Entry conditions
Use -topoi only when your objects are homotopy types and you need sheaf-like gluing at the level of spaces.
Definitions
An -topos is a presentable -category that satisfies higher analogues of the sheaf conditions and descent.
Vocabulary (plain language)
- Presentable: large enough to have colimits and a set of generators.
- Descent: the higher-categorical version of gluing.
Symbols used
- : an -topos.
Intuition
Ordinary topoi are like categories of sheaves of sets. Higher topoi are categories of sheaves of spaces, where equivalences carry homotopy data.
Worked example
Sheaves of spaces on a topological space form an -topos.
How to recognize the structure
- You need sheaves valued in spaces, not sets.
- You need homotopy-coherent gluing.
Common mistakes
- Using -topoi when ordinary sheaf topoi suffice.