Rosetta Stones

A Rosetta Stone lines up one concept across the fields that each named it differently, so the shared essence shows through the vocabulary. Every row is the same object seen from another domain, and the last column records where the conventions diverge.

The tables below collect recurring correspondences. They are identifications up to a dictionary rather than loose analogies, since each row is the same operator or construction carried between algebraic topology, differential geometry, and the cellular-sheaf models used across this wiki. Reading a concept through its row is often faster than reading its definition, because the column that is already familiar fixes the meaning of the rest.

Laplacians

The essence is a second-order operator that measures variation, whose kernel is the harmonic data that does not vary.

domain	operator	definition	acts on
graph theory	graph Laplacian	degree minus adjacency	functions on vertices
Hodge theory	Hodge Laplacian	exterior plus co-differential	-forms
differential geometry	connection Laplacian	adjoint of the covariant derivative	sections of a bundle
cellular sheaves	sheaf Laplacian	adjoint of the coboundary	0-cochains

The variants degenerate into one another. Trivial restriction maps collapse the sheaf Laplacian to the graph Laplacian , a continuum limit carries it to the connection Laplacian, and the Weitzenböck identity ties the Hodge and connection forms together through a curvature term.

Coboundary and codifferential

The essence is the adjoint of the degree-raising differential, although the naming and the direction flip between fields.

domain	name	symbol	direction
algebraic topology	coboundary
differential geometry	codifferential
cellular sheaves	coboundary

The direction is the trap. The topologist’s coboundary raises degree, whereas the geometer’s codifferential lowers it, so the shared symbol points opposite ways. The sheaf coboundary follows the topological convention, subtracting the two restrictions on each overlap.

Sections and fields

The essence is an assignment of data to each point of a base, subject to a coherence condition.

domain	name	object	coherence
sheaf theory	global section		required
fibre bundles	section, field	,	continuity
physics	field		the field equation
cellular sheaves	section

A section is the special 0-cochain that already agrees across every incidence, so it is the subspace . An ordinary node feature is a 0-cochain under no such constraint, and sheaf diffusion is the flow that drives a general 0-cochain toward this consistent subspace.

Cocycles and curvature

The essence is the failure of transport to be trivial around a loop.

domain	name	triviality condition
cohomology	cocycle	closes it
differential geometry	curvature	is flat
fibre bundles	holonomy	is flat
cellular sheaves	holonomy of the restrictions around a cycle	identity is flat

When transport around every cycle is trivial the connection is flat, and the space of consistent sections is then largest. A nontrivial loop product is exactly discrete curvature, which is what lets a sheaf carry harmonic data richer than the constants a trivial sheaf allows.

Face maps and transport

The essence is moving data along an incidence.

domain	name	direction
simplicial sets	face map
sheaf theory	functor on a morphism	along the arrow
fibre bundles	parallel transport	along a path
cellular sheaves	restriction map	cell to incident cell
graph networks	message passing	vertex to vertex

The continuous and discrete pictures share one dictionary. A connection fixes parallel transport along a path, and its discrete counterpart is a restriction map assigned to each incidence, with the connection coefficients playing the role of the restriction data.

Hodge decompositions

The essence is the orthogonal splitting of a space into exact, co-exact, and harmonic parts.

domain	decomposition	harmonic part
Hodge theory		de Rham cohomology representatives
graph theory	$\mathbb R^{	V
cellular sheaves		sections

On a graph only degrees 0 and 1 exist, so the middle term is absent and the splitting has two pieces rather than three. Extending the base to higher cells restores the full three-term decomposition.

A companion table

The recurring prefixes of category theory admit the same treatment, gathered in categorical-modifier-prefixes.