# nLab semi-strict infinity-category

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

In higher category theory a notion of $\infty$-categories or $n$-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 $n \leq 2$, even strict n-categories are semi-strict, but this does not hold for $n \gt 2$.

For $n \leq 3$ 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.

## 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)