Timeline for Is there a correction to the failure of geometric morphisms to preserve internal homs?
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Oct 20 at 17:53 | comment | added | Christopher Townsend | $f:\mathcal{E}\rightarrow\mathcal{F}$ gives rise to an adjunction $\Sigma_f \dashv f^*: \mathbf{Loc}_{\mathcal{E}} \rightarrow \mathbf{Loc}_{\mathcal{F}}$. this adjunction stably satisfies Frobenius reciprocity and $f^*$ preserves the Sierp\{'}nski locale. And any adjunction with these properties comes from a unique (up to iso.) geometric morphism $f$. So $f^*:\mathbf{Loc}_{\mathcal{F}} \rightarrow \mathbf{Loc}_{\mathcal{E}}$ preserves any exponential objects that exist. I.e. you can represent geometric morphisms so that the desired property holds; but the exponentiation is not in the topos. | |
| Jun 17 at 13:48 | comment | added | Ingo Blechschmidt | Yes indeed, there is a well-defined pullback operation, but this does NOT proceed by applying the inverse image functor $f^*$ to the frame object. (Applying $f^*$ to a poset will yield a poset again, but even if it was complete before after applying $f^*$ it usually won't be.) Instead, locales need to be pulled back as follows: Pick a generating system for its frame, pull back that generating system using $f^*$, and then complete this generating system into a full-fledged frame again. Details are on page 25 of Steve's paper | |
| Jun 11 at 16:04 | comment | added | Cameron | Late reply to this but thank you Ingo! This isn't something I'm familiar with so I've been reading up various expositions. Hopefully this will help but in the meantime I've marked as useful. Gavin Wraith's Localic Groups gives the simplest way of forming the "function space" of two plain objects. The elephant seems to say these aren't preserved by pullback on the nose, but that there is a pullback operation that preserves the frames, if not the objects | |
| Apr 25 at 20:25 | history | answered | Ingo Blechschmidt | CC BY-SA 4.0 |