Entry conditions
Use sheaves of spaces when your local data is itself homotopy-coherent (e.g., spaces or -groupoids).
Definitions
A sheaf of spaces assigns to each object in a site a space (or -groupoid) and satisfies higher descent: compatible local data glues up to coherent homotopy.
A geometric morphism between higher topoi is given by adjoint functors respecting the higher structure.
Vocabulary (plain language)
- Space: a homotopy type, not just a set.
- Higher descent: gluing with coherent homotopies, not strict equality.
Symbols used
- -groupoid: a space considered as a higher groupoid.
Intuition
Sheaves of spaces keep track of homotopies during gluing, so local compatibility is up to homotopy rather than exact equality.
Worked example
Assign to each open set the space of maps into a fixed space . The gluing condition uses homotopies to identify overlaps.
How to recognize the structure
- You need local data with higher homotopy information.
- Equality is too strict; coherence up to homotopy is required.
Common mistakes
- Treating space-valued sheaves as set-valued sheaves.