Entry conditions
Use infinity-categories only when ordinary categories are too rigid and you need morphisms between morphisms.
Definitions
An infinity-category has objects, 1-morphisms, 2-morphisms, and so on, with composition associative up to higher coherent homotopies.
A -category is an infinity-category where all -morphisms for are invertible up to higher morphisms.
Vocabulary (plain language)
- Higher morphism: a morphism between morphisms.
- Coherence: compatibility conditions between compositions, up to homotopy.
Symbols used
- : indicates invertibility above level 1.
Intuition
Infinity-categories generalize categories by allowing maps between maps, so equations are replaced by homotopies.
Worked example
The homotopy category of spaces forgets higher data. An -category of spaces keeps that higher data, tracking homotopies between maps.
How to recognize the structure
- You need morphisms between morphisms to represent your equivalences.
- Associativity holds only up to coherent homotopy.
Common mistakes
- Using -language when ordinary categories suffice.