In higher category theory a notion of -categories or -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 , even strict n-categories are semi-strict, but this does not hold for .
For two alternative semi-strictifications are known:
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 -category is -equivalent to a dg-category. This is at least rouhgly the stable (∞,1)-category analog of the above statement.
A review, some references and further discussion is at