Infinity Category Theory

Schematic definitions

An $(\infty ,1)$-category¹ can be defined in various ways. The schematic definition defines it as categories ‘weakly’ enriched in infinity groupoids. The ambiguity comes from the notion of weak enrichment, which is not precisely and consistently defined.

Models

Models of $(\infty ,1)$-categories are precise mathematical objects, which include:

• Quasi-Categories (Weak Kan Complexes)
• Complete Segal Spaces
• Segal Categories
• Naturally Marked Simplicial Sets (1-trivial saturated weak complicial sets) A non-schematic development of $(\infty ,1)$-categories requires choosing a model.

Riehl’s approach to infinity categories riehl2019-infinity-categories is axiomatic and blind to choice of model. It defines infinity categories using a homotopy 2-category of all infinity categories (implicitly up to some universe level).

Concepts

Infinity Category Adjunction Equivalence Between Infinity Categories

Sources

riehl2019-infinity-categories

Footnotes

1. Weak-Infinity Category ↩