nLab
semi-strict infinity-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

In higher category theory a notion of \infty-categories or nn-categories is said to be semi-strict, if these higher categories are, somewhat vaguely, as strict as possible while still being equivalent to general weak higher categories – a kind of rectification statement.

For n2n \leq 2, even strict n-categories are semi-strict, but this does not hold for n>2n \gt 2.

For n3n \leq 3 two alternative semi-strictifications are known:

  1. Gray-semistrictness: horizontal composition is strict, but the exchange laws are nontrivial; see Gray-category.

  2. Simpson-semistrictness: everything except the unit laws hold strictly; see Simpson's conjecture.

Examples for strictification of horizontal composition

References

A review, some references and further discussion is at

Revised on September 4, 2013 14:05:55 by Urs Schreiber (212.238.84.235)