nLab Cat



Category theory

Categories of categories

2-category theory



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

This is also the archetypical 2-topos.


To avoid set-theoretic problems related to Russell's paradox, it is typical to restrict CatCat 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 CatCat in the literature.

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

Here the vertical composition of 2-morphisms is the evident composition of component maps of natural transformations, whereas the horizontal composition is given by their Godement product.

Finally, we can use Cat for the bicategory with:

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


Cartesian closed structure

The category CatCat, at least in its traditional version comprising small categories only, is cartesian closed: the exponential objects are functor categories. Direct proofs can be found in:

A more indirect proof could proceed by identifying CatCat via the nerve construction as a reflective subcategory of sSet, which is cartesian closed as it is a presheaf category, and showing that this subcategory is an exponential ideal.

Size issues

As a 22-category, CatCat could even include (some) large categories without running into Russell’s paradox. More precisely, if UU is a Grothendieck universe such that Set\Set is the category of all UU-small sets, then you can define Cat\Cat to be the 2-category of all UU'-small categories, where UU' is some Grothendieck universe containing UU. That way, you have SetCat\Set \in \Cat without contradiction. (This can be continued to higher categories.)

By the axiom of choice, the two definitions of CatCat as a 22-category are equivalent. In contexts without choice, it is usually better to use anafunctors all along; if necessary, use StrCatStr Cat for the strict 22-category. Even without choice, a functor or anafunctor between categories is an equivalence in the anabicategory CatCat 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 StrCatStr Cat. We can regard CatCat as obtained from StrCatStr Cat using homotopy theory by “formally inverting” the essentially surjective and fully faithful functors as weak equivalences.


(n+1,r+1)(n+1,r+1)-categories of (n,r)-categories


See also the references at category and category theory.

Discussion of (certain) pushouts in CatCat is in

  • John Macdonald, Laura Scull, Amalgamations of categories (pdf)
category: category

Last revised on June 10, 2021 at 08:50:18. See the history of this page for a list of all contributions to it.