nLab
strict n-category

A strict $n$-category is a strict omega-category all whose k-morphisms for $k>n$ are identities.

The category $n\mathrm{Cat}$ of strict $n$-categories can also be defined inductively by

• starting by setting $0\mathrm{Cat}:=$ Set;

• noticing that Set is canonically a (symmetric, in fact cartesian) closed monoidal category such that one can consider categories enriched over it;

• noticing that for $V$ any complete and cocomplete closed monoidal category, also $V\mathrm{Cat}$ has these same properties;

• finally setting, recursively,

  $$(n+1)\mathrm{Cat}:=n\mathrm{Cat}\mathrm{Cat}.$$(n+1)Cat := n Cat Cat \,.

The category $\mathrm{Str}\omega \mathrm{Cat}$ of strict $\omega$-categories can then in turn be defined as a suitable limit of the categories $n\mathrm{Cat}$.

A strict 1-category is just a category. Strict 2-categories are also very important, because the coherence theorem for bicategories? states that any weak 2-category is equivalent to a strict one, and also because many 2-categories, such as Cat, are naturally strict. However, for $n\ge 3$, these two properties fail, so that strict $n$-categories become less useful (though not useless). Instead, one needs to use (at least) semistrict categories.