Hi David Since i could not e-mail you directly, i got delivery failure, i send my response here.
One thing that I have found is that one has to develop one's own feeling for categories.
I wouldn't say I am terribly good at abstract category theory (monads and algebras and
Kan extensions and so forth), but I work with categories more in the style of Ehresmann
- my thesis is essentially on homotopy ideas.
I have developed many perspectives on categories As the study of algebraic structures with several objects As the study of primitive mathematical universe or space (not as fancy as a topos) As an unifying tool in mathematics As a foundation of mathematics (that is structural) As an abstarction of an abstarction of an abstarction of ... (if you go to higher categories) As a generalized theory of representations If i have missed someone please let me know. I don't think i understand in what style Ehresmann worked in.
Here is a real, famous unsolved problem, which Michael Batanin is on his way to solving: Prove the homotopy theory of \infty-groupoids is equivalent to the homotopy theory of
spaces and the related
Should it not be weak oo-groupoids? I think you also mean homotopy category of spaces instead of "homotopy theory of spaces andthe related". Spaces is a bit vague but i encounter this sometimes in category theory. Usually in such statements it is ment a topological space. Is there a categorical definition of a space (not a topological space)?
But I am sure someone has already mentioned these Prove the homotopy theory of n-groupoids is equivalent to the homotopy theory
of n-types
I have seen similar problems and maby this one also. John Baez mentioned a bunch of such problems on an internet page and Ronnie Brown also in explaining pursuing stacks. Thank you for the problems. Best regards Rafael Borowiecki [For admin and other information see: http://www.mta.ca/~cat-dist/ ]