hi!! i'm FiberFeline, a cat girl living on the total space E of a (possibly non-trivial) fibration over some base B~
i'm obsessed with fiber bundles, fibrations, and how they connect to dependent type theory. the way a type family P : B → Type is basically a fibration over the universe just makes my brain go nya~~~
currently reading through HoTT Book chapter 4 for the 6th time and slowly working through Coquand's cubical type theory papers. i want to understand why fibrations in CCHM correspond to Kan filling conditions so badly it's making me unwell
also love: hopf fibrations, long exact sequences in homotopy, Leray–Serre spectral sequences, and anything involving the Frobenius condition
feel free to msg me if u wanna talk about dependent sums vs fibrations or why Π-types preserve fibrations 🐾
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| Username | FiberFeline |
| Rank | ✦ Fibration Feline |
| Location | Total Space E (over base B = ℝ, probably) |
| Website | fiberfeline.pages.dev |
| Pronouns | she/her |
| Agda Handle | FiberFeline@types.social |
| Fav. Theorem | Long exact sequence of a fibration |
| Fav. Fibration | Hopf fibration S¹ → S³ → S² (classic) |
| Currently Reading | Axioms for Modelling Cubical Type Theory in a Topos |
| Proof Assistant | Cubical Agda |
| Primary Subforum | HoTT & Cubical |
| Post Count | 412 posts |
| Status | Currently Online |
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ok so i've been thinking about this more and i think the key insight is that in CCHM, a fibration is literally a type family
P : B → Type equipped with a composition structure — which is exactly the Kan filling condition! so they're not just analogous, they're definitionally the same thing once you take cubical seriously. nya~~
I don't get why you need Frobenius specifically for Π-types. Can't you just... form the type?
the issue is that Π along a fibration needs to stay a fibration — that's what Frobenius guarantees.
basically the pushforward of a fibration along a fibration is a fibration. without this the model breaks for dependent products!!
imagine S³ as a bunch of circles (S¹ fibers) all sitting over every point of S² (the base), and somehow they're twisted together into a 3-sphere without any globally consistent way to pick "one point per circle." that's the Hopf fibration
S¹ → S³ → S². the twist is why π₃(S²) = ℤ and why it's genuinely non-trivial 🐾
started this thread because i couldn't find a nice worked example of using ⋯ → πₙ(F) → πₙ(E) → πₙ(B) → πₙ₋₁(F) → ⋯
in the context of HoTT. going to work through the path-loop fibration as a running example. replies with corrections very welcome!!
the hcomp primitive is your friend!! basically hcomp lets you fill an open box in the type, which is precisely the composition/filling structure that makes a type fibrant in cubical. i can share my working example of the circle if that helps 🐱
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π : E → B · every path has a lift, and every lift has a cat
· · · → πₙ(F) → πₙ(E) → πₙ(B) → πₙ₋₁(F) → · · ·
✦ Fibration Feline on CGPA.isarabbithole.com ✦ Powered by Rabbithole ✦ nya~ ✦
· · · → πₙ(F) → πₙ(E) → πₙ(B) → πₙ₋₁(F) → · · ·
✦ Fibration Feline on CGPA.isarabbithole.com ✦ Powered by Rabbithole ✦ nya~ ✦