Vaughen wrote: My question to Steve and the list as a whole would be, if you had been assigned the task of writing an explanation section following the formal definition section, where would you have put the emphasis: on how the definition facilitates a first-order characterization of the notion of "subobject", or on the geometric morphism perspective? My answer: Topos theory, like other theories, is about the interaction of objects (toposes) and morphisms (geometric morphisms). I see it as inherently an aspect of category theory. We use category theory to define it and I believe the most effective expositions on topos theory are those that are based on category theory. Agreed, a strength of the definition of topos is that it allows a multi-faceted approach (three blind men ...) but my personal view is that this must confuse any newcomer. A topos is a type of category which [insert definition here]. Some of the basic results of topos theory are [insert categorical lemmas here]. Once these categorical foundations are in place one is able to (a) investigate/research toposes and (b) learn more about other (say logical) aspects of toposes. I don't want to detract from the importance of (b), but (a) can be carried out without (b). Christopher -----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Vaughan Pratt Sent: 11 March 2008 07:33 To: Categories list Subject: categories: Re: Heyting algebras and Wikipedia Steve Vickers wrote:
The gorilla in the cage is the topos in the classical world.
According to http://en.wikipedia.org/wiki/Gorilla, gorillas in captivity tend to obesity and earlier maturation of females, and have been taught sign language. It doesn't mention any other differences, and the rest of the article is about gorillas in the wild. Does this make a "topos in the classical world" a gros topos and the wild ones petit? The analogy is rather on the colorful side for me. The first half of http://en.wikipedia.org/wiki/Topos is about Grothendieck topoi [sic], the rest about elementary toposes (was topoi but I changed it out of deference to PTJ's strong feelings in the matter). The Explanation section (my contribution, intended as a response to the "respectful awe" tone of the comments on the article's talk page reacting to the bald definition, i.e. the commenters seemed largely mystified but accepted this as par for the course for anything this far beyond rocket science) is presented from the point of view of elementary toposes as a solution to the problem of characterizing the notion of subobject in elementary terms. My question to Steve and the list as a whole would be, if you had been assigned the task of writing an explanation section following the formal definition section, where would you have put the emphasis: on how the definition facilitates a first-order characterization of the notion of "subobject", or on the geometric morphism perspective? Vaughan