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### 1 Category of Matrices

#### 1.1 Constructors

##### 1.1-1 MatrixCategory
 ‣ MatrixCategory( F ) ( attribute )

Returns: a category

The argument is a homalg field F. The output is the matrix category over F. Objects in this category are non-negative integers. Morphisms from a non-negative integer m to a non-negative integer n are given by m \times n matrices.

##### 1.1-2 VectorSpaceMorphism
 ‣ VectorSpaceMorphism( S, M, R ) ( operation )

Returns: a morphism in \mathrm{Hom}(S,R)

The arguments are an object S in the category of matrices over a homalg field F, a homalg matrix M over F, and another object R in the category of matrices over F. The output is the morphism S \rightarrow R in the category of matrices over F whose underlying matrix is given by M.

##### 1.1-3 VectorSpaceObject
 ‣ VectorSpaceObject( d, F ) ( operation )

Returns: an object

The arguments are a non-negative integer d and a homalg field F. The output is an object in the category of matrices over F of dimension d.

#### 1.2 GAP Categories

##### 1.2-1 IsVectorSpaceMorphism
 ‣ IsVectorSpaceMorphism( object ) ( filter )

Returns: true or false

The GAP category of morphisms in the category of matrices of a field F.

##### 1.2-2 IsVectorSpaceObject
 ‣ IsVectorSpaceObject( object ) ( filter )

Returns: true or false

The GAP category of objects in the category of matrices of a field F.

#### 1.3 Attributes

##### 1.3-1 UnderlyingFieldForHomalg
 ‣ UnderlyingFieldForHomalg( alpha ) ( attribute )

Returns: a homalg field

The argument is a morphism \alpha in the matrix category over a homalg field F. The output is the field F.

##### 1.3-2 UnderlyingMatrix
 ‣ UnderlyingMatrix( alpha ) ( attribute )

Returns: a homalg matrix

The argument is a morphism \alpha in a matrix category. The output is its underlying matrix M.

##### 1.3-3 UnderlyingFieldForHomalg
 ‣ UnderlyingFieldForHomalg( A ) ( attribute )

Returns: a homalg field

The argument is an object A in the matrix category over a homalg field F. The output is the field F.

##### 1.3-4 Dimension
 ‣ Dimension( A ) ( attribute )

Returns: a non-negative integer

The argument is an object A in a matrix category. The output is the dimension of A.

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