isomorphismes

John Baez:

“There must be hundreds of mathematicians roaming the earth thinking operads are difficult and abstruse, because they’ve seen the definition without any pictures.

These people should be grateful they weren’t taught how to tie their shoes in an equally wrong-headed manner.”

What is an operad?

• A collection of  N→1 operators
• “associatively composable”
• permutable

Why should someone interested in homotopy care about operads?

• the main way to probe a space is by looking at the maps from circle, sphere, 3-sphere, 4-sphere, … onto it.
• “Most homotopy theorists would gladly sell their souls for the ability to compute the homotopy groups of an arbitrary space”
• Operads organise the information of the restricted higher homotopies.

What is a fibration?

by Niles Johnson

• ℝ×ℝ = plane (infinitely big square)
• ℝ×𝕊 = cylinder (infinitely long)
• 𝕊×𝕊 = torus
• (Remember: 𝕊×𝕊≠𝕊² ! Because of the North Pole & South Pole. But neither does [0,1]×𝕊≠𝕊² since the 𝕊¹ needs to come together into points at either end 𝕊⁰.)
• projection from 𝔸×𝔹 → 𝔹 is 𝔸×{b∈𝔹} ↦{b∈𝔹} for a given {b∈𝔹}. Here 𝔸 is the fibre.
• a cylinder is locally an interval [0,1] or vertical stick |
  
  crossed with a circle
• a Möbius band is locally an interval [0,1] or vertical stick | but twisted once
  
• a Hopf ring is locally an interval [0,1] or vertical stick | but twisted twice
  
  
  
  
  
  
  
  
• Fibration 𝔽 → total space 𝔼 → 𝔹 base space

• Hopf map: 𝕊¹→𝕊³→𝕊²
• 𝕊⁰→𝕊¹→𝕊¹
• 𝕊³→𝕊⁷→𝕊⁴
• 𝕊⁷→𝕊¹⁵→𝕊⁸
• That’s it. That’s all the fibrations of spheres by spheres over spheres.
• Any quaternion q∈ℍ times its complement (flip signs on all the “weirdo” i,j,k terms) has magnitude one
• q•k•q⁻¹ zeroes out the first term (reducing dimensionality from 4→3) and still the magnitude 1—meaning ℍ~ℝ⁴→ℝ³→𝕊².
• boom.
• Stereographic projection.
  
• ℝ² is the same as the open unit disk (btw: disk is filled in whereas circle is not) with a point at ∞ — think of “bubbling up”
• "arctan is a great function to use for mapping the real line (without ±∞) down to a finite interval.” (See also the video of why Bicontinuity is the right condition for topological sameness.)
• “So, um, just imagine the three-sphere…. OK, that was easy. Now…”
  
• Some stuff I couldn’t see which was pretty important.
• Minute 46. Rock out to the Hopf links.

(por Eddie Beck)

differential topology lecture by John W. Milnor from the 1960’s: Topology from the Differentiable Viewpoint

• A function that’s problematic for analytic continuations:
  
  
• Definitions of smooth manifold, diffeomorphism, category of smooth manifolds
• bicontinuity condition
• two Euclidean spaces are diffeomorphic iff they have the same dimension
• torus ≠ sphere but compact manifolds are equivalence-classable by genus
• Moebius band is not compact
• Four categories of topology, which were at first thought to be the same, but by the 60’s seen to be really different (and the maps that keep you within the same category):
  
  diffeomorphisms on smooth manifolds;
  
  
  

  
  

  
  
  piecewise-linear maps on simplicial complexes;
  
  
  
  
  homeomorphisms on sets (point-set topology)
  
  
  
  
  
  
  
  
• Those three examples of categories helped understand category and functor in general. You could work for your whole career in one category—for example if you work on fluid dynamics, you’re doing fundamentally different stuff than computer scientists on type theory—and this would filter through to your vocabulary and the assumptions you take for granted. Eg “maps” might mean “smooth bicontinuous maps” in fluid dynamics but non-surjective, discontinuous maps are possible all the time in logic or theoretical comptuer science. Functor being the comparison between the different subjects.
• The fourth, homotopy theory, was invented in the 1930’s because topology itself was too hard.
  
  

• Minute 38-40. A pretty slick proof. I often have a hard time following, but this is an exception.
• Minute 43. He misspeaks! In defining the hypercube.
• Minute 47. Homology groups relate the category of topological-spaces-with-homotopy-classes-of-mappings, to the category of groups-with-homomorphisms.

That’s the first of three lectures. Also Milnor’s thoughts almost half a century later on how differential topology had evolved since the lectures:

Hat tip to david a edwards.

What I really loved about this talk was the categorical perspective. The talks are really structured so that three categories — smooth things, piecewise things, and points/sets — are developed in parallel. Better than development of the theory of categories in the abstract, I like having these specific examples of categories and how “sameness” differs from category to category.

(Source: simonsfoundation.org)