nLab
Practical Foundations of Mathematics

The book

is a description of how (in Taylor's opinion) the foundations of mathematics should really be done, with an eye towards matching how mathematics is done in practice (with the consequence that the system is no stronger than necessary).

The result is actually a series of foundations, most constructive, suitable for different sorts of mathematics. Ultimately, these are described as logic in categories defined by sketches and equipped with distinguished pullback-stable classes of display morphisms.

It is available online in a somewhat unreadable format.

A useful survey of some of the topics discussed there is also in

which is an exposition of Taylor’s Abstract Stone Duality.

category: reference