Seams and Signals
Chains are what you build from. Cochains are what you measure with.
Topology begins where local cancellation stops being the whole story. Click around. Form a chain. Take its boundary. Puncture the space and watch a loop refuse to fill.
A cycle is closed geometry.
A cocycle is consistent measurement.
A boundary is a cycle with a filling.
A coboundary is a cocycle with an explanation.
Chain side
what you build from
Cochain side
what you measure with
k-chain
signed assembly of k-pieces
k-cochain
measurement rule on k-chains
boundary ∂
exposed seam operator
coboundary δ
inconsistency operator
cycle
chain with no exposed boundary
cocycle
measurement with no local inconsistency
boundary
cycle that fills from one dimension up
coboundary
measurement explained by a potential
homology
cycles modulo boundaries
cohomology
cocycles modulo coboundaries