nLab
Cat

Context

Category theory

Categories of categories

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

categories of categories

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 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 matural 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.

Properties

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.

Colimits

References

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

Revised on July 27, 2014 01:34:08 by Urs Schreiber (89.204.153.39)