Infinity-Categories

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 $(\infty,1)$-category is an infinity-category where all $k$-morphisms for $k>1$ 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

• $(\infty,1)$: 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 $(\infty,1)$-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 $(\infty,1)$-language when ordinary categories suffice.