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Gershom (Oct 08 2022 at 00:48):Begin with the following:
a) (X, x ∈ X) is pointed sets, fibered over Set by first projection.
b) (X, S ⊆ X) is propositions on sets, fibered over Set by first projection.
Now, consider the category given by objects (X, L ⊆ P(X)), for some class of Ls. Morphisms are given by set functions f which act pulling back sets in L' to sets in L given as their preimage in f.
My question is: under what conditions is (X, L ⊆ P(X)), "pre-covers on sets" fibered over Set by first projection?
By conditions, I'm thinking along the lines of: "L is closed under intersection" or "L, when viewed as a partially ordered set by inclusion is a distributive lattice" or simply "L contains the empty and maximal sets", etc.
Matteo Capucci (he/him) (Oct 08 2022 at 18:01):As long as Ls pull back to Ls, you're fine. This condition is better given for families of morphisms than for families of objects though. See [[display map]].
Matteo Capucci (he/him) (Oct 08 2022 at 18:01):Also you might want to check out [[Grothendieck coverage]], which formalizes this idea of 'covering'
Notification Bot (Oct 08 2022 at 18:02):This topic was moved here from #general: mathematics > When is (X,S ⊆ P(X)) -> X a fibration? by Matteo Capucci (he/him).
Gershom (Oct 09 2022 at 05:44):Nice, thanks! The displayed perspective is helpful here.
Gershom (Oct 09 2022 at 05:51):Related question: what can be said about classes of morphisms that "push forward" Ls or indeed classes that do both?
Gershom (Oct 09 2022 at 06:02):Additionally, a grothondieck coverage is i thought just another name for the topos generalization of a topology, no? So that has a bunch of conditions on it that amount to, when you're giving it on a discrete category (i.e. a set) that it is closed under certain operations. Note that what I'm asking about puts _no conditions_ on the collection of "pre-covers" and instead is asking what if any conditions must be there to get a (cartesian) fibration. Examples of "set systems" I have in mind that are not going to satisfy the conditions for topologies include greedoids, antimatroids, convexity spaces, etc.
Matteo Capucci (he/him) (Oct 10 2022 at 15:39):Gershom said:
Note that what I'm asking about puts _no conditions_ on the collection of "pre-covers" and instead is asking what if any conditions must be there to get a (cartesian) fibration.
In a sense, this what Grothendieck coverages do too: they ask for the minimum structure such that things work out. Probably you can phraise such a condition fibrationally? I don't remember much about that.
Matteo Capucci (he/him) (Oct 10 2022 at 15:40):But maybe I now see what you are getting at, you're seeing as structure on !
Matteo Capucci (he/him) (Oct 10 2022 at 15:40):So you're asking for conditions for the category of such structured spaces to be fibred over
Matteo Capucci (he/him) (Oct 10 2022 at 15:42):So yeah, probably the display maps perspective doesn't help much either... :thinking:
Matteo Capucci (he/him) (Oct 10 2022 at 15:45):Surely you need the morphisms picking out structure to be stable under pullback. For instance, topological spaces have this thing: given a map of sets and a topological structure on , there is a universal topology on making the map continuous, i.e. the initial topology (it also works the other way around, to is also an opfibration!).
Matteo Capucci (he/him) (Oct 10 2022 at 15:46):Since your structures are going to be given by some axiomatic theory there's probably more interesting things to say about when these axioms guarantee you the existence of an initial/final structure, i.e. when the corresponding structure pullback/pushforward with maps of the underlying sets