Edit detail for SandBoxMonad revision 2 of 1

This may be a better definition of Monad in Axiom.

Ref: http://en.wikibooks.org/wiki/Haskell/Category_theory

A monad is an endofunctor

In Axiom a functor is a domain constructor. An endofunctor takes a domain in some category and constructs a domain in that same category.

--)abbrev category MONADC MonadCat
#pile
#include "axiom"
MonadCat(A : Type, M: Type -> Type): Category == with
  if A has SetCategory then SetCategory
  fmap: (A->A, %) -> %
  unit: A -> %
  join: M % -> %
  mult: (%, %) -> %

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/monadcat.as 
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   The )library system command was not called after compilation.

)show MonadCat

   The )show system command is used to display information about types 
      or partial types. For example, )show Integer will show 
      information about Integer .

      MonadCat is not the name of a known type constructor. If you want
      to see information about any operations named MonadCat , issue
                        )display operations MonadCat

--)abbrev domain MONADL MonadList
#pile
#include "axiom"
#library lmonadcat "monadcat.ao"
import from lmonadcat
MonadList(A:Type): MonadCat(A,List) == add
  Rep == List A
  import from Rep
  rrep(x:%):Rep == x pretend Rep
  --perr(x:Rep):% == x pretend %

  fmap(f:A->A,x:%):% == per(map(f,rep x)$ListFunctions2(A,A))
  unit(x:A):% == per list(x)
  join(x:List %):% == per(concat(map(rrep,x)$ListFunctions2(%,Rep)))
  mult(x:%,y:%):% == per concat(rep x,rep y)
  if A has SetCategory then
    coerce(x:%):OutputForm == rep(x)::OutputForm
    ((x:%) = (y:%)):Boolean ==
      L:List Boolean := map(_=$A,rep x,rep y)$ListFunctions3(A,A,Boolean)
      reduce(_and,L,false)$ListFunctions2(Boolean,Boolean)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2630045957127378990-25px003.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   The )library system command was not called after compilation.

)show MonadList(Integer)

   The )show system command is used to display information about types 
      or partial types. For example, )show Integer will show 
      information about Integer .

      MonadList is not the name of a known type constructor. If you 
      want to see information about any operations named MonadList , 
      issue
                        )display operations MonadList 

      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

L1 := unit(1)$MonadList(Integer)

   There are no library operations named MonadList 
      Use HyperDoc Browse or issue
                             )what op MonadList
      to learn if there is any operation containing " MonadList " in 
      its name.

   Cannot find a definition or applicable library operation named 
      MonadList with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

)abbrev domain MAYBE2 Maybe2
Maybe2(R:SetCategory):SetCategory
  == add
    Rep := Union(R,"failed")
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4607644844684909994-25px006.spad
      using old system compiler.
   MAYBE2 abbreviates domain Maybe2 
------------------------------------------------------------------------
   initializing NRLIB MAYBE2 for Maybe2 
   compiling into NRLIB MAYBE2 
   compiling local rep : $ -> Rep
      MAYBE2;rep is replaced by x 
Time: 0 SEC.

   compiling local per : Rep -> $
      MAYBE2;per is replaced by x 
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Maybe2| REDEFINED

;;;     ***       |Maybe2| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] rep: pretendRep -- should replace by @
      [2] per: pretend$ -- should replace by @

   Cumulative Statistics for Constructor Maybe2
      Time: 0 seconds

   finalizing NRLIB MAYBE2 
   Processing Maybe2 for Browser database:
--->-->Maybe2(): Missing Description
; compiling file "/var/aw/var/LatexWiki/MAYBE2.NRLIB/MAYBE2.lsp" (written 06 APR 2022 03:04:34 AM):

; /var/aw/var/LatexWiki/MAYBE2.NRLIB/MAYBE2.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
   Maybe2 is now explicitly exposed in frame initial 
   Maybe2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MAYBE2.NRLIB/MAYBE2

)abbrev domain MAYBE2 Maybe2
Maybe2(R:SetCategory):MonadCat(R,Maybe2)
  == add
    Rep ==> Union(R,"failed")
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    fmap(f:R->R,x:%):% ==
      rep(x) case R => per f(rep x)
      per "failed"
    unit(x:R):% = per x
    join(x:Maybe %):% ==
      rep(x) case "failed" => per "failed"
      rep(rep x) case "failed" => per "failed"
      x

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2461590120764584468-25px007.spad
      using old system compiler.
   MAYBE2 abbreviates domain Maybe2 
------------------------------------------------------------------------
   initializing NRLIB MAYBE2 for Maybe2 
   compiling into NRLIB MAYBE2 
   cannot produce category object:
(|MonadCat| R |Maybe2|)

   >> Apparent user error:
   cannot produce category object

)show Maybe

   The )show system command is used to display information about types 
      or partial types. For example, )show Integer will show 
      information about Integer .

      Maybe is not the name of a known type constructor. If you want to
      see information about any operations named Maybe , issue
                          )display operations Maybe

x:Maybe(Integer):= unit(-1)

   There are no library operations named Maybe 
      Use HyperDoc Browse or issue
                               )what op Maybe
      to learn if there is any operation containing " Maybe " in its 
      name.

   Cannot find a definition or applicable library operation named Maybe
      with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.