-categories of (n,r)-categories
This is also the archetypical 2-topos.
To be explicit, define Cat to be the category with:
composition of morphisms the evident composition of functors.
This is probably the most common meaning of in the literature.
We more often use Cat to stand for the strict 2-category with:
Finally, we can use Cat for the bicategory with:
To be really careful, this version of is an anabicategory.
As a -category, could even include (some) large categories without running into Russell’s paradox. More precisely, if is a Grothendieck universe such that is the category of all -small sets, then you can define to be the 2-category of all -small categories, where is some Grothendieck universe containing . That way, you have without contradiction. (This can be continued to higher categories.)
By the axiom of choice, the two definitions of as a -category are equivalent. In contexts without choice, it is usually better to use anafunctors all along; if necessary, use for the strict -category. Even without choice, a functor or anafunctor between categories is an equivalence in the anabicategory iff it is essentially surjective and fully faithful. However, the weak inverse of such a functor may not be a functor, so it need not be an equivalence in . We can regard as obtained from using homotopy theory by “formally inverting” the essentially surjective and fully faithful functors as weak equivalences.
Discussion of (certain) pushouts in is in