ogposets.OgMap

class rewalt.ogposets.OgMap(source, target, mapping=None, **params)

Bases: object

Class for (partial) maps of oriented graded posets, compatible with boundaries.

To define a map on one element, it must have been defined on all elements below it. The assignment can be made all at once, or element by element. Once the map has been defined on an element, the assignment cannot be modified.

Parameters

source (OgPoset) – The source (domain) of the map.
target (OgPoset) – The target (codomain) of the map.
mapping (list[list[El]], optional) – Data specifying the partial map as follows: mapping[n][k] == El(m, j) if the map sends El(n, k) to El(m, j), and None if the map is undefined on El(n, k) (default is the nowhere defined map).

Keyword Arguments

wfcheck (bool) – Check whether the data defines a well-formed map compatible with all boundaries (default is True).

Notes

Objects of the class are callable on objects of type El (returning the image of an element) and of type GrSubset and GrSet (returning the image of a subset of their source).

Examples

Let us create two simple oriented graded posets, the “point” and the “arrow”.

>>> point = OgPoset.point()
>>> arrow = point >> point

We define the map that collapses the arrow onto the point. First we create a nowhere defined map.

>>> collapse = OgMap(arrow, point)
>>> assert not collapse.istotal

We declare the assignment first on the 0-dimensional elements, then on the single 1-dimensional element. Trying to do otherwise results in a ValueError.

>>> collapse[El(0, 0)] = El(0, 0)
>>> collapse[El(0, 1)] = El(0, 0)
>>> collapse[El(1, 0)] = El(0, 0)

We can check various properties of the map.

>>> assert collapse.istotal
>>> assert collapse.issurjective
>>> assert not collapse.isinjective

Alternatively, we could have defined the map all at once, as follows.

>>> mapping = [[El(0, 0), El(0, 0)], [El(0, 0)]]
>>> assert collapse == OgMap(arrow, point, mapping)

Methods

`bot`()	Functorial extension of `OgPoset.bot()` to maps.
`boundary`([sign, dim])	Returns the map restricted to a specified boundary of its source.
`co`()	Returns the dual in all even dimensions.
`dual`(ogmap, *dims)	Functorial extension of `OgPoset.dual()` to maps of oriented graded posets.
`gray`(*maps)	Functorial extension of `OgPoset.gray()` to maps of oriented graded posets.
`hasse`(**params)	Bound version of `hasse.draw()`.
`image`()	Returns the image of the map.
`inv`()	Returns the inverse of the map if it is an isomorphism.
`isdefined`(element)	Returns whether the map is defined on a given element.
`join`(*maps)	Functorial extension of `OgPoset.join()` to maps of oriented graded posets.
`op`()	Returns the dual in all odd dimensions.
`then`(other, *others)	Returns the composite with other maps or pairs of maps of oriented graded posets, when defined.

Attributes

`input`	Alias for `boundary('-')`.
`isinjective`	Returns whether the map is injective.
`isiso`	Returns whether the map is an isomorphism, that is, total, injective, and surjective.
`issurjective`	Returns whether the map is surjective.
`istotal`	Returns whether the map is total.
`mapping`	Returns the data specifying the map's assignments.
`output`	Alias for `boundary('+')`.
`source`	Returns the source (domain) of the map.
`target`	Returns the target (codomain) of the map.

property source

Returns the source (domain) of the map.

Returns: source – The source of the map.
Return type: OgPoset

property target

Returns the target (codomain) of the map.

Returns: target – The target of the map.
Return type: OgPoset

property mapping

Returns the data specifying the map’s assignments.

Returns: mapping – The mapping data.
Return type: list[list[El]]

property istotal

Returns whether the map is total.

Returns: istotal – True if and only if the map is total.
Return type: bool

property isinjective

Returns whether the map is injective.

Returns: isinjective – True if and only if the map is injective.
Return type: bool

property issurjective

Returns whether the map is surjective.

Returns: issurjective – True if and only if the map is surjective.
Return type: bool

property isiso

Returns whether the map is an isomorphism, that is, total, injective, and surjective.

Returns: isiso – True if and only if the map is an isomorphism.
Return type: bool

isdefined(element)

Returns whether the map is defined on a given element.

Parameters: element (El) – The element to check.
Returns: isdefined – True if and only if the map is defined on the element.
Return type: bool

then(other, *others)

Returns the composite with other maps or pairs of maps of oriented graded posets, when defined.

If given an OgMapPair as argument, it returns the pair of composites of the map with each map in the pair.

Parameters

other (OgMap | OgMapPair) – The first map or pair of maps to follow.
*others (OgMap | OgMapPair, optional) – Any number of other maps or pair of maps to follow.

Returns

composite – The composite with all the other arguments.

Return type

OgMap | OgMapPair

Notes

If all the maps have type shapes.ShapeMap, their composite has the same type.

inv()

Returns the inverse of the map if it is an isomorphism.

Returns: inv – The inverse of the map, if defined.
Return type: OgMap
Raises: ValueError – If the map is not an isomorphism.

image()

Returns the image of the map.

Returns: image – The image of the source through the map.
Return type: Closed

boundary(sign=None, dim=None)

Returns the map restricted to a specified boundary of its source.

Parameters

sign (str, optional) – Orientation: '-' for input, '+' for output, None (default) for both.
dim (int, optional) – Dimension of the boundary (default is self.dim - 1).

Returns

boundary – The map restricted to the requested boundary.

Return type

OgMap

property input: Alias for boundary('-').

property output: Alias for boundary('+').

bot()

Functorial extension of OgPoset.bot() to maps.

Returns: bot – The map extended to a map from source.bot to target.bot.
Return type: OgMap

static gray(*maps)

Functorial extension of OgPoset.gray() to maps of oriented graded posets.

This method can be called with the math operator *, that is, fst * snd is equivalent to gray(fst, snd).

This static method can also be used as a bound method, that is, fst.gray(*maps) is equivalent to gray(fst, *maps).

Parameters: *maps (OgMap) – Any number of maps of oriented graded posets.
Returns: gray – The Gray product of the arguments.
Return type: OgMap

Notes

If all the arguments have type shapes.ShapeMap, so does their Gray product.

static join(*maps)

Functorial extension of OgPoset.join() to maps of oriented graded posets.

This method can be called with the shift operators >> and <<, that is, fst >> snd is equivalent to join(fst, snd) and fst << snd is equivalent to join(snd, fst).

This static method can also be used as a bound method, that is, fst.join(*maps) is equivalent to join(fst, *maps).

Parameters: *maps (OgMap) – Any number of maps of oriented graded posets.
Returns: join – The join of the arguments.
Return type: OgMap

Notes

If all the arguments have type shapes.ShapeMap, so does their join.

static dual(ogmap, *dims)

Functorial extension of OgPoset.dual() to maps of oriented graded posets.

The dual in all dimensions can also be called with the negation operator ~, that is, ~ogmap is equivalent to ogmap.dual().

This static method can be also used as a bound method, that is, self.dual(*dims) is equivalent to dual(self, *dims).

Parameters

ogmap (OgMap) – A map of oriented graded posets.
*dims (int) – Any number of dimensions; if none, defaults to all dimensions.

Returns

dual – The map dualised in the given dimensions.

Return type

OgMap

Notes

If the map is a ShapeMap, so is its dual.

op(): Returns the dual in all odd dimensions.

co(): Returns the dual in all even dimensions.

hasse(**params)

Bound version of hasse.draw().

Calling f.hasse(**params) is equivalent to calling hasse.draw(f, **params).