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Double complexes -- the user's interface

We briefly review the mathematical notion of a double complex. Let \mathcal A be an Abelian category. A double complex D_{\bullet, \bullet} consists of a collection of objects D_{i, j} in \mathcal A with indices (i, j) \in \mathbb Z^2 and usually arranged in a matrix-like grid, together with two collections of morphisms D_{i, j} \to D_{i \pm 1, j}, the horizontal morphisms, and D_{i, j} \to D_{i, j \pm 1}, the vertical morphisms, so that both the rows and the columns of D_{\bullet, \bullet} are complexes in the classical sense and such that all resulting squares of maps commute.

In practice one usually encounters complexes which are bounded in the sense that outside some specified area of indices (i, j) \in \mathbb Z^2 the entries D_{i, j} are all zero. Such entries are then usually omitted.

Basic getters and attributes

In OSCAR the generic functionality for double complexes is declared for the abstract type AbsDoubleComplexOfMorphisms. These functions comprise

  getindex(D::AbsDoubleComplexOfMorphisms, i::Int, j::Int) # Get the `(i,j)`-th entry of `D`
horizontal_map(dc::AbsDoubleComplexOfMorphisms, i::Int, j::Int)
vertical_map(dc::AbsDoubleComplexOfMorphisms, i::Int, j::Int)

In which direction the maps in the rows and columns go can be asked with the following methods:

horizontal_direction(dc::AbsDoubleComplexOfMorphisms)
vertical_direction(dc::AbsDoubleComplexOfMorphisms)

Double complexes can be bounded or unbounded. It is important to note that even if such bounds exist and are known, this is a priori not related to whether or not certain entries are computable! I.e. even in the case of a bounded complex dc it might still be valid to call dc[i, j] beyond that bound. In general, one should use the following functions to determine whether or not it is legitimate to ask for a specific entry.

    has_index(D::AbsDoubleComplexOfMorphisms, i::Int, j::Int)
    can_compute_index(D::AbsDoubleComplexOfMorphisms, i::Int, j::Int)
    has_horizontal_map(dc::AbsDoubleComplexOfMorphisms, i::Int, j::Int)
    can_compute_horizontal_map(dc::AbsDoubleComplexOfMorphisms, i::Int, j::Int)
    has_vertical_map(dc::AbsDoubleComplexOfMorphisms, i::Int, j::Int)
    can_compute_vertical_map(dc::AbsDoubleComplexOfMorphisms, i::Int, j::Int)

Explicitly known bounds for the non-zero entries of a complex are nevertheless relevant for various generic functionalities. For example, computing a total complex is only possible in practice if one has an a priori estimate where the non-zero entries are located. For such purposes, we provide the following functionality:

has_upper_bound(D::AbsDoubleComplexOfMorphisms)
has_lower_bound(D::AbsDoubleComplexOfMorphisms)
has_right_bound(D::AbsDoubleComplexOfMorphisms)
has_left_bound(D::AbsDoubleComplexOfMorphisms)

If they exist, these bounds can be asked for using

right_bound(D::AbsDoubleComplexOfMorphisms)
left_bound(D::AbsDoubleComplexOfMorphisms)
upper_bound(D::AbsDoubleComplexOfMorphisms)
lower_bound(D::AbsDoubleComplexOfMorphisms)

It is also possible to query whether or not a double complex is already complete in the sense that it knows about all of its non-zero entries.

is_complete(D::AbsDoubleComplexOfMorphisms)

Generic functionality

total_complex(D::AbsDoubleComplexOfMorphisms)

Constructors

    tensor_product(C1::ComplexOfMorphisms{ChainType}, C2::ComplexOfMorphisms{ChainType}) where {ChainType}