natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In classical algebraic topology we have four Hopf fibrations (of spheres):
These can be constructed in homotopy type theory as part of a more general construction:
An H-space structure on a pointed type gives a fibration over via the Hopf construction. This fibration can be written classically as: where is the join of and . This is all done in the HoTT book. Note that can be written as a homotopy pushout , and there is a lemma in the HoTT book allowing you to construct a fibration on a pushout (the equivalence needed is simply the multiplication from the H-space ).
Thus the problem of constructing a Hopf fibration reduces to finding a H-space structure on the spheres: the , and .
The space is not connected so we cannot perform the construction from the book on it. However it is very easy to construct a family with fiber by induction on . (Note: loop maps to where is the equivalence of negation and is the univalence axiom.
For Peter Lumsdaine gave the construction in 2012 and Guillaume Brunerie proved it was correct in 2013. By induction on the circle we can define the multiplication: , and where is also defined by circle induction: and . denotes functional extensionality.
For Buchholtz-Rijke 16 solved this through a homotopy theoretic version of the Cayley-Dickson construction. This has been formalised in Lean.
It is still an open problem to show that these are the only spheres to have a H-space structure. This would be done by showing these are the only spheres with Hopf invariant which has been defined in Brunerie 2016
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Guillaume Brunerie, On the homotopy groups of spheres in homotopy type theory arxiv:1606.05916
Ulrik Buchholtz, Egbert Rijke, The Cayley-Dickson Construction in Homotopy Type Theory (arXiv:1610.01134)
Last revised on June 15, 2022 at 16:55:16. See the history of this page for a list of all contributions to it.