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Jun 26, 2020 at 1:19 comment added Praphulla Koushik For anyone else looking for more references, they can see “17.7 Internal Hom” of Masaki Kashiwara and Pierre Schapira’s book titled “Categories and Sheaves”.
Jun 24, 2020 at 3:22 comment added Praphulla Koushik Sorry. I was looking at table of contents and was lost in 8+ table of contents. Thanks for the reference. I feel like I might enjoy reading parts of this book.
Jun 24, 2020 at 1:49 comment added Dmitri Pavlov @PraphullaKoushik: Sections 1.57–1.65.
Jun 24, 2020 at 0:54 comment added Praphulla Koushik That book looks interesting (too lengthy though:)) but I do not see where does they talk about the internal hom description :O
Jun 23, 2020 at 23:54 history bounty ended Praphulla Koushik
Jun 23, 2020 at 23:23 comment added Dmitri Pavlov @PraphullaKoushik: Patrick Iglesias-Zemmour, Diffeology. It's all about diffeological spaces (and sheaves of sets on manifolds).
Jun 23, 2020 at 16:07 comment added Praphulla Koushik Yes, I saw that explanation only recently.. I am still trying to look at the motivation behind such description of the inner-hom/internal-hom $\text{Hom}(M,N)$.. I am reading MacLane and Moedijk's Sheaves in Geometry and Logic (3.6 first properties of categories of sheaves).. Do you have any suggestion on any other book that I should be looking at?
Jun 23, 2020 at 14:51 comment added Dmitri Pavlov @PraphullaKoushik: Just to be clear, Hom(M,N) denotes the sheaf whose S-points are smooth maps S⨯M→N. In particular, the points of Hom(M,N) are precisely the smooth maps M→N. Hom is right adjoint to the (categorical cartesian) product of sheaves of sets, defined objectwise: (F⨯G)(S)=F(S)⨯G(S).
Jun 22, 2020 at 21:04 vote accept Praphulla Koushik
Jun 22, 2020 at 21:05
Jun 21, 2020 at 4:42 comment added Praphulla Koushik I have some other questions, but I should at least read the Freed-Hopkins paper first :) Thanks for this answer (really).. :) :)
Jun 21, 2020 at 4:39 comment added Praphulla Koushik This is interesting.. For manifolds $M.N$, as the mapping space $\text{Map}(M,N)$ is not expected to have a nice smooth structure, you would like to see $M$ and $N$ as objects of the category $Sh(\text{Man})$ and taking then take the internal-hom object of $M,N$ (I do not know why it exists, I will check details; I guess here "the tensor" is tensor product of sheaves)..
Jun 20, 2020 at 22:09 history edited Dmitri Pavlov CC BY-SA 4.0
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Jun 20, 2020 at 21:57 history answered Dmitri Pavlov CC BY-SA 4.0