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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 2nd 2014

created finite homotopy type, just for completeness.

This just a distraction when I saw that it was missing,while I was really going to create an entry on truncated homotopy types with finite homotopy groups.

The main problem about them is that nobody agrees on how to call them ;-)

In groupoid cardinality they have been called “tame”, some call them $\pi$-finite,I suppose, and homological algebra suggests “of finite type”, which in itself is good, however rather badly goes together with the crucially different “finite homotopy type”.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeFeb 2nd 2014
• (edited Feb 2nd 2014)

The same sort of problem occurs with profinite homotopy types. This is complicated by the fact that simplicial profinite spcaes and pro-simplicial finite spaces, and pro-finite homotopy types all look to be almost the same ….. but the morphisms go all over the place. (There is a paper by Isaksen that discusses this, but it seems to have been controversial.)

Urs: do you know the paper: G. J. Ellis, Spaces with finitely many non-trivial homotopy groups all of which are finite, Topology, 36, (1997), 501–504, ISSN 0040-9383.

This might be relevant.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 2nd 2014

thanks for the reference. I have put it into a stub homotopy type with finite homotopy groups for the moment. But that needs to be expanded.

If you feel like adding a comment on how the terminology is even more of a mess for pro-theory, please feel invited.

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeFeb 2nd 2014

I would have to sort out the mess to my own satisfaction first!!! :-( but I should do that.