structure chain_complex : Type :=
(boundary_ops : ) -- A collection of linear maps ∂ᵢ
(C : ) -- F2-vector spaces Cᵢ

• Chain complex C of length n + 1.

  • Fulfilling ∂ᵢ ∂ᵢ₊₁ = 0.

  • All vector spaces Ci are equipped with a basis such that the boundary operators ∂i can be interpreted as matrices.

• Chain complexes come from algebraic topology.

  • The basis vectors of Ci are called i-cells and elements i-chains.

  • An i-cycle is an i-chain with trivial boundary so an element in ker(∂i),

  • whereas an i-boundary is an i-chain in the image of the boundary operator so an element in im(∂ᵢ₊₁).

  • The i-th homology Hi(C) = ker(∂ᵢ)/im(∂ᵢ₊₁) of C is the vector space of i-cycles modulo i-boundaries.

  • Dually, one defines i-cocycles, i-coboundaries and the i-th cohomology Hi(X) = ker(∂tr )/im(∂tr).

References

• Breuckmann2021;

• [Cowtan2023]