Consider the category TOP of topological spaces. A morphism $s : X \rightarrow B$ is said to be exponentiable in TOP if $(X, s)$ is an exponentiable object in TOP/B, i.e. the functor $(-)$ x $(X, s)$ has a right adjoint. Please give explicit discription of right adjoint functor of the fuctor $(-)$ x $(X, s)$.
$\begingroup$
$\endgroup$
2
-
$\begingroup$ Is this a homework question? $\endgroup$– David Roberts ♦Commented Dec 5, 2012 at 1:27
-
2$\begingroup$ I doubt it's homework. But it's not really clear how to pitch an answer, since no background information is given. @Hina: please click on "how to ask" above. A good MO question will generally indicate what thinking the poster has done on the problem, where the difficulty seems to lie, and (it would be nice) why the interest. For example, I don't know if you know what the underlying sets are of the exponentials you are asking about. $\endgroup$– Todd TrimbleCommented Dec 5, 2012 at 15:31
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Since you don't indicate what you know and what you don't know about this problem, I'll simply refer you to the work of Susan Niefield. You can start reading here, and consult the references therein, particularly
- S. B. Niefield, Cartesianness, topological spaces, uniform spaces, and affine schemes, J. Pure Appl. Alg. 23, 1982, 147–167.
where she extends the Day-Kelley characterization of exponentiable objects in $Top$ to slice categories $Top/B$. Much of this is phrased in the language of continuous lattices.
answered Dec 5, 2012 at 15:24