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5 Category 2-Cells
 5.1 Attributes for the Type of 2-Cells
 5.2 Identity 2-Cell and Composition of 2-Cells
 5.3 Well-Definedness for 2-Cells

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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