nLab
Cat

Contents

Idea

Cat is a name for the category or 2-category of all categories.

This is also the archetypical 2-topos.

Definition

To avoid set-theoretic problems related to Russell’s paradox, it is typical to restrict Cat to small categories. But see CAT for alternatives.

To be explicit, define Cat to be the category with:

This is probably the most common meaning of Cat in the literature.

We more often use Cat to stand for the strict 2-category with:

Here, the horizontal composition? of natural transformations is given by their Godement product.

Finally, we can use Cat for the bicategory with:

To be really careful, this version of Cat is an anabicategory.

As a 2-category, Cat could even include (some) large categories without running into Russell’s paradox. More precisely, if U is a Grothendieck universe such that Set is the category of all U-small sets, then you can define Cat to be the 2-category of all U-small categories, where U is some Grothendieck universe containing U. That way, you have SetCat without contradiction. (This can be continued to higher categories.)

By the axiom of choice, the two definitions of Cat as a 2-category are equivalent. In contexts without choice, it is usually better to use anafunctors all along; if necessary, use StrCat for the strict 2-category. Even without choice, a functor or anafunctor between categories is an equivalence in the anabicategory Cat 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 StrCat. We can regard Cat as obtained from StrCat using homotopy theory by “formally inverting” the essentially surjective and fully faithful functors as weak equivalences.

category: category