nLab
semi-strict infinity-category

Idea

In higher category theory a notion of $\mathrm{\infty }$-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.

For $n\le 2$, even strict n-categories are semi-strict, but this does not hold for $n>2$.

For $n\le 3$ 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 law?s hold strictly; see Simpson's conjecture.

Examples for strictification of horizontal composition

• Every quasi-category is equivalently modeled by a simplicially enriched category, which is a model for an (∞,1)-category in which all horizontal composition is strict. See relation between quasi-categories and simplicial categories.

• A dg-category is an A-infinity-category in which horizontal composition is defined strictly. Every $A_{\mathrm{\infty }}$-category is $A_{\mathrm{\infty }}$-equivalent to a dg-category. This is at least rouhgly the stable (∞,1)-category analog of the above statement.

References

A review, some references and further discussion is at

• Semistrict $\mathrm{\infty }$-categories and $\omega$-Semi-Categories