Categories, Allegories

This brilliant idiosyncratic book conveys a crystalline insight into category theoretic accounts of structure. The most original aspect is the theory of relations. The approach is spartan in its simplicity. In effect we assume that each relation Rxy has an opposite (as "x is less than y" has opposite "x is greater than y") and suitably matched relations have composites (as the composite of "x is brother to y" with "y is a parent of z" is "x is uncle to z"). We make a few more assumptions none of which involve set theory, and we get a very powerful theory with applications in many situations in logic and category theory.