Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the global sections a sheaf local categories on topological space. stiff embeds into such sections. An infinitary version these theorems also holds in spatial case. These representation results are functorial and subsume Lambek–Moerdijk–Awodey for toposes, Stone Boolean algebras, Takahas...

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.