Cellular Sheaf

A cellular sheaf (in Justin Curry’s sense) attaches data to the cells of a space and glues it along incidences. Fix a base index category , a partially-ordered-set of cells or a small category, and a fibre category in which each stalk lives, such as vector spaces or Hilbert spaces. A sheaf is a functor

assigning to each object its stalk and to each inclusion a restriction , contravariantly and functorially. It is a copresheaf valued in read on .

For a cover the cochain spaces are (with ), and the degree-zero coboundary compares the two restrictions on each overlap :

The global sections are . When is Hilbertian the restrictions have adjoints and give the sheaf Laplacian , whose kernel is again . Hansen & Ghrist develop the spectral theory of this Laplacian. When the base is a poset with its Alexandrov topology the sheaves form a topos, whose internal language lets each sheaf be handled as a single object rather than an indexed family. Its coboundary, Laplacian, and sections line up with the same objects in topology and geometry, a correspondence collected in the Rosetta Stones.

Proposition. Descent is an equalizer

Gluing over a cover of is the statement that

is an equalizer. On a finite poset with its Alexandrov topology every functor already satisfies the gluing axiom, so sheafhood on the nerve is automatic and carries no information; descent for a Grothendieck topology on is the substantive question. The nerve of a cover has the nonempty intersections as cells; functoriality makes pairwise agreement on overlaps propagate to all higher intersections, so only nerve edges carry a condition.

Definition. Local system

If every restriction is invertible, is a local system. Transport is then an isomorphism, so no information is destroyed along a morphism, and the comparison map from a global object to its family of restrictions is injective. A separation failure, meaning distinct global objects with identical restrictions, can occur only when some restriction is non-invertible.

References

• Curry, “Sheaves, Cosheaves and Applications,” PhD thesis, University of Pennsylvania (2014)
• Shepard, “A Cellular Description of the Derived Category of a Stratified Space,” PhD thesis, Brown University (1985)
• Hansen & Ghrist, “Toward a Spectral Theory of Cellular Sheaves,” Journal of Applied and Computational Topology (2019)