====== BigObject Morphism<Addition> ======
//from application [[..:tropical|tropical]]//\\
\\
 A morphism is a function between cycles which is locally affine linear and respects the lattices. It is defined by a ''[[..:tropical:Morphism#DOMAIN |DOMAIN]]'', which is a cycle, and values on this domain, ''[[..:tropical:Morphism#VERTEX_VALUES |VERTEX_VALUES]]'' and ''[[..:tropical:Morphism#LINEALITY_VALUES |LINEALITY_VALUES]]'', much like ''[[..:tropical:TropicalRationalFunction |TropicalRationalFunction]]''. Alternatively, it can be defined as a global affine linear function by giving a matrix and a  translation vector.

  ? Type Parameters:
  :: ''Addition'':  Mode of tropical addition, must be ''[[..:common#Min |Min]]'' or ''[[..:common#Max |Max]]''. There is on purpose no default value for it.
===== Properties =====

==== Defining morphisms and functions ====
 These properties are used to define morphisms or rational functions on a Cycle.
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{{anchor:domain:}}
  ?  **''DOMAIN''**
  :: This property describes the domain of the morphism. I.e. the morphism is defined on this complex and is locally affine integral linear. 
    ? Type:
    :''[[..:tropical:Cycle |Cycle]]<Addition>''
    ? from extension:
    : [[:external_software|bundled:atint]]


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{{anchor:is_globally_affine_linear:}}
  ?  **''IS_GLOBALLY_AFFINE_LINEAR''**
  :: This is TRUE, if the morphism is defined on the full projective torus by a ''[[..:tropical:Morphism#MATRIX |MATRIX]]'' and a ''[[..:tropical:Morphism#TRANSLATE |TRANSLATE]]'' The rules do not actually check for completeness of the ''[[..:tropical:Morphism#DOMAIN |DOMAIN]]''.  This property will be set to TRUE, if the morphism is only defined by ''[[..:tropical:Morphism#MATRIX |MATRIX]]'' and ''[[..:tropical:Morphism#TRANSLATE |TRANSLATE]]'',  otherwise it is false (or you can set it upon creation).
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? from extension:
    : [[:external_software|bundled:atint]]


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{{anchor:lineality_values:}}
  ?  **''LINEALITY_VALUES''**
  :: The vector in row i describes the function value (slope) of ''[[..:tropical:Morphism#DOMAIN |DOMAIN]]''->LINEALITY_SPACE->row(i)
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
    ? from extension:
    : [[:external_software|bundled:atint]]


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{{anchor:matrix:}}
  ?  **''MATRIX''**
  :: If the morphism is a global affine linear map x |-> Ax+v, then this contains the matrix A. Note that this must be well-defined on tropical projective coordinates, so the sum of the columns of A must be a multiple of the (1,..,1)-vector. If ''[[..:tropical:Morphism#TRANSLATE |TRANSLATE]]'' is set, but this property is not set, then it is the identity by default.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
    ? from extension:
    : [[:external_software|bundled:atint]]


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{{anchor:translate:}}
  ?  **''TRANSLATE''**
  :: If the morphism is a global affine linear map x |-> Ax+v, then this contains the translation vector  v. If ''[[..:tropical:Morphism#MATRIX |MATRIX]]'' is set, but this property is not set, then it is the zero vector by default.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Rational |Rational]]>''
    ? from extension:
    : [[:external_software|bundled:atint]]


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{{anchor:vertex_values:}}
  ?  **''VERTEX_VALUES''**
  :: The vector at row i describes the function value of vertex ''[[..:tropical:Morphism#DOMAIN |DOMAIN]]''->SEPARATED_VERTICES->row(i).  (In tropical homogenous coordinates, but without leading coordinate). More precisely, if the corresponding vertex is not a far ray, it describes its function value.  If it is a directional ray, it describes the slope on that ray.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
    ? from extension:
    : [[:external_software|bundled:atint]]


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===== Methods =====

==== Affine and projective coordinates ====
 These methods deal with affine and projective coordinates, conversion between those  and properties like dimension that change in projective space.
----
{{anchor:affine_representation:}}
  ?  **''affine_representation([[..:common#Int |Int]] domain_chart, [[..:common#Int |Int]] target_chart)''**
  :: Computes the representation of a morphism (given by ''[[..:tropical:Morphism#MATRIX |MATRIX]]'' and ''[[..:tropical:Morphism#TRANSLATE |TRANSLATE]]'') on tropical affine coordinates.
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''domain_chart'': Which coordinate index of the homogenized domain is shifted to zero to identify it with the domain of the affine function. 0 by default.
    :: ''[[..:common#Int |Int]]'' ''target_chart'': Which coordinate of the homogenized target space is shifted to zero to identify it with the target of the affine function. 0 by default.
    ? Returns:
    :''[[..:common#Pair |Pair]]<[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]]>,[[..:common#Vector |Vector]]<[[..:common#Rational |Rational]]%%>>%%''
    ? from extension:
    : [[:external_software|bundled:atint]]


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==== Morphisms ====
 These are general methods that deal with morphisms and their arithmetic.
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{{anchor:after:}}
  ?  **''after([[..:tropical:Morphism |Morphism]] g)''**
  :: Computes the composition of another morphism g and this morphism. This morphism comes after g.
    ? Parameters:
    :: ''[[..:tropical:Morphism |Morphism]]'' ''g''
    ? Returns:
    :''[[..:tropical:Morphism |Morphism]]''
    ? from extension:
    : [[:external_software|bundled:atint]]


----
{{anchor:before:}}
  ?  **''before([[..:tropical:Morphism |Morphism]] g)''**
  :: Computes the composition of this morphism and another morphism g. This morphism comes before g.
    ? Parameters:
    :: ''[[..:tropical:Morphism |Morphism]]'' ''g''
    ? Returns:
    :''[[..:tropical:Morphism |Morphism]]''
    ? from extension:
    : [[:external_software|bundled:atint]]


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{{anchor:restrict:}}
  ?  **''restrict([[..:tropical:Cycle |Cycle]] Some)''**
  :: Computes the restriction of the morphism to a cycle. The cycle need not be contained in the ''[[..:tropical:Morphism#DOMAIN |DOMAIN]]'' of the morphism, the restriction will be computed on the intersection.
    ? Parameters:
    :: ''[[..:tropical:Cycle |Cycle]]'' ''Some'': cycle living in the same ambient space as the ''[[..:tropical:Morphism#DOMAIN |DOMAIN]]''
    ? Returns:
    :''[[..:tropical:Morphism |Morphism]]''
    ? from extension:
    : [[:external_software|bundled:atint]]


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==== Visualization ====
 These methods are for visualization.
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{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualizes the domain of the morphism. Works exactly as VISUAL of WeightedComplex, but has additional option
    ? Options:
    : 
    :: ''[[..:common#String |String]]'' ''FunctionLabels'': If set to "show", textual function representations are diplayed on cones. False by default
    : option list ''[[..:tropical#Visual_Cycle_FunctionDecorations |Visual::Cycle::FunctionDecorations]]''
    ? from extension:
    : [[:external_software|bundled:atint]]


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