GAP (CAP) - Chapter 4: Category 2-Cells

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4 Category 2-Cells
4.1 Attributes for the Type of 2-Cells
4.1-1 Source
4.1-2 Range
4.2 Identity 2-Cell and Composition of 2-Cells
4.2-1 IdentityTwoCell
4.2-2 AddIdentityTwoCell
4.2-3 HorizontalPreCompose
4.2-4 AddHorizontalPreCompose
4.2-5 HorizontalPostCompose
4.2-6 AddHorizontalPostCompose
4.2-7 VerticalPreCompose
4.2-8 AddVerticalPreCompose
4.2-9 VerticalPostCompose
4.2-10 AddVerticalPostCompose
4.3 Well-Definedness for 2-Cells
4.3-1 IsWellDefinedForTwoCells
4.3-2 AddIsWellDefinedForTwoCells

### 4 Category 2-Cells

#### 4.1 Attributes for the Type of 2-Cells

##### 4.1-1 Source

`‣ Source`( c ) | ( attribute )

Returns: a morphism

The argument is a \(2\)-cell \(c: \alpha \rightarrow \beta\). The output is its source \(\alpha\).

##### 4.1-2 Range

`‣ Range`( c ) | ( attribute )

Returns: a morphism

The argument is a \(2\)-cell \(c: \alpha \rightarrow \beta\). The output is its range \(\beta\).

#### 4.2 Identity 2-Cell and Composition of 2-Cells

##### 4.2-1 IdentityTwoCell

`‣ IdentityTwoCell`( alpha ) | ( attribute )

Returns: a \(2\)-cell

The argument is a morphism \(\alpha\). The output is its identity \(2\)-cell \(\mathrm{id}_{\alpha}: \alpha \rightarrow \alpha\).

##### 4.2-2 AddIdentityTwoCell

`‣ AddIdentityTwoCell`( 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 `IdentityTwoCell`. \(F: \alpha \mapsto \mathrm{id}_{\alpha}\).

##### 4.2-3 HorizontalPreCompose

`‣ HorizontalPreCompose`( c, d ) | ( operation )

Returns: a \(2\)-cell

The arguments are two \(2\)-cells \(c: \alpha \rightarrow \beta\), \(d: \gamma \rightarrow \delta\) between morphisms \(\alpha, \beta: a \rightarrow b\) and \(\gamma, \delta: b \rightarrow c\). The output is their horizontal composition \(d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta)\).

##### 4.2-4 AddHorizontalPreCompose

`‣ AddHorizontalPreCompose`( 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 `HorizontalPreCompose`. \(F: (c,d) \mapsto d \ast c\).

##### 4.2-5 HorizontalPostCompose

`‣ HorizontalPostCompose`( d, c ) | ( operation )

Returns: a \(2\)-cell

The arguments are two \(2\)-cells \(d: \gamma \rightarrow \delta\), \(c: \alpha \rightarrow \beta\) between morphisms \(\alpha, \beta: a \rightarrow b\) and \(\gamma, \delta: b \rightarrow c\). The output is their horizontal composition \(d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta)\).

##### 4.2-6 AddHorizontalPostCompose

`‣ AddHorizontalPostCompose`( 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 `HorizontalPostCompose`. \(F: (d,c) \mapsto d \ast c\).

##### 4.2-7 VerticalPreCompose

`‣ VerticalPreCompose`( c, d ) | ( operation )

Returns: a \(2\)-cell

The arguments are two \(2\)-cells \(c: \alpha \rightarrow \beta\), \(d: \beta \rightarrow \gamma\) between morphisms \(\alpha, \beta, \gamma: a \rightarrow b\). The output is their vertical composition \(d \circ c: \alpha \rightarrow \gamma\).

##### 4.2-8 AddVerticalPreCompose

`‣ AddVerticalPreCompose`( 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 `VerticalPreCompose`. \(F: (c,d) \mapsto d \circ c\).

##### 4.2-9 VerticalPostCompose

`‣ VerticalPostCompose`( d, c ) | ( operation )

Returns: a \(2\)-cell

The arguments are two \(2\)-cells \(d: \beta \rightarrow \gamma\), \(c: \alpha \rightarrow \beta\) between morphisms \(\alpha, \beta, \gamma: a \rightarrow b\). The output is their vertical composition \(d \circ c: \alpha \rightarrow \gamma\).

##### 4.2-10 AddVerticalPostCompose

`‣ AddVerticalPostCompose`( 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 `VerticalPostCompose`. \(F: (d,c) \mapsto d \circ c\).

#### 4.3 Well-Definedness for 2-Cells

##### 4.3-1 IsWellDefinedForTwoCells

`‣ IsWellDefinedForTwoCells`( c ) | ( operation )

Returns: a boolean

The argument is a \(2\)-cell \(c\). The output is `true` if \(c\) is well-defined, otherwise the output is `false`.

##### 4.3-2 AddIsWellDefinedForTwoCells

`‣ AddIsWellDefinedForTwoCells`( 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 `IsWellDefinedForTwoCells`. \(F: c \mapsto \mathtt{IsWellDefinedForMorphisms}( c )\).

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