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### 3 Morphisms

Any GAP object satisfying IsCapCategoryMorphism can be added to a category and then becomes a morphism in this category. Any morphism can belong to one or no category. After a GAP object is added to the category, it knows which things can be computed in its category and to which category it belongs. It knows categorical properties and attributes, and the functions for existential quantifiers can be applied to the morphism.

#### 3.1 Attributes for the Type of Morphisms

##### 3.1-1 CapCategory
 ‣ CapCategory( alpha ) ( attribute )

Returns: a category

The argument is a morphism \alpha. The output is the category \mathbf{C} to which \alpha was added.

##### 3.1-2 Source
 ‣ Source( alpha ) ( attribute )

Returns: an object

The argument is a morphism \alpha: a \rightarrow b. The output is its source a.

##### 3.1-3 Range
 ‣ Range( alpha ) ( attribute )

Returns: an object

The argument is a morphism \alpha: a \rightarrow b. The output is its range b.

#### 3.2 Categorical Properties of Morphisms

 ‣ AddIsMonomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsMonomorphism. F: \alpha \mapsto \mathtt{IsMonomorphism}(\alpha).

 ‣ AddIsEpimorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsEpimorphism. F: \alpha \mapsto \mathtt{IsEpimorphism}(\alpha).

 ‣ AddIsIsomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsIsomorphism. F: \alpha \mapsto \mathtt{IsIsomorphism}(\alpha).

 ‣ AddIsSplitMonomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsSplitMonomorphism. F: \alpha \mapsto \mathtt{IsSplitMonomorphism}(\alpha).

 ‣ AddIsSplitEpimorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsSplitEpimorphism. F: \alpha \mapsto \mathtt{IsSplitEpimorphism}(\alpha).

 ‣ AddIsOne( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsOne. F: \alpha \mapsto \mathtt{IsOne}(\alpha).

 ‣ AddIsIdempotent( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsIdempotent. F: \alpha \mapsto \mathtt{IsIdempotent}(\alpha).

#### 3.3 Non-Categorical Properties of Morphisms

Non-categorical properties are not stable under equivalences of categories.

##### 3.3-1 IsIdenticalToIdentityMorphism
 ‣ IsIdenticalToIdentityMorphism( alpha ) ( property )

Returns: a boolean

The argument is a morphism \alpha: a \rightarrow b. The output is true if \alpha = \mathrm{id}_a, otherwise the output is false.

 ‣ AddIsIdenticalToIdentityMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsIdenticalToIdentityMorphism. F: \alpha \mapsto \mathtt{IsIdenticalToIdentityMorphism}(\alpha).

##### 3.3-3 IsIdenticalToZeroMorphism
 ‣ IsIdenticalToZeroMorphism( alpha ) ( property )

Returns: a boolean

The argument is a morphism \alpha: a \rightarrow b. The output is true if \alpha = 0, otherwise the output is false.

 ‣ AddIsIdenticalToZeroMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsIdenticalToZeroMorphism. F: \alpha \mapsto \mathtt{IsIdenticalToZeroMorphism }(\alpha).

 ‣ AddIsEndomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsEndomorphism. F: \alpha \mapsto \mathtt{IsEndomorphism}(\alpha).

 ‣ AddIsAutomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsAutomorphism. F: \alpha \mapsto \mathtt{IsAutomorphism}(\alpha).

#### 3.4 Equality and Congruence for Morphisms

##### 3.4-1 IsCongruentForMorphisms
 ‣ IsCongruentForMorphisms( alpha, beta ) ( operation )

Returns: a boolean

The arguments are two morphisms \alpha, \beta: a \rightarrow b. The output is true if \alpha \sim_{a,b} \beta, otherwise the output is false.

 ‣ AddIsCongruentForMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsCongruentForMorphisms. F: (\alpha, \beta) \mapsto \mathtt{IsCongruentForMorphisms}(\alpha, \beta).

##### 3.4-3 IsEqualForMorphisms
 ‣ IsEqualForMorphisms( alpha, beta ) ( operation )

Returns: a boolean

The arguments are two morphisms \alpha, \beta: a \rightarrow b. The output is true if \alpha = \beta, otherwise the output is false.

 ‣ AddIsEqualForMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsEqualForMorphisms. F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphisms}(\alpha, \beta).

##### 3.4-5 IsEqualForMorphismsOnMor
 ‣ IsEqualForMorphismsOnMor( alpha, beta ) ( operation )

Returns: a boolean

The arguments are two morphisms \alpha: a \rightarrow b, \beta: c \rightarrow d. The output is true if \alpha = \beta, otherwise the output is false.

 ‣ AddIsEqualForMorphismsOnMor( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsEqualForMorphismsOnMor. F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphismsOnMor}(\alpha, \beta).

#### 3.5 Basic Operations for Morphisms in Ab-Categories

##### 3.5-1 IsZeroForMorphisms
 ‣ IsZeroForMorphisms( alpha ) ( operation )

Returns: a boolean

The argument is a morphism \alpha: a \rightarrow b. The output is true if \alpha \sim_{a,b} 0, otherwise the output is false.

 ‣ AddIsZeroForMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsZeroForMorphisms. F: \alpha \mapsto \mathtt{IsZeroForMorphisms}(\alpha).

 ‣ AdditionForMorphisms( alpha, beta ) ( operation )

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

The arguments are two morphisms \alpha, \beta: a \rightarrow b. The output is the addition \alpha + \beta.

 ‣ AddAdditionForMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation AdditionForMorphisms. F: (\alpha, \beta) \mapsto \alpha + \beta.

 ‣ AdditiveInverseForMorphisms( alpha ) ( operation )

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

The argument is a morphism \alpha: a \rightarrow b. The output is its additive inverse -\alpha.

 ‣ AddAdditiveInverseForMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation AdditiveInverseForMorphisms. F: \alpha \mapsto -\alpha.

##### 3.5-7 ZeroMorphism
 ‣ ZeroMorphism( a, b ) ( operation )

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

The arguments are two objects a and b. The output is the zero morphism 0: a \rightarrow b.

 ‣ AddZeroMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ZeroMorphism. F: (a,b) \mapsto (0: a \rightarrow b).

#### 3.6 Subobject and Factorobject Operations

Subobjects of an object c are monomorphisms with range c and a special function for comparision. Similarly, factorobjects of an object c are epimorphisms with source c and a special function for comparision.

##### 3.6-1 IsEqualAsSubobjects
 ‣ IsEqualAsSubobjects( alpha, beta ) ( operation )

Returns: a boolean

The arguments are two subobjects \alpha: a \rightarrow c, \beta: b \rightarrow c. The output is true if there exists an isomorphism \iota: a \rightarrow b such that \beta \circ \iota \sim_{a,c} \alpha, otherwise the output is false.

 ‣ AddIsEqualAsSubobjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsEqualAsSubobjects. F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsSubobjects}(\alpha,\beta).

##### 3.6-3 IsEqualAsFactorobjects
 ‣ IsEqualAsFactorobjects( alpha, beta ) ( operation )

Returns: a boolean

The arguments are two factorobjects \alpha: c \rightarrow a, \beta: c \rightarrow b. The output is true if there exists an isomorphism \iota: b \rightarrow a such that \iota \circ \beta \sim_{c,a} \alpha, otherwise the output is false.

 ‣ AddIsEqualAsFactorobjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsEqualAsFactorobjects. F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsFactorobjects}(\alpha,\beta).

##### 3.6-5 IsDominating
 ‣ IsDominating( alpha, beta ) ( operation )

Returns: a boolean

In short: Returns true iff \alpha is smaller than \beta. \\ Full description: The arguments are two subobjects \alpha: a \rightarrow c, \beta: b \rightarrow c. The output is true if there exists a morphism \iota: a \rightarrow b such that \beta \circ \iota \sim_{a,c} \alpha, otherwise the output is false.

 ‣ AddIsDominating( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsDominating. F: (\alpha, \beta) \mapsto \mathtt{IsDominating}(\alpha,\beta).

##### 3.6-7 IsCodominating
 ‣ IsCodominating( alpha, beta ) ( operation )

Returns: a boolean

In short: Returns true iff \alpha is smaller than \beta. \\ Full description: The arguments are two factorobjects \alpha: c \rightarrow a, \beta: c \rightarrow b. The output is true if there exists a morphism \iota: b \rightarrow a such that \iota \circ \beta \sim_{c,a} \alpha, otherwise the output is false.

 ‣ AddIsCodominating( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsCodominating. F: (\alpha, \beta) \mapsto \mathtt{IsCodominating}(\alpha,\beta).

#### 3.7 Identity Morphism and Composition of Morphisms

##### 3.7-1 IdentityMorphism
 ‣ IdentityMorphism( a ) ( attribute )

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

The argument is an object a. The output is its identity morphism \mathrm{id}_a.

 ‣ AddIdentityMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IdentityMorphism. F: a \mapsto \mathrm{id}_a.

##### 3.7-3 PreCompose
 ‣ PreCompose( alpha, beta ) ( operation )

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

The arguments are two morphisms \alpha: a \rightarrow b, \beta: b \rightarrow c. The output is the composition \beta \circ \alpha: a \rightarrow c.

##### 3.7-4 PreCompose
 ‣ PreCompose( L ) ( operation )

Returns: a morphism in \mathrm{Hom}(a_1, a_{n+1})

This is a convenience method. The argument is a list of morphisms L = ( \alpha_1: a_1 \rightarrow a_2, \alpha_2: a_2 \rightarrow a_3, \dots, \alpha_n: a_n \rightarrow a_{n+1} ). The output is the composition \alpha_{n} \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) ) ).

 ‣ AddPreCompose( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation PreCompose. F: (\alpha, \beta) \mapsto \beta \circ \alpha.

##### 3.7-6 PostCompose
 ‣ PostCompose( beta, alpha ) ( operation )

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

The arguments are two morphisms \beta: b \rightarrow c, \alpha: a \rightarrow b. The output is the composition \beta \circ \alpha: a \rightarrow c.

##### 3.7-7 PostCompose
 ‣ PostCompose( L ) ( operation )

Returns: a morphism in \mathrm{Hom}(a_1, a_{n+1})

This is a convenience method. The argument is a list of morphisms L = ( \alpha_n: a_n \rightarrow a_{n+1}, \alpha_{n-1}: a_{n-1} \rightarrow a_n, \dots, \alpha_1: a_1 \rightarrow a_2 ). The output is the composition ((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1.

 ‣ AddPostCompose( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation PostCompose. F: (\alpha, \beta) \mapsto \alpha \circ \beta.

#### 3.8 Well-Definedness of Morphisms

##### 3.8-1 IsWellDefinedForMorphisms
 ‣ IsWellDefinedForMorphisms( alpha ) ( operation )

Returns: a boolean

The argument is a morphism \alpha. The output is true if \alpha is well-defined, otherwise the output is false.

 ‣ AddIsWellDefinedForMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsWellDefinedForMorphisms. F: \alpha \mapsto \mathtt{IsWellDefinedForMorphisms}( \alpha ).

#### 3.9 Basic Operations for Morphisms in Abelian Categories

##### 3.9-1 LiftAlongMonomorphism
 ‣ LiftAlongMonomorphism( iota, tau ) ( operation )

Returns: a morphism in \mathrm{Hom}(t,k)

The arguments are a monomorphism \iota: k \hookrightarrow a and a morphism \tau: t \rightarrow a such that there is a morphism u: t \rightarrow k with \iota \circ u \sim_{t,a} \tau. The output is such a u.

 ‣ AddLiftAlongMonomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation LiftAlongMonomorphism. The function F maps a pair (\iota, \tau) to a lift u if it exists, and to fail otherwise.

##### 3.9-3 ColiftAlongEpimorphism
 ‣ ColiftAlongEpimorphism( epsilon, tau ) ( operation )

Returns: a morphism in \mathrm{Hom}(c,t)

The arguments are an epimorphism \epsilon: a \rightarrow c and a morphism \tau: a \rightarrow t such that there is a morphism u: c \rightarrow t with u \circ \epsilon \sim_{a,t} \tau. The output is such a u.

 ‣ AddColiftAlongEpimorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ColiftAlongEpimorphism. The function F maps a pair (\epsilon, \tau) to a lift u if it exists, and to fail otherwise.

#### 3.10 Lift/ Colift

• For any pair of morphisms \alpha: a \rightarrow c, \beta: b \rightarrow c, we call each morphism \alpha / \beta: a \rightarrow b such that \beta \circ (\alpha / \beta) \sim_{a,c} \alpha a lift of \alpha along \beta.

• For any pair of morphisms \alpha: a \rightarrow c, \beta: a \rightarrow b, we call each morphism \alpha \backslash \beta: c \rightarrow b such that (\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta a colift of \beta along \alpha.

Note that such lifts (or colifts) do not have to be unique. So in general, we do not expect that algorithms computing lifts (or colifts) do this in a functorial way. Thus the operations \mathtt{Lift} and \mathtt{Colift} are not regarded as categorical operations, but only as set-theoretic operations.

##### 3.10-1 Lift
 ‣ Lift( alpha, beta ) ( operation )

Returns: a morphism in \mathrm{Hom}(a,b) + \{ \mathtt{fail} \}

The arguments are two morphisms \alpha: a \rightarrow c, \beta: b \rightarrow c such that there is a lift \alpha / \beta: a \rightarrow b of \alpha along \beta, i.e., a morphism such that \beta \circ (\alpha / \beta) \sim_{a,c} \alpha. The output is such a lift or \mathtt{fail} if it doesn't exist.

 ‣ AddLift( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation Lift. The function F maps a pair (\alpha, \beta) to a lift \alpha / \beta if it exists, and to fail otherwise.

##### 3.10-3 Colift
 ‣ Colift( alpha, beta ) ( operation )

Returns: a morphism in \mathrm{Hom}(c,b) + \{ \mathtt{fail} \}

The arguments are two morphisms \alpha: a \rightarrow c, \beta: a \rightarrow b such that there is a colift \alpha \backslash \beta: c \rightarrow b of \beta along \alpha., i.e., a morphism such that (\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta. The output is such a colift or \mathtt{fail} if it doesn't exist.

 ‣ AddColift( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation Colift. The function F maps a pair (\alpha, \beta) to a colift \alpha \backslash \beta if it exists, and to fail otherwise.

#### 3.11 Inverses

Let \alpha: a \rightarrow b be a morphism. An inverse of \alpha is a morphism \alpha^{-1}: b \rightarrow a such that \alpha \circ \alpha^{-1} \sim_{b,b} \mathrm{id}_b and \alpha^{-1} \circ \alpha \sim_{a,a} \mathrm{id}_a.

 ‣ AddInverse( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation Inverse. F: \alpha \mapsto \alpha^{-1}.

#### 3.12 Tool functions for caches

##### 3.12-1 IsEqualForCacheForMorphisms
 ‣ IsEqualForCacheForMorphisms( phi, psi ) ( operation )

Returns: true or false

Compares two objects in the cache

 ‣ AddIsEqualForCacheForMorphisms( c, F ) ( operation )