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.
