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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 29th 2012
• (edited Jun 29th 2012)

I felt it was time for another table: homotopy-homology-cohomology

The structure is just a first attempt, begun in a brief moment of leisure. I’ll try to think about how to improve on it. Let me know what you think.

I have started to include this into relevant entries.

• CommentRowNumber2.
• CommentAuthorjim_stasheff
• CommentTimeJun 29th 2012
If cohomology is from maps to a general A
then homotopy should not be restricted to maps from a sphere

Ext and Tor are functors of two variables
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 29th 2012
• (edited Jun 29th 2012)

If cohomology is from maps to a general A then homotopy should not be restricted to maps from a sphere

Yes, we play around with that observation in homotopy (as an operation). One may generally speak about homotopy with “co-coefficients”. Unfortunately this is very much non-standard, so I am not sure if it would do the table much good.

Also, for any two arbitary $B$ and $A$, the homotopy of $A$ with co-coefficients in $B$ is the cohomology of $B$ with coefficients in $A$. So it may not be worth adding more terminology here.

Ext and Tor are functors of two variables

Yes, that’s why $Ext$ appears in two slots in the table. In order to have Tor appear in 2 slots we would have to make it clear that we allow the tensor product to be non-symmetric. That’s maybe a bit too heavy for an overview table like this.

• CommentRowNumber4.
• CommentAuthorjim_stasheff
• CommentTimeJun 30th 2012
@ Urs 3: the homotopy of A with co-coefficients in B

the existing terminology is the homotopy of A with coefficients...
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 30th 2012
• (edited Jun 30th 2012)

Really? Do you ever say “The homotopy of a space with coefficients in the $n$-sphere” for the ordinary $n$th homotopy group?

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeJun 30th 2012

I have seen it in Russian textbooks at least (5), By the way, saying that homological algebra has Ext and homotopy theory has RHom is not entirely fair. Old fashioned homological algebra has Ext-s, which in totality form RHom, as seen already in derived category picture.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 30th 2012
• (edited Jun 30th 2012)

Let’s see, what is it that is not fair?

The entry say, it seems, that what in homological algebra is called Ext is in homotopical category theory called the derived hom. Seems okay to me. How would you want to change the table?