nLab Cat

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:

• small categories as objects,

• functors as morphisms;

• composition of morphisms the evident composition of functors.

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:

• small categories as objects,
• functors as morphisms.
• natural transformations as 2-morphisms.

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:

• small categories as objects,
• anafunctors as morphisms.
• ananatural transformations as 2-morphisms.

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

Properties

Cartesian closed structure

The category $Cat$, 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:

• p. 98 of Categories Work, 2nd ed.
• remark below Definition 4.3.9 in Category Theory in Context
• Steve Awodey, Category Theory, Second Edition, Sections 7.5-7.7.

A more indirect proof could proceed by identifying $Cat$ 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 $2$-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 $\Set \in \Cat$ 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 $Str Cat$ 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 $Str Cat$. We can regard $Cat$ as obtained from $Str Cat$ 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 $\mathcal{C}_{(-)} \,\colon\, I \to Cat$ is computed componentwise: the sets of objects and of morphisms of the limiting category $\underset{\longleftarrow}{\lim} F$ are the limits in Set of $Obj(\mathcal{C}_{(-)}) \,\colon\, I \to Set$ and of $Mor(\mathcal{C}_{(-)}) \,\colon\, I \to Set$, respectively, equipped with the universally induced structure maps.

2. The colimit of a diagram $\mathcal{C}_{(-)} \,\colon\, I \to Cat$ 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.

Related concepts

• elementary theory of the category of categories

• canonical model structure on Cat

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

• Pos

• Set

  • Rel

• Grpd, ∞Grpd

• Cat

  • Ho(Cat)

  • AccCat

  • PrCat

  • LexCat

  • MonCat

  • VCat

  • CatAdj

  • Prof

  • Operad

• 2Cat

• ModCat, CombModCat

• (∞,1)Cat

  • Pr(∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

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:

• William Lawvere, The Category of Categories as a Foundation for Mathematics, pp. 1-20 in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) [doi:10.1007/978-3-642-99902-4]

• John Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, 391, Springer (1974) [doi:10.1007/BFb0061280]

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

• Horst Schubert, §3 in: Categories, Springer (1972) [doi:10.1007/978-3-642-65364-3]

  (with size issues dealt with via Grothendieck universes)

• Saunders MacLane, §II.5 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

  (size issues ignored by considering only the category of small categories)

On colimits in the 1-category of small categories:

coequalizers:

• Marek A. Bednarczyk, Andrzej M. Borzyszkowski, Wieslaw Pawlowski, Section 4 of: Generalized congruences – epimorphisms in $\mathcal{C}at$, Theory and Applications of Categories 5 11 (1999) 266-280 [tac:5-11, dml:120226]

certain pushouts:

• John MacDonald, Laura Scull, Amalgamations of categories, Canad. Math. Bull. 52 2 (2009) 273–284 [pdf, journal:pdf]

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

• François Foltz, Christian Lair, G. M. Kelly, Algebraic categories with few monoidal biclosed structures or none, Journal of Pure and Applied Algebra 17 2 (1980) 171-177 [pdf, doi:10.1016/0022-4049(80)90082-1]

category: category