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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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