^^Elementary Theory of the Category of Sets.

F. William Lawvere


Even in set theory and elementary mathematics the substance of mathematics resides:

As in algebra and topology, here again the concrete technical machinery for the precise expression and efficient handling of these ideas is provided by the Eilenberg-Mac Lane theory of categories, functors, and natural transformations.


The undefined terms of our theory are mappings, domain, co-domain, and composition. (pag 7)



Todd Trimble


Sets are tree-like constructions, from ∅. The ZFC axioms are very rich, having to do with incredibly powerful operations on trees, and the combinatorial results are extremely complicated.


rob: mi ero un po' dimenticato questo aspetto. Per ricordarmelo i 2 esempi fondamentali sono:


by treating membership not as a global endo-relation on sets, but as local and relative, we effectively eliminate all the extraneous dreck and driftwood which one rightly ignores when examining the mathematics of ZFC.


Fregean language of first-order logic.


A partially ordered set (poset for short) is a set equipped with a partial order.


If X, Y are groups or monoids regarded as one-object categories, then a functor between them amounts to a group or monoid homomorphism.



Category theory.