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:
Gray-semistrictness: horizontal composition is strict, but the exchange laws are nontrivial; see Gray-category.
Simpson-semistrictness: everything except the unit laws hold strictly; see Simpson's conjecture.
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 $A_\infty$-category is $A_\infty$-equivalent to a dg-category. This is at least roughly the stable (∞,1)-category analog of the above statement.
A review, some references and further discussion is at
Last revised on May 29, 2023 at 05:34:05. See the history of this page for a list of all contributions to it.