nLab Cat

category theory

Applications

Categories of categories

$\left(n+1,r+1\right)$-categories of (n,r)-categories

Higher category theory

higher category theory

1-categorical presentations

categories of categories

$\left(n+1,r+1\right)$-categories of (n,r)-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 $\mathrm{Cat}$ to small categories. But see CAT for alternatives.

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 $\mathrm{Cat}$ 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 $\mathrm{Cat}$ is an anabicategory.

Properties

Size issues

As a $2$-category, $\mathrm{Cat}$ 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 $U\prime$-small categories, where $U\prime$ is some Grothendieck universe containing $U$. That way, you have $Set\in Cat$ without contradiction. (This can be continued to higher categories.)

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

References

Discussion of (certain) pushouts in $\mathrm{Cat}$ is in