GAP (CAP) - Chapter 5: Category 2-Cells

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5 Category 2-Cells
5.1 Attributes for the Type of 2-Cells
5.1-1 Source
5.1-2 Range
5.2 Identity 2-Cell and Composition of 2-Cells
5.2-1 IdentityTwoCell
5.2-2 AddIdentityTwoCell
5.2-3 HorizontalPreCompose
5.2-4 AddHorizontalPreCompose
5.2-5 HorizontalPostCompose
5.2-6 AddHorizontalPostCompose
5.2-7 VerticalPreCompose
5.2-8 AddVerticalPreCompose
5.2-9 VerticalPostCompose
5.2-10 AddVerticalPostCompose
5.3 Well-Definedness for 2-Cells
5.3-1 IsWellDefinedForTwoCells
5.3-2 AddIsWellDefinedForTwoCells

### 5 Category 2-Cells

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

##### 5.1-1 Source

`‣ Source`( c ) | ( attribute )

Returns: a morphism

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

##### 5.1-2 Range

`‣ Range`( c ) | ( attribute )

Returns: a morphism

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

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

##### 5.2-1 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.

##### 5.2-2 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}.

##### 5.2-3 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).

##### 5.2-4 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.

##### 5.2-5 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).

##### 5.2-6 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.

##### 5.2-7 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.

##### 5.2-8 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.

##### 5.2-9 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.

##### 5.2-10 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.

#### 5.3 Well-Definedness for 2-Cells

##### 5.3-1 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`.

##### 5.3-2 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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