Could not include topos theory - contents
basic constructions:
strong axioms
The Elementary Theory of the Category of Sets (Lawvere 65), or ETCS for short, is a formulation of set-theoretic foundations in a category-theoretic spirit. As such, it is the prototypical structural set theory.
More in detail, ETCS is a first-order theory axiomatizing elementary toposes and specifically those which are well-pointed, have a natural numbers object and satisfy the axiom of choice. The idea is, first of all, that traditional mathematics naturally takes place “inside” such a topos, and second that by varying the axioms much of mathematics may be done inside more general toposes: for instance omitting the well-pointedness and the axiom of choice but adding the Kock-Lawvere axiom gives a smooth topos inside which synthetic differential geometry takes place.
Modern mathematics with emphasis on concepts of homotopy theory would more directly be founded in this spirit by an axiomatization not just of elementary toposes but of elementary (∞,1)-toposes. This is roughly what univalent homotopy type theory accomplishes – for more on this see at relation between type theory and category theory – Univalent HoTT and Elementary infinity-toposes.
Instead of increasing the higher categorical dimension (n,r) in the first argument, one may also, in this context of elementary foundations, consider raising the second argument. The case $(2,2)$ is the elementary theory of the 2-category of categories? (ETCC).
The axioms of ETCS can be summed up in one sentence as:
The category of sets is the topos which
is a well-pointed topos
has a natural numbers object
and satisfies the axiom of choice.
For more details see
Todd Trimble has a series of expositional writings on ETCS which provide a very careful introduction and at the same time a wealth of useful details.
Todd Trimble, ZFC and ETCS: Elementary Theory of the Category of Sets (nLab entry, original blog entry)
Todd Trimble, ETCS: Internalizing the logic (nLab entry, original blog entry)
Todd Trimble, ETCS: Building joins and coproducts (nLab entry, original blog entry)
ETCS was proposed in
An undergraduate set-theory textbook using it is
An informative discussion of the pros and cons of Lawvere’s approach can be found in
Erik Palmgren has a constructive predicative variant of ETCS, which can be summarized as: $Set$ is a well-pointed $\Pi$-pretopos with a NNO and enough projectives (i.e. COSHEP is satisfied). Here “well-pointed” must be taken in its constructive sense, as including that the terminal object is indecomposable and projective.