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 $Cat$ 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

See also discussion at MO:q/272479.

Related concepts

• elementary theory of the category of categories

• canonical model structure on 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]