$Y]$List(OutputForm)) project1(x:%):X == rep(x).a project2(y:%):Y == rep(y).b product(A:Type,f:A->X,g:A->Y):(A->%) == (x:A):% +-> per [f(x),g(x)] -- Product3(X:SetCategory,Y:SetCategory,Z:SetCategory): with construct: (X,Y,Z) -> % coerce: % -> OutputForm product: (A:Type, A->X,A->Y,A->Z) -> (A->%) project1: % -> X project2: % -> Y project3: % -> Z == add Rep == Record(a:X,b:Y,c:Z) import from Rep -- construct(x:X,y:Y,z:Z):% == per [x,y,z] coerce(x:%):OutputForm == bracket([coerce(rep(x).a)$X, coerce(rep(x).b)$Y, coerce(rep(x).c)$Z]$List(OutputForm)) project1(x:%):X == rep(x).a project2(y:%):Y == rep(y).b project3(z:%):Z == rep(z).c product(A:Type,f:A->X,g:A->Y,h:A->Z):(A->%) == (x:A):% +-> per [f(x),g(x),h(x)] aldor  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/limits.as using AXIOM-XL compiler and options -O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y$AXIOM/algebra -I $AXIOM/algebra Use the system command )set compiler args to change these options. #1 (Warning) Could not use archive file libaxiom.al'. #2 (Warning) Could not use archive file libaxiom.al'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 4: import from AxiomLib; ............^ [L4 C13] #3 (Error) No meaning for identifier AxiomLib'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 15: import { true: %, false: % } from Boolean; ..................................^ [L15 C35] #4 (Error) No meaning for identifier Boolean'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 17: string: Literal -> %; ........................^.......^ [L17 C25] #5 (Error) No meaning for identifier Literal'. [L17 C33] #6 (Error) There are no suitable meanings for the operator ->'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 18: } from String; .......^ [L18 C8] #8 (Error) No meaning for identifier String'. "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/limits.as", line 21: Product3(X:SetCategory,Y:SetCategory,Z:SetCategory): with ...........^ [L21 C12] #9 (Error) No meaning for identifier SetCategory'. "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/limits.as", line 22: construct: (X,Y,Z) -> % ..................^.^ [L22 C19] #12 (Error) There are 0 meanings for Y' in this context. The possible types were: Y: SetCategory, a local The context requires an expression of type Tuple(Type). [L22 C21] #10 (Error) There are 0 meanings for Z' in this context. The possible types were: Z: SetCategory, a local The context requires an expression of type Tuple(Type). "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/limits.as", line 27: project3: % -> Z ................^ [L27 C17] #11 (Error) There are no suitable meanings for the operator ->'. [L27 C17] #13 (Fatal Error) Too many errors (use -M emax=n' or -M no-emax' to change the limit). The )library system command was not called after compilation. fricas )show Product2(Integer,Float) The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer . Product2 is not the name of a known type constructor. If you want to see information about any operations named Product2 , issue )display operations Product2 Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Float is not the name of a known type constructor. If you want to see information about any operations named Float , issue )display operations Float  fricas )show Product3(Integer,Integer,Integer) The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer . Product3 is not the name of a known type constructor. If you want to see information about any operations named Product3 , issue )display operations Product3 Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Equalizer is a limit. Given two domains X, Y and two functions f:X->Y, g:X->Y, the Equalizer is a new domain E and a function eq:E->X for which f eq = g eq such that for any other domain M and function m:M->X for which f m = g m, there exists a unique function u:M->E such that u eq = m. reference In Spad Equalizer can be defined as a SubDomain as follows: spad )abbrev domain EQLZR Equalizer Equalizer(X:SetCategory,Y:SetCategory,f:X->Y, g:X->Y): SetCategory with eq: % -> X == SubDomain(X,f #1 = g #1) add -- eq(x:%):X == x pretend X )abbrev package EQLZF EqualizerFunctions EqualizerFunctions(X:SetCategory,Y:SetCategory,f:X->Y,g:X->Y, M:SetCategory,m:M->X): with u: M->Equalizer(X,Y,f,g) == add -- u(x:M):Equalizer(X,Y,f,g) == m(x) pretend Equalizer(X,Y,f,g) spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3871883559109604243-25px003.spad using old system compiler. EQLZR abbreviates domain Equalizer ------------------------------------------------------------------------ initializing NRLIB EQLZR for Equalizer compiling into NRLIB EQLZR compiling exported eq :$ -> X
EQLZR;eq;$X;1 is replaced by x Time: 0.01 SEC. (time taken in buildFunctor: 0) ;;; *** |Equalizer| REDEFINED ;;; *** |Equalizer| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor Equalizer Time: 0.01 seconds finalizing NRLIB EQLZR Processing Equalizer for Browser database: --->-->Equalizer(constructor): Not documented!!!! --->-->Equalizer((eq (X %))): Not documented!!!! --->-->Equalizer(): Missing Description ; compiling file "/var/aw/var/LatexWiki/EQLZR.NRLIB/EQLZR.lsp" (written 31 JUL 2013 03:11:54 PM): ; /var/aw/var/LatexWiki/EQLZR.NRLIB/EQLZR.fasl written ; compilation finished in 0:00:00.018 ------------------------------------------------------------------------ Equalizer is now explicitly exposed in frame initial Equalizer will be automatically loaded when needed from /var/aw/var/LatexWiki/EQLZR.NRLIB/EQLZR EQLZF abbreviates package EqualizerFunctions ------------------------------------------------------------------------ initializing NRLIB EQLZF for EqualizerFunctions compiling into NRLIB EQLZF compiling exported u : M -> Equalizer(X,Y,f,g) Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |EqualizerFunctions| REDEFINED ;;; *** |EqualizerFunctions| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor EqualizerFunctions Time: 0 seconds finalizing NRLIB EQLZF Processing EqualizerFunctions for Browser database: --->-->EqualizerFunctions(constructor): Not documented!!!! --->-->EqualizerFunctions((eq (X %))): Not documented!!!! --->-->EqualizerFunctions(constructor): Not documented!!!! --->-->EqualizerFunctions((u ((Equalizer X Y f g) M))): Not documented!!!! --->-->EqualizerFunctions(): Missing Description ; compiling file "/var/aw/var/LatexWiki/EQLZF.NRLIB/EQLZF.lsp" (written 31 JUL 2013 03:11:54 PM): ; /var/aw/var/LatexWiki/EQLZF.NRLIB/EQLZF.fasl written ; compilation finished in 0:00:00.013 ------------------------------------------------------------------------ EqualizerFunctions is now explicitly exposed in frame initial EqualizerFunctions will be automatically loaded when needed from /var/aw/var/LatexWiki/EQLZF.NRLIB/EQLZF fricas )show Equalizer(Float,Float,sin,cos) Equalizer(Float,Float,theMap(FLOAT;sin;2$;13,303),theMap(FLOAT;cos;2$;15,303)) is a domain constructor. Abbreviation for Equalizer is EQLZR This constructor is exposed in this frame. ------------------------------- Operations -------------------------------- ?=? : (%,%) -> Boolean coerce : % -> OutputForm eq : % -> Float hash : % -> SingleInteger latex : % -> String ?~=? : (%,%) -> Boolean hashUpdate! : (HashState,%) -> HashState ## Co-limits Sum is a co-limit in the sense of category theory. Given a set of domains X, Y, ... it constructs a new domain and a function (called inject) from each domain to the new domain such that for any other domain A and functions f:X->A, g:Y->A, ... there exists a unique function called their sum from the new domain into A which commutes with the inject functions. reference fricas p:=[2,3.14]$Product2(Integer,Float)
There are no library operations named Product2
Use HyperDoc Browse or issue
)what op Product2
to learn if there is any operation containing " Product2 " in its
name.
Cannot find a definition or applicable library operation named
Product2 with argument type(s)
Type
Type
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. q:=[2,3.14,"abc"]$Product3(Integer,Float,String)
There are no library operations named Product3
Use HyperDoc Browse or issue
)what op Product3
to learn if there is any operation containing " Product3 " in its
name.
Cannot find a definition or applicable library operation named
Product3 with argument type(s)
Type
Type
Type
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. r:=["a","b","c"]$Product3(String,String,String)
There are no library operations named Product3
Use HyperDoc Browse or issue
)what op Product3
to learn if there is any operation containing " Product3 " in its
name.
Cannot find a definition or applicable library operation named
Product3 with argument type(s)
Type
Type
Type
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. project3 r There are no library operations named project3 Use HyperDoc Browse or issue )what op project3 to learn if there is any operation containing " project3 " in its name. Cannot find a definition or applicable library operation named project3 with argument type(s) Variable(r) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

aldor
#pile
#include "axiom"
Sum2(X:SetCategory,Y:SetCategory): with
coerce: % -> OutputForm
inject1: X -> %
coerce: X -> %
inject2: Y -> %
coerce: Y -> %
Rep == Union(a:X,b:Y)
import from Rep, Integer
--
coerce(x:%):OutputForm ==
rep(x) case a => sub(coerce(rep(x).a)$X,outputForm 1) rep(x) case b => sub(coerce(rep(x).b)$Y,outputForm 2)
never
inject1(x:X):% == per [a==x]
coerce(x:X):% == per [a==x]
inject2(y:Y):% == per [b==y]
coerce(y:Y):% == per [b==y]
-- Note: There is a problem with dependent domains in FriCAS so
-- it is necessary to define 'sum' in a separate package.
Sum2Functions(X:SetCategory,Y:SetCategory,A:SetCategory): with
sum: (X->A,Y->A) -> (Sum2(X,Y) -> 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
-- Requires same Rep as domain Sum2!
macro rep(x) == x pretend Union(a:X,b:Y)
import from Union(a:X,b:Y)
--
sum(f:X->A,g:Y->A):(Sum2(X,Y)->A) ==
(x:Sum2(X,Y)):A +->
rep(x) case a => f(rep(x).a)
rep(x) case b => g(rep(x).b)
never
--
Sum3(X:SetCategory,Y:SetCategory,Z:SetCategory): with
coerce: % -> OutputForm
inject1: X -> %
coerce: X -> %
inject2: Y -> %
coerce: Y -> %
inject3: Z -> %
coerce: Z -> %
Rep == Union(a:X,b:Y,c:Z)
import from Rep, Integer
--
coerce(x:%):OutputForm ==
rep(x) case a => sub(coerce(rep(x).a)$X,outputForm 1) rep(x) case b => sub(coerce(rep(x).b)$Y,outputForm 2)
rep(x) case c => sub(coerce(rep(x).c)$Z,outputForm 3) never inject1(x:X):% == per [a==x] coerce(x:X):% == per [a==x] inject2(y:Y):% == per [b==y] coerce(y:Y):% == per [b==y] inject3(z:Z):% == per [c==z] coerce(z:Z):% == per [c==z] Sum3Functions(X:SetCategory,Y:SetCategory,Z:SetCategory,A:SetCategory): with sum: (X->A,Y->A,Z->A) -> (Sum3(X,Y,Z) -> A) -- Given three functions, each with one of the domains in this Sum -- and having a common co-domain A: returns the unique function with -- domain Sum and co-domain A == add -- Requires same Rep as domain Sum3! macro rep(x) == x pretend Union(a:X,b:Y,c:Z) import from Union(a:X,b:Y,c:Z) -- sum(f:X->A,g:Y->A,h:Z->A):(Sum3(X,Y,Z)->A) == (x:Sum3(X,Y,Z)):A +-> rep(x) case a => f(rep(x).a) rep(x) case b => g(rep(x).b) rep(x) case c => h(rep(x).c) never aldor  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/colimits.as using AXIOM-XL compiler and options -O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y$AXIOM/algebra -I $AXIOM/algebra Use the system command )set compiler args to change these options. #1 (Warning) Could not use archive file libaxiom.al'. #2 (Warning) Could not use archive file libaxiom.al'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 4: import from AxiomLib; ............^ [L4 C13] #5 (Error) No meaning for identifier AxiomLib'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 15: import { true: %, false: % } from Boolean; ..................................^ [L15 C35] #6 (Error) No meaning for identifier Boolean'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 17: string: Literal -> %; ........................^.......^ [L17 C25] #7 (Error) No meaning for identifier Literal'. [L17 C33] #8 (Error) There are no suitable meanings for the operator ->'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", line 18: } from String; .......^ [L18 C8] #10 (Error) No meaning for identifier String'. "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/colimits.as", line 33: macro rep(x) == x pretend Union(a:X,b:Y) ..............^ [L33 C15] #3 (Warning) Definition of macro rep' hides an outer definition. "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/colimits.as", line 42: Sum3(X:SetCategory,Y:SetCategory,Z:SetCategory): with .......^ [L42 C8] #11 (Error) No meaning for identifier SetCategory'. "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/colimits.as", line 46: inject2: Y -> % .............^ [L46 C14] #14 (Error) There are 0 meanings for Y' in this context. The possible types were: Y: SetCategory, a local The context requires an expression of type Tuple(Type). "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/colimits.as", line 48: inject3: Z -> % .............^.^ [L48 C14] #12 (Error) There are 0 meanings for Z' in this context. The possible types were: Z: SetCategory, a local The context requires an expression of type Tuple(Type). [L48 C16] #13 (Error) There are no suitable meanings for the operator ->'. "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/colimits.as", line 73: macro rep(x) == x pretend Union(a:X,b:Y,c:Z) ..............^ [L73 C15] #4 (Warning) Definition of macro rep' hides an outer definition. [L73 C15] #15 (Fatal Error) Too many errors (use -M emax=n' or -M no-emax' to change the limit). The )library system command was not called after compilation. fricas )show Sum2(Integer,Float) The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer . Sum2 is not the name of a known type constructor. If you want to see information about any operations named Sum2 , issue )display operations Sum2 Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Float is not the name of a known type constructor. If you want to see information about any operations named Float , issue )display operations Float  fricas )show Sum2Functions(Integer,Float,Integer) The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer . Sum2Functions is not the name of a known type constructor. If you want to see information about any operations named Sum2Functions , issue )display operations Sum2Functions Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Float is not the name of a known type constructor. If you want to see information about any operations named Float , issue )display operations Float Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer  fricas )show Sum3(Integer,Float,String) The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer . Sum3 is not the name of a known type constructor. If you want to see information about any operations named Sum3 , issue )display operations Sum3 Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Float is not the name of a known type constructor. If you want to see information about any operations named Float , issue )display operations Float String is not the name of a known type constructor. If you want to see information about any operations named String , issue )display operations String  fricas )show Sum3Functions(Integer,Float,String) The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer . Sum3Functions is not the name of a known type constructor. If you want to see information about any operations named Sum3Functions , issue )display operations Sum3Functions Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer Float is not the name of a known type constructor. If you want to see information about any operations named Float , issue )display operations Float String is not the name of a known type constructor. If you want to see information about any operations named String , issue )display operations String 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. fricas inject1(-1)$Sum2(Integer,Float)
There are no library operations named Sum2
Use HyperDoc Browse or issue
)what op Sum2
to learn if there is any operation containing " Sum2 " in its
name.
Cannot find a definition or applicable library operation named Sum2
with argument type(s)
Type
Type
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. inject2(-1.1)$Sum2(Integer,Float)
There are no library operations named Sum2
Use HyperDoc Browse or issue
)what op Sum2
to learn if there is any operation containing " Sum2 " in its
name.
Cannot find a definition or applicable library operation named Sum2
with argument type(s)
Type
Type
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. inject3("c")$Sum3(String,String,String)
There are no library operations named Sum3
Use HyperDoc Browse or issue
)what op Sum3
to learn if there is any operation containing " Sum3 " in its
name.
Cannot find a definition or applicable library operation named Sum3
with argument type(s)
Type
Type
Type
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. fricas f:Sum2(Integer,Float)->Integer:=sum(abs$Integer,wholePart$Float) There are no library operations named Sum2 Use HyperDoc Browse or issue )what op Sum2 to learn if there is any operation containing " Sum2 " in its name. Cannot find a definition or applicable library operation named Sum2 with argument type(s) Type Type Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
f(-1)
There are no library operations named f
Use HyperDoc Browse or issue
)what op f
to learn if there is any operation containing " f " in its name.
Cannot find a definition or applicable library operation named f
with argument type(s)
Integer
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. f(3.14) There are no library operations named f Use HyperDoc Browse or issue )what op f to learn if there is any operation containing " f " in its name. Cannot find a definition or applicable library operation named f with argument type(s) Float Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.