In the context of foundations of mathematics or mathematical logic one studies formal systems – theories – that allow us to formalize much if not all of mathematics (and hence, by extension, at least aspects of mathematical fields such as fundamental physics).
There are two different attitudes to what a desirable or interesting foundation should achieve:
In proof-theoretic foundations the emphasis is on seeing which formal systems, however convoluted they may be conceptually, allow us to formalize and prove which theorems.
The archetypical such system is ZFC set theory. Other formal systems of interest here are elementary function arithmetic and second order arithmetic, because they are proof-theoretically weak, and still can derive “almost all of undergraduate mathematics” (Harrington).
In practical foundations (following a term introduced in (Taylor)) the emphasis is on conceptually natural formalizations that concentrate the essence of practice and in turn use the result to guide practice (Lawvere), as in (Eilenberg-Steenrod, Harper).
Formal systems of interest here are ETCS or flavors of type theory, which allow natural expressions for central concepts in mathematics (notably via their categorical semantics and the conceptual strength of category theory).
There are two big questions about category theory and the logical foundations of mathematics:
These questions also apply to higher category theory, which also involves the relation between them. (2-categories as a foundation for categories, for example.)
For a philosophical treatment of category theoretic foundations see foundations and philosophy.
The problem with mathematical foundations of category theory is that in category theory we frequently speak of large categories, which it is tricky to deal with rigorously in the usual sort of set theories.
One common view seems to be to found category theory on a theory of sets and classes; see the English Wikipedia's definition, for example. But the standard reference, Saunders Mac Lane's Categories for the Working Mathematician, assumes the existence of a universe (an inaccessible cardinal) instead. Both of these approaches rely on a distinction between small and large categories. There is a category of all small categories, but this category is not itself small; there is no category of all categories.
Alexander Grothendieck used more (although apparently he did not need to); he used what we now call Grothendieck universess. He assumed that every set is contained within a universe; that is, for every cardinal number , there is a cardinal inaccessible from . (This is still a rather moderate axiom, compared to some of the large-cardinal axioms studied by set theorists.) Now one has a relative notion of small and large; the category of all -small categories (where is some universe) is -large but must be -small for some other universe , and there exists a category (which is both -large and -large) of all -small categories.
If one does not accept the axiom of choice, then there are additional complications in general category theory. In particular, one must distinguish between a universal property (for example, having all products) and having a universal structure realising that property (in the example, a functor taking each pair of objects to a specific product cone ). This difficulty was overcome by Michael Makkai using anafunctors, but these have not been widely adopted, even by constructivists.
For a summary of the mathematical foundations of category theory, see Mike Shulman, Set theory for category theory, arXiv:0810.1279.
One way to think of category theory is as a framework in which the idea is formalized that every kind of equality is really secretly a choice of isomorphism or equivalence. In some sense the notion of identity is potentially evil, in a technical sense.
Bill Lawvere proposed to found mathematics on a first-order axiomatisation of the category of categories. This has not been very successful, but his other proposal, a first-order axiomatisation of the category of sets, works well. These and related approaches to foundations may be called structural or categorial (or categorical, which is more common but clashes with another sense of ‘categorical’ in logic).
Lawvere's system ETCS (for ‘the Elementary Theory of the Category of Sets’) essentially states that the category of sets is a topos with certain properties, in particular a well-pointed topos. This can be stated in elementary (first-order) terms; indeed, Lawvere invented the now-default notion of elementary topos (in contrast to the original notion of Grothendieck topos) to do this.
It is also possible to found mathematics on the internal language of a topos. In this case, the topos need not be well-pointed (and indeed, the condition that a topos be well-pointed cannot be stated in its own internal language; or if you prefer, every topos is well-pointed internally). This is equivalent to a certain formulation of type theory, so it is (in a sense) nothing new, although it leads to new perspectives, as in the next paragraph.
Categories (not just toposes) can serve as models of type theories, each type theory corresponding to a certain class of categories. Toposes correspond directly to a constructive but impredicative type theory; to make the theory predicative (in the constructivists' sense) you generalise to a pretopos (maybe locally cartesian closed), to make the theory nonconstructive you specialise to a Boolean topos, and so on. More specifically, every category's internal language is a type theory (with many odd constants), and every type theory (of appropriate form) defines a category (its free model); this is an adjunction between categories and type theories. Paul Taylor's book Practical Foundations of Mathematics is essentially all about this subject, as is (at a more advanced level) most of the career of Michael Makkai.
Certain ‘strong’ axioms of set theory (those involving quantification over all sets) are difficult to state in category-theoretic (or type-theoretic) terms, but this can be overcome in a theory like ETCS; talk to Mike Shulman. (Ironically, this makes it harder to do foundations with categorial foundations!)
In contrast, many of the optional or controversial axioms of set theory (such as the axiom of choice) can be stated quite directly in ETCS. These can be examined quite well in a naïve set-theoretic language that never need be precise about whether one's foundations are traditional (membership-based), categorial, or whatever.
A big picture intro to the comparison between set theory, type theory and topos/category theory as approaches to foundations is in
It may seem on first sight that foundational questions in mathematics are remote from “real mathematics”. This is not quite so. For a list of “real world” problems that do depend on foundations see
A foundation for algebraic topology is this practical spirit is laid out in
Some old discussions from this page archived here