^^Elementary Theory of the Category of Sets.

F. William Lawvere

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

• not in Substance
as it is made to seem when ∈ is the irreducible predicate, with the accompanying necessity of defining all concepts in terms of a rigid elementhood relation
• but in Form
as is clear when the guiding notion is isomorphism-invariant structure, as defined, for example, by universal mapping properties.

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)

http://topologicalmusings.wordpress.com/2008/06/22/basic-category-theory-i/

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:

• numeri naturali
• coppie ordinate

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.