isomorphismes

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)

The category of categories as a model for the Platonic World of Forms by David A Edwards & Marilyn L Edwards

• Thales (7th cent. BC) made the first universal statement (proof w/o regard to the gods or mythology, just from pure reason)
• pre-Greek mathematics was essentially engineering maths.
• I owe ya a post on the illiterates in chapter 2 of James Gleick’s The Information. He tells the story of some illiterates in outer Soviet Union. According to the tale, they basically do not abstract at all. No abstract reasoning, no properties ascribed to members of a class, and so on.
  
  It sounds kind of idyllic in the way of NYT tales of the Pirahã or Jill Bolte Taylor’s story of losing the logical half of her brain. I’m not sure if Thales set us on the path to Hell or Heaven.
• “Plato set for himself the [goal] of extending geometry [beyond] triangles and circles and such, to all of human thought. He failed, but his vision has come to pass.”
• Why did Lawvere succeed where Plato and Whitehead failed?
• He had Descartes’ already-abstract notion of a function, along with
• Eilenberg & Mac Lane’s notions of category and functor.
• The definition of function for infinite sets is already implicit in the choice of “which set theory”.
• Category theory, unlike earlier formalisations (think Peano arithmetic and Goedel’s proof), is stable to the “meta” step: you do 2-categories, you do n-categories … the abstraction is ultimately a k → k+1 kind of deal rather than a “And this is the ultimate finality!” kind of deal.