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@Ryuya Hora in Grothendieck topoi with a left adjoint to a left adjoint to a left adjoint to the global sections functor in Theorem 4.17 describes the construction of forming families of a topos by adding a strict initial object to its site.
Restricting to presheaf toposes, , where is equipped with a new formal initial object.
Say I want to generalize the presheaf values to some other category than , say . Is there an expression for ?
I feel like the final question of "Is there an expression" is a little unclear. What do you mean by it exactly?
Shouldn't that be ?
I would think that would be equipped with a new formal terminal object.
Anyway, if is with a formal initial object, then I think is the comma category where is the constant diagram functor.
Yes, that's right @Mike Shulman , should be .
And to clear up what that comma category is: it's a -shaped diagram in for some , and morphisms from to consist of an and . It's very much a "family" construction in the internal sense where families are represented by bundles. It's not particularly an external families construction where you have a bunch of presheaves indexed by a set, though.
James Deikun said:
What do you mean by it exactly?
Perhaps better to ask whether there's an expression that might reasonably be used for the construction, like . Is it reasonable to think of objects here as a kind of family?
It may be useful for something I'm doing to work with - valued presheaves (quasi-Borel space). Would it be reasonable to speak of a -family of presheaves?
James Deikun said:
It's not particularly an external families construction where you have a bunch of presheaves indexed by a set, though.
Right. But perhaps for some values of , like some concrete category, is it reasonable to speak of a structured family?
It's reasonable to speak of a structured family when you're in a category where maps at least conceptually have fibers that look like objects of the category. I'm not familiar enough with quasi-Borel spaces to have an immediate intuition of whether this is the case.
Basically the more topos-like the category is, the more reasonable that is.
E.g. is it extensive?
It's a quasitopos it turns out, so it probably is reasonable.
Speaking rather loosely, could one take the following to be a useful construction when looking to have spatially-indexed quantities?
When you have some category of quantities, , Lawvere tell us that copresheaves are generalized quantities. More generally, take some topos-like and form . Then it's useful to have 'spatially'-indexed such generalized quantities, so form , which will generate quantities indexed over spatial objects of , the images of the strict terminal object.
If we take circuits as a quantity-like resource, perhaps this is happening in: