Struct WalkingCategory 

pub struct WalkingCategory();

The walking category as a VDC.

The walking category is the simplest example of a virtual double category that has units (and in fact all composites). Specifically, the walking category is the unital VDC freely generated by a single object, here called ().

The concept of a category can be interpreted in any virtual double category: a category object** in a VDC is a functor from the walking category into that VDC. In particular, a category object in spans is a category in the ordinary sense.

Trait Implementations

impl VDCWithComposites for WalkingCategory

fn composite_ext(&self, path: Path<Self::Ob, Self::Pro>) -> Option<Self::Cell>

In the walking category, every cell is an extension.

fn through_composite( &self, n: Self::Cell, range: Range<usize>, ) -> Option<Self::Cell>

Factorizes a cell through a composite of proarrows.

fn has_composite(&self, path: &Path<Self::Ob, Self::Pro>) -> bool

Does the path of proarrows have a chosen composite?

fn has_unit(&self, x: &Self::Ob) -> bool

Does the object have a chosen unit?

fn composite2_ext(&self, m: Self::Pro, n: Self::Pro) -> Option<Self::Cell>

Gets the chosen cell witnessing a composite of two proarrows, if there is one.

fn composite(&self, path: Path<Self::Ob, Self::Pro>) -> Option<Self::Pro>

Gets the chosen composite for a path of proarrows, if there is one.

fn composite2(&self, m: Self::Pro, n: Self::Pro) -> Option<Self::Pro>

Gets the chosen composite for a pair of consecutive proarrows, if there is one.

fn unit_ext(&self, x: Self::Ob) -> Option<Self::Cell>

Gets the chosen extension cell for an object, if there is one.

fn unit(&self, x: Self::Ob) -> Option<Self::Pro>

Gets the chosen unit for an object, if there is one.

fn unit_arrow(&self, f: Self::Arr) -> Option<Self::Cell>

Constructs the unit cell on an arrow, if there is one.

fn through_unit(&self, cell: Self::Cell, index: usize) -> Option<Self::Cell>

Factorizes a cell through the unit proarrow for an object.

impl VDblCategory for WalkingCategory

type Ob = ()

Type of objects in the VDC.

type Arr = ()

Type of arrows (tight morphisms) in the VDC.

type Pro = ()

Type of proarrows (loose morphisms) in the VDC.

type Cell = usize

Type of cells in the VDC.

fn has_ob(&self, _: &Self::Ob) -> bool

Does the object belong to the VDC?

fn has_arrow(&self, _: &Self::Arr) -> bool

Does the arrow belong to the VDC?

fn has_proarrow(&self, _: &Self::Pro) -> bool

Does the proarrow belong to the VDC?

fn has_cell(&self, _: &Self::Cell) -> bool

Does the cell belong to the VDC?

fn dom(&self, _: &Self::Arr) -> Self::Ob

Gets the domain of an arrow.

fn cod(&self, _: &Self::Arr) -> Self::Ob

Gets the codomain of an arrow.

fn src(&self, _: &Self::Pro) -> Self::Ob

Gets the source of a proarrow.

fn tgt(&self, _: &Self::Pro) -> Self::Ob

Gets the target of a proarrow.

fn cell_dom(&self, n: &Self::Cell) -> Path<Self::Ob, Self::Pro>

Gets the domain of a cell, a path of proarrows.

fn cell_cod(&self, _: &Self::Cell) -> Self::Pro

Gets the codomain of a cell, a single proarrow.

fn cell_src(&self, _: &Self::Cell) -> Self::Arr

Gets the source of a cell, an arrow.

fn cell_tgt(&self, _: &Self::Cell) -> Self::Arr

Gets the target of a cell, an edge.

fn compose(&self, _: Path<Self::Ob, Self::Arr>) -> Self::Arr

Composes a path of arrows in the VDC.

fn compose_cells( &self, tree: DblTree<Self::Arr, Self::Pro, Self::Cell>, ) -> Self::Cell

Composes a tree of cells in the VDC.

fn arity(&self, cell: &Self::Cell) -> usize

Gets the arity of a cell.

fn compose2(&self, f: Self::Arr, g: Self::Arr) -> Self::Arr

Composes a pair of arrows with compatible (co)domains.

fn id(&self, x: Self::Ob) -> Self::Arr

Constructs the identity arrow at an object.

fn compose_cells2( &self, αs: impl IntoIterator<Item = Self::Cell>, β: Self::Cell, ) -> Self::Cell

where Self: Sized,

Composes a two-layer pasting of cells.

fn id_cell(&self, m: Self::Pro) -> Self::Cell

Constructs the identity cell on a proarrow.