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1. • CommentRowNumber1.
   • CommentAuthorPatrick
   • CommentTimeDec 26th 2015
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   Author: Patrick
   Format: MarkdownItexI recently found the nLab [cohomology](https://ncatlab.org/nlab/show/cohomology) entry and was intrigued by the claim that any reasonable kind of cohomology ought to "have a natural interpretation in terms of connected components of hom-spaces in $(\infty,1)$-categories." Does quantum cohomology, as developed by Fulton and Pandharipande in their [Notes on stable maps and quantum cohomology](http://arxiv.org/abs/alg-geom/9608011), fit into this scheme? It is not obvious to me that it does. I checked the entries on quantum sheaf cohomology and Gromov-Witten invariants, but they do not address this question.

   I recently found the nLab cohomology entry and was intrigued by the claim that any reasonable kind of cohomology ought to “have a natural interpretation in terms of connected components of hom-spaces in $(\infty,1)$-categories.”

   Does quantum cohomology, as developed by Fulton and Pandharipande in their Notes on stable maps and quantum cohomology, fit into this scheme? It is not obvious to me that it does. I checked the entries on quantum sheaf cohomology and Gromov-Witten invariants, but they do not address this question.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 4th 2016
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   Author: Urs
   Format: MarkdownItexNot sure if this helps, but just for the record I'll say that what is called quantum cohomology is a certain graded direct sum of sheaf cohomology groups, and equipped with a certain Frobenius algebra structure. So to some extent this is just plain sheaf cohomology with extra structure. Of course it may still be that this has a nice more abstract description, not sure.

   Not sure if this helps, but just for the record I’ll say that what is called quantum cohomology is a certain graded direct sum of sheaf cohomology groups, and equipped with a certain Frobenius algebra structure. So to some extent this is just plain sheaf cohomology with extra structure. Of course it may still be that this has a nice more abstract description, not sure.