Bill Halchin wrote:
This is actually a "dual" question. Basically I want to do the dual of the construction gives the notion of an exponential or map object.
Suppose we have a category C with sums. Then we build the following category from C.
object: T+X<-----Y
map: from T+X<-----Y to T+X'<-------Y is a C map "alpha" such that we have the following diagram:
I-sub-T+alpha T+X---------------------------->T+X' ^ ^ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \/ Y
Then suppose there exists a C-object called Y**T such that T+Y**T<-------Y is the initial object of the category just built above. What significance does Y**T have opposed to the concept of an exponential???? If I did everything correctly it (Y**T) should be the dual of T**Y.
Forgive my ignorance: is T+Y**T<----Y to be initial because the usual construction for the exponential has the relevant arrow as terminal? Can you suggest to me a reference for that construction of exponentials? As for significance: my own research into co-exponentials is in terms of them as characteristic of lattices dual to Heyting algebras. These dual-Heyting algebras work well as algebras for a brand of paraconsistent logic (they have a complement operator which has in general that an element and its complement overlap). Alternatively, you can think of co-exponentials as productive of the interesting topological notions associated with closed sets, like boundary. My feeling is that co-exponentials count as useful in more interesting kinds of maths than turn up simply in those categories dual to toposes. William James