In



> 
> Lawvere, F. W., 1966, “The Category of
>  Categories as a Foundation for
>  Mathematics”, Proceedings of the
>  Conference on Categorical Algebra, La
>  Jolla, New York: Springer-Verlag,
>  1–21.
> 
> 
> 


Lawvere proposed an elementary theory of the category of categories which can serve as a foundation for mathematics.


So far I have heard from several sources that there are some flaws with this theory so that it does not completely work as proposed.


So my question is whether there is currently any (accepted) elementary theory of the category of categories that is rich enough so that one can formulate, say, the following things in the theory:


* The category of sets.
* Basic notions of category theory (functor categories, adjoints, Kan extensions, etc.).
* Other important categories (like the category of rings or the category of schemes).


The elementary theory I am looking for should allow me to identify what should be called a category of commutative rings (at best I would like to see this category defined by a universal 2-categorical property) or how to work with this category. I am not interested in defining groups, rings, etc. as special categories as this seems to be better done in an elementary theory of sets.


P.S.: The same question has an analogue one level higher. Assume that we have constructed an object in the category of categories (=: CAT) which can serve as a, say, category C of spaces. Classically, we can associate to each space X in C the sheaf topos over it. In the picture I have in mind, one should ask whether there is a similar elementary theory of the category of 2-categories (=: 2-CAT). Then one should be able to lift the object C from CAT to 2-CAT (as one is able to form the discrete category from a set), define an object T in 2-CAT that serves as the 2-category of toposes, and a functor C -> T in 2-CAT.



Category of categories as a foundation of mathematics
My personal opinion is that one should consider the *2-category* of categories, rather than the 1-category of categories. I think the axioms one wants for such an "ET2CC" will be something like:


* Firstly, some exactness axioms amounting to its being a "2-pretopos" in the sense I described here: <http://ncatlab.org/michaelshulman/show/2-categorical+logic> . This gives you an "internal logic" like that of an ordinary (pre)topos.
* Secondly, the existence of certain exponentials (this is optional).
* Thirdly, the existence of a "classifying discrete opfibration" $el\to set$ in the sense introduced by Mark Weber ("Yoneda structures from 2-toposes") which serves as "the category of sets," and internally satisfies some suitable axioms.
* Finally, a "well-pointedness" axiom saying that the terminal object is a generator, as is the case one level down with in ETCS. This is what says you have a 2-category of categories, rather than (for instance) a 2-category of stacks.


Once you have all this, you can use finite 2-categorical limits and the "internal logic" to construct all the usual concrete categories out of the object "set". For instance, "set" has finite products internally, which means that the morphisms $set \to 1$ and $set \to set \times set$ have right adjoints in our 2-category Cat (i.e. "set" is a "cartesian object" in Cat). The composite $set \to set\times set \to set$ of the diagonal with the "binary products" morphism is the "functor" which, intuitively, takes a set $A$ to the set $A\times A$. Now the 2-categorical limit called an "inserter" applied to this composite and the identity of "set" can be considered "the category of sets $A$ equipped with a function $A\times A\to A$," i.e. the category of magmas.


Now we have a forgetful functor $magma \to set$, and also a functor $magma \to set$ which takes a magma to the triple product $A\times A \times A$, and there are two 2-cells relating these constructed from two different composites of the inserter 2-cell defining the category of magmas. The "equifier" (another 2-categorical limit) of these 2-cells it makes sense to call "the category of semigroups" (sets with an associative binary operation). Proceeding in this way we can construct the categories of monoids, groups, abelian groups, and eventually rings.


A more direct way to describe the category of rings with a universal property is as follows. Since $set$ is a cartesian object, each hom-category $Cat(X,set)$ has finite products, so we can define the category $ring(Cat(X,set))$ of rings internal to it. Then the category $ring$ is equipped with a forgetful functor $ring \to set$ which has the structure of a ring in $Cat(ring,set)$, and which is universal in the sense that we have a natural equivalence $ring(Cat(X,set)) \simeq Cat(X,ring)$. The above construction then just shows that such a representing object exists whenever Cat has suitable finitary structure.


One can hope for a similar elementary theory of the 3-category of 2-categories, and so on up the ladder, but it's not as clear to me yet what the appropriate exactness properties will be.


