Shapes of homological complexes #

We define a structure complex_shape ι for describing the shapes of homological complexes indexed by a type ι. This is intended to capture chain complexes and cochain complexes, indexed by either ℕ or ℤ, as well as more exotic examples.

Rather than insisting that the indexing type has a succ function specifying where differentials should go, inside c : complex_shape we have c.rel : ι → ι → Prop, and when we define homological_complex we only allow nonzero differentials d i j from i to j if c.rel i j. Further, we require that { j // c.rel i j } and { i // c.rel i j } are subsingletons. This means that the shape consists of some union of lines, rays, intervals, and circles.

Convenience functions c.next and c.prev provide these related elements when they exist, and return their input otherwise.

This design aims to avoid certain problems arising from dependent type theory. In particular we never have to ensure morphisms d i : X i ⟶ X (succ i) compose as expected (which would often require rewriting by equations in the indexing type). Instead such identities become separate proof obligations when verifying that a complex we've constructed is of the desired shape.

If α is an add_right_cancel_semigroup, then we define up α : complex_shape α, the shape appropriate for cohomology,so d : X i ⟶ X j is nonzero only when j = i + 1, as well as down α : complex_shape α, appropriate for homology, so d : X i ⟶ X j is nonzero only when i = j + 1. (Later we'll introduce cochain_complex and chain_complex as abbreviations for homological_complex with one of these shapes baked in.)