nLab Cat

Contents

Context

Category theory

Categories of categories

2-category theory

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 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.

Properties

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.

Limits and colimits

Remark

(ordinary limits and colimits of categories)
The 1-category Cat of small categories is bicomplete:

  1. The limit of a diagram 𝒞 ():ICat\mathcal{C}_{(-)} \,\colon\, I \to Cat is computed componentwise: the sets of objects and of morphisms of the limiting category limF\underset{\longleftarrow}{\lim} F are the limits in Set of Obj(𝒞 ()):ISetObj(\mathcal{C}_{(-)}) \,\colon\, I \to Set and of Mor(𝒞 ()):ISetMor(\mathcal{C}_{(-)}) \,\colon\, I \to Set, respectively, equipped with the universally induced structure maps.

  2. The colimit of a diagram 𝒞 ():ICat\mathcal{C}_{(-)} \,\colon\, I \to Cat is on objects still the colimit of the underlying diagram of sets of objects, but on morphisms it is more complicated, since in the naive colimit of sets of morphisms some morphisms may become composable that were not composable before.

    An exception is the case of coproducts of categories, which are just given componentwise by disjoint union. With this, it is (by this Prop) sufficient that coequalizers of functors exist.

    Explicit formulas for coequalizers of categories are given in Bednarczyk, Borzyszkowski & Pawlowski 1999, §4.

    Moreover, formulas for pushouts in Cat of injective functors are discussed in MacDonald & Scull 2009.

See also discussion at MO:q/272479.

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

References

The structure of the 2-category of categories (vertical composition, horizontal composition and whiskering of natural transformations) was first described in:

  • Roger Godement, Appendix (pp. 269) of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

(towards the goal of describing the standard resolution of abelian sheaves).

Dedicated discussion in the spirit of formal category theory:

See also most references at category and at category theory, such as:

On colimits in the 1-category of small categories:

coequalizers:

certain pushouts:

Proof that the funny tensor product of categories is the only other symmetric closed monoidal structure on Cat besides the cartesian monoidal structure:

category: category

Last revised on November 15, 2023 at 16:35:25. See the history of this page for a list of all contributions to it.