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