$(n+1,r+1)$-categories of (n,r)-categories
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Cat is a name for the category or 2-category of all categories.
This is also the archetypical 2-topos.
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:
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:
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 $Cat$ is an anabicategory.
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:
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.
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.
$(n+1,r+1)$-categories of (n,r)-categories
William Lawvere, The Category of Categories as a Foundation for Mathematics, pp.1-20 in Eilenberg, Harrison, MacLane, Röhrl (eds.), Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer Heidelberg 1966 (doi:10.1007/978-3-642-99902-4_1)
John Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, Vol. 391.
Springer 1974 (doi:10.1007/BFb0061280)
See also the references at category and category theory.
Discussion of (certain) pushouts in $Cat$ is in
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