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Stream: learning: questions

Topic: Generalizing the family construction


view this post on Zulip David Corfield (May 25 2025 at 18:26):

@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, Fam(PSh(C))PSh(C)Fam(PSh(\mathcal{C})) \cong PSh(\mathcal{C}^{\triangleright}), where C\mathcal{C}^{\triangleright} is C\mathcal{C} equipped with a new formal initial object.

Say I want to generalize the presheaf values to some other category than SetSet, say D\mathcal{D}. Is there an expression for PSh(C,D)PSh(\mathcal{C}^{\triangleright}, \mathcal{D})?

view this post on Zulip James Deikun (May 25 2025 at 19:21):

I feel like the final question of "Is there an expression" is a little unclear. What do you mean by it exactly?

view this post on Zulip Mike Shulman (May 25 2025 at 19:24):

Shouldn't that be CC^{\triangleleft}?

view this post on Zulip Mike Shulman (May 25 2025 at 19:25):

I would think that CC^{\triangleright} would be CC equipped with a new formal terminal object.

view this post on Zulip Mike Shulman (May 25 2025 at 19:27):

Anyway, if CC^{\triangleleft} is CC with a formal initial object, then I think [(C)op,D][(C^{\triangleleft})^{\rm op}, D] is the comma category [Cop,D]Δ[C^{\rm op},D] \downarrow \Delta where Δ:D[Cop,D]\Delta : D \to [C^{\rm op},D] is the constant diagram functor.

view this post on Zulip David Corfield (May 25 2025 at 19:27):

Yes, that's right @Mike Shulman , should be C\mathcal{C}^{\triangleleft}.

view this post on Zulip James Deikun (May 25 2025 at 19:34):

And to clear up what that comma category is: it's a CopC^\mathrm{op}-shaped diagram in D/dD/d for some dd, and morphisms from (d,X)(d,X) to (d,X)(d',X') consist of an f:ddf : d \to d' and x:f!XXx : f_!X \to X'. 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.

view this post on Zulip David Corfield (May 25 2025 at 19:38):

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 FamD(PSh(C,D))Fam_{\mathcal{D}}(PSh(\mathcal{C}, \mathcal{D})). 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 QBSQBS- valued presheaves (quasi-Borel space). Would it be reasonable to speak of a QBSQBS-family of presheaves?

view this post on Zulip David Corfield (May 25 2025 at 19:43):

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 D\mathcal{D}, like some concrete category, is it reasonable to speak of a structured family?

view this post on Zulip James Deikun (May 25 2025 at 19:46):

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.

view this post on Zulip Mike Shulman (May 25 2025 at 19:53):

Basically the more topos-like the category is, the more reasonable that is.

view this post on Zulip Mike Shulman (May 25 2025 at 19:53):

E.g. is it extensive?

view this post on Zulip James Deikun (May 25 2025 at 19:54):

It's a quasitopos it turns out, so it probably is reasonable.

view this post on Zulip David Corfield (May 27 2025 at 08:48):

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, CC, Lawvere tell us that copresheaves [C,Set][C, Set] are generalized quantities. More generally, take some topos-like DD and form [C,D][C, D]. Then it's useful to have 'spatially'-indexed such generalized quantities, so form [C,D][C^{\triangleright}, D], which will generate quantities indexed over spatial objects of DD, the images of the strict terminal object.

If we take circuits as a quantity-like resource, perhaps this is happening in: