Recently I started learning "theory of locales" (point-free topology) by my-self. While being a very beautiful, natural subject and parallel to point-set topology, some of its notions are bit hard to understand due to the way map between locales has defined and (imho) heavy use of category theory.
- An open continuous map $f : X\to Y$ between locales means that the corresponding formal dual frame homomorphism $f^*: \operatorname{Open}(Y)\to \operatorname{Open}(X)$ has a left adjoint $f_!: \operatorname{Open}(X)\to \operatorname{Open}(Y)$ satisfying the Frobenius reciprocity condition: $f_!(a\land f^*(b))=f_!(a)\land b.$
- Further, if $f$ is a local homeomorphism, then $f^*$ has a right adjoint $f_*.$ Also, $f_!$ is equal to $f_*$ (correct me if I am wrong in this second point).
Here is my question:
Does this means that open $f$ has a right adjoint (as a map of locales)?
And, locale homeomorphisms have both adjoints?
I am interested in this question because of the following sub-questions: Suppose $a, b\in X,$ and $f$ is a local homeomorphism.
- The pullback (in fact, any limit or colimit would work here) $\downarrow a\times_X\downarrow b$ as an open sublocale of $X.$ Do we have $$f\left(\downarrow a\underset{X}{\times}\downarrow b\right)\simeq \downarrow f(a)\underset{\downarrow f(X)}{\times}\downarrow f(a)\, ?$$
- Since monomorphisms are preserved by any right adjoint functor, is the inclusion $\downarrow f(a)\to \downarrow f(X)$ a monomorphism?
I think "Yes" to all these questions. But something seems wrong, and therefore thought to seek opinions from experts.
Thank you for any help you can provide.