Re: co-exponential question

[William James <w.james@latrobe.edu.au>]

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