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 HorizontalPreCompose
4.2-3 HorizontalPostCompose
4.2-4 VerticalPreCompose
4.2-5 VerticalPostCompose
4.3 Well-Definedness for 2-Cells
4.3-1 IsWellDefinedForTwoCells

### 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 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).

##### 4.2-3 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).

##### 4.2-4 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.

##### 4.2-5 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.

#### 4.3 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`.

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