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:
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 -category is -equivalent to a dg-category. This is at least roughly the stable (∞,1)-category analog of the above statement.
Globular proof assistant
A review, some references and further discussion is at
Last revised on December 7, 2018 at 19:38:22. See the history of this page for a list of all contributions to it.