Timeline for The definition of homotopy in algebraic topology
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 12, 2010 at 7:59 | comment | added | Dylan Wilson | I think the best exposition on compactly generated spaces is the original one in Steenrod's paper ("A Convenient Category of Topological Spaces")... May's is very terse. Another good one (very close to Steenrod) is Gray's exposition. | |
| Aug 12, 2010 at 7:46 | vote | accept | Mark | ||
| Aug 11, 2010 at 21:15 | comment | added | Peter LeFanu Lumsdaine | A category-theoretic gloss on this would be: $\hom(X,Y^I)$ may be what we first think of, but whenever it exists and behaves well, it'll be isomorphic to $\hom(X \times I, Y)$, and the latter behaves well in many more categories; so we transfer our intuition for the former over to the latter (via the isomorphism that “should exist”), and use the latter instead. | |
| Aug 11, 2010 at 19:40 | comment | added | Tim Porter | @Mark You mentioned manifolds and CW complexes, but as suggested in another comment C(I,X) is usually not as nice as X. Even if you want smooth paths and smooth homotopies in a smooth manifold the second definition is easier to tweek to make acceptable and useful, whilst the function space definition hits quite big problems. | |
| Aug 11, 2010 at 19:18 | comment | added | Mark | Fair enough, though I don't think the students would find the compact-open topology confusing once you tell them that it's just the topology of uniform convergence on compact sets (for metrizable spaces, which is anyway the case most of the time), which they have probably encountered several times in calculus courses. | |
| Aug 11, 2010 at 18:13 | comment | added | Andy Putman | @Mark : Working with $X^{Y \times Z}$ is often more better than working with $(X^Y)^Z$ because it makes it easier to work with topological properties of $Y \times Z$. Also, when you're writing an intro textbook it frees you from having to discuss the compact-open topology, which many students find confusing. | |
| Aug 11, 2010 at 17:52 | comment | added | Mark | Thanks for the answer and the references. What do you think about my second question, though? According to my (limited) experience with algebraic topology, it seems that all objects which one is interested in (CW complexes, topological manifolds,...) are compactly generated. Hell, I think they are all even metrizable. So I don't see a good reason not to specialize to compactly generated spaces, which allows us to use the compact-open topology, as you mentioned. | |
| Aug 11, 2010 at 17:23 | history | answered | Andy Putman | CC BY-SA 2.5 |