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# map -- make a map

- Usage:
    `map(Y,X,d) or map(Y,X)`
- Inputs:
  - Y, an object, such as a ring, module, or chain complex
  - X, another object of the same type
  - d, a specification, such as a function, list, or matrix, or if omitted, understood to specify the identity map
- Optional inputs:
  - Degree => ..., default value null, specify the degree of a map
  - DegreeLift => ..., default value null, make a ring map
  - DegreeMap => ..., default value null, make a ring map
- Outputs:
  - a map to Y from X defined by data d.

## Description

The function map provides a general mechanism for constructing a map (homomorphism) between rings (a ring map), modules (a matrix), or between objects in other categories.

See also the function matrix, which focuses on creating new matrices from rectangular arrays of ring elements or matrices.

### Menu

#### Getting information about a map

- source -- source of a map
- target -- target of a map

#### Creating a map (matrix) between modules

- map(Module,Module,Function) -- create a matrix by specifying a function that gives each entry
- map(Module,Module,List) -- create a matrix by giving a sparse or dense list of entries
- map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
- map(Module,Module,Matrix) -- create the matrix induced on generators by a given matrix

#### Creating a map between modules, where the source module is computed

- map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function that gives each entry
- map(Module,Nothing,List) -- create a matrix by giving a doubly nested list of ring elements
- map(Module,ZZ,List) -- create a matrix by giving a sparse or dense list of entries
- map(Module,Nothing,Matrix) -- recast a matrix to have a new target, and a free module as source

#### Creating a map with a different degree

- map(Matrix) -- make a matrix with a different degree

#### Creating a map between modules over different rings

- map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings

#### Creating a map between rings

- map(Ring,Ring) -- make a ring map, using the names of the variables
- map(Ring,Ring,List) -- make a ring map
- map(Ring,Ring,Matrix) -- make a ring map

#### The Hom and End modules

- Hom -- module of homomorphisms
- End -- module of endomorphisms

## Ways to use map:

- map(GaloisField,GaloisField) (missing documentation)
- map(Matrix) -- make a matrix with a different degree
- map(Module) -- identity map
- map(Module,Module,Function) -- create a matrix by specifying a function that gives each entry
- map(Module,Module,List) -- create a matrix by giving a sparse or dense list of entries
- map(Module,Module,Matrix) -- create the matrix induced on generators by a given matrix
- `map(Module,Module,Number)` -- see map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
- map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
- `map(Module,Module,ZZ)` -- see map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
- `map(Module,ZZ,ZZ)` -- see map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
- `map(Module,Module,RingMap,List)` -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
- map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
- `map(Module,Nothing,RingMap,List)` -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
- `map(Module,Nothing,RingMap,Matrix)` -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
- `map(Module,RingMap)` -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
- map(Module,Nothing,List) -- create a matrix by giving a doubly nested list of ring elements
- map(Module,Nothing,Matrix) -- recast a matrix to have a new target, and a free module as source
- map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function that gives each entry
- map(Module,ZZ,List) -- create a matrix by giving a sparse or dense list of entries
- map(Ring,Matrix) -- make a ring map
- map(Ring,Ring) -- make a ring map, using the names of the variables
- map(Ring,Ring,List) -- make a ring map
- map(Ring,Ring,Matrix) -- make a ring map
- `map(Ring,Ring,RingMap)` -- see map(Ring,Ring,Matrix) -- make a ring map

## For the programmer

The object map is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/map-doc.m2:65:0.