(Infinity,1)-Topoi

Entry conditions

Use $(\infty,1)$-topoi only when your objects are homotopy types and you need sheaf-like gluing at the level of spaces.

Definitions

An $(\infty,1)$-topos is a presentable $(\infty,1)$-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

• $\mathcal{X}$: an $(\infty,1)$-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 $(\infty,1)$-topos.

How to recognize the structure

• You need sheaves valued in spaces, not sets.
• You need homotopy-coherent gluing.

Common mistakes

• Using $(\infty,1)$-topoi when ordinary sheaf topoi suffice.