Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
created internal sheaf
Mainly it was bugging me that I didn’t find a piece of literature that said it quite explicitly the way I do there, so I wanted to have that written down. To be expanded, eventually.
I have further expanded it (added two basic Propositions, more details in the definitions) and tried to prettify a bit more.
I added the pedestrian definitions of presheaf and sheaf (as internal diagrams).
Thanks!
I moved your new paragraph to become a subsection of the Definition-section, made my previous material there also a subsection, and added at the beginning of the Definition-section a little lead-in on how we will present two versions of the definition, one abstract, one more explicit.
Of course much of the explicitness of the second definition (the one you added) is out-sourced to the entry internal diagram- But I guess that’s okay.
In this spirit I have also edited the very very last sentence of the entry (at then end of the References) making it now point to the two Definition-sections where previously it just vaguely referred to the explicit defintition. Good.
Thanks, much better! I agree it’s not a problem that the actual explicitness is one further click away.
Do you see a direct way to expand the concept to an internal generalization of non-set valued sheaves ? In my understanding this entry generalizes the set-valued sheaves only.
You can certainly have presheaves valued in locally internal categories over the base topos, see the last section of internal diagram (where this is detailed in the language of indexed categories). Using the internal language, these look exactly like ordinary non-Set-valued presheaves; so I’d guess that to obtain a sensible notion of a sheaf, one could simply formulate the usual sheaf condition in the internal language.
The formulation In terms of external 2-sheaves works by passing to the 2-category of internal categories. In there you can do all of category theory and topos theory as you did externally.
1 to 9 of 9