Category 2-Cells
Attributes for the Type of 2-Cells a morphism The argument is a 2-cell c: \alpha \rightarrow \beta. The output is its source \alpha. a morphism The argument is a 2-cell c: \alpha \rightarrow \beta. The output is its range \beta.
Identity 2-Cell and Composition of 2-Cells a 2-cell The argument is a morphism \alpha. The output is its identity 2-cell \mathrm{id}_{\alpha}: \alpha \rightarrow \alpha. 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}. 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). 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. 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). 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. 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. 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. 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. 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.
Well-Definedness for 2-Cells a boolean The argument is a 2-cell c. The output is true if c is well-defined, otherwise the output is false. 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 ).