cat-dist@mta.ca writes:
- Category theorists are keen on statements to the effect that structures are - defined by their universal properties. A typical book on topos theory may - define an elementary topos as a category with finite limits and power - objects. It then goes on to show that any topos has internal-homs. How? - By defining the function space as a certain set of sets of ordered pairs...
Nonsense. How can a set of sets of ordered pairs be an object of a topos. In fact, there is at least one book that defines a topos as a category with finite limits and power objects and constructs the internal homs as a limit of two arrows between two power objects. The construction is hidden inside a cotriple, but that is what it amounts to.
<sigh/smirk> My desire for a little irony got the better of my pedantic instincts. I'm well aware that the phrase "representative of a subobject of the power-object of a product" would have be more precise than the morally equivalent set-theoretic terminology. It doesn't effect the point that I was trying to make; that whether one defines certain things by explicit construction or via an axiomatisation is fairly independent of whether one prefers to work in a logical, categorical or set-theoretic foundation. Neither is this effected by the availability of alternative constructions. Of course, for some people, it would be a perfectly sensible point of view that topos theory is interesting precisely because of results to the effect that a couple of elementary constructs are sufficient to justify the fairly general use of set-theoretic notation, such as refering to a `set of sets of ordered pairs'. Ralph.