On 11/09/17 13:19, Joyal, Andr? wrote:
Dear John, and category theorists,
The fact that every category has an opposite introduces a symmetry in mathematics that would not be there otherwise.
The category of sets is not self dual, but a disjoint union of sets is a coproduct, dual to a product.
Thurston does not show esteem for logic. Most mathematicians are taking logic for granted; they just use it as a part of their natural language. It is obvious that human understanding depends on the the laws of thought, on logic. In a sense, category theory is a branch of mathematical logic, since it greatly improves mathematical thinking in general. A category theorist might say (not too loudly) that mathematical logic is a branch of category theory.
Best, andr?
The opposite category (*) may look a senseless obscurity and make some people nauseous, but it seems to me it made an important contribution to the understanding of mathematics. It took a long time to form part of mathematical thinking (and still is). For example, Bourbaki treatment of limits (of sets say) define and develops basic properties of projective limits, including the universal property. Later does the same for inductive limits, and includes a proof of the dual statements !!. He had to do so since it had not incorporated categories and the opposite category. He states what a universal property is, but can not state that the respective universal properties (for limits and colimits) are one the dual of the other. (*) The axioms of a category are self dual. Another examples are abelian categories, and a very subtle one, namely, Quillen's model categories. Many categories are not self dual, and this is underneath the duality between algebra and geometry. best e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]