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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 23rd 2018
   • (edited Jun 17th 2019)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIf you recall, I was interested a while ago in which modalities are induced by the units and counits of the (differential) cohesive modalities. So we noticed that the [[jet comonad]] behaves rather like necessity since both are compositions of adjoints induced by counits, $X \to 1$ and $X \to \Im(X)$. I was wondering what else might arise in this way, and observed that Awodey and Kishida's [work](http://aiml.net/volumes/volume9/Awodey-Kishida.pdf) on models for first-order modal logic relied on modalities induced over the unit $\flat(X) \to X$. But this latter is surely very close to an instance of the first appearance of the monad concept as "standard construction" by Godement in 1958. At [wikipedia: Godement resolution](https://en.wikipedia.org/wiki/Godement_resolution) you can see $Gode = p_{\ast} \circ p^{-1}$, where $p$ maps into a space from its discretized version. They mention a 1973 book, but this is the 3rd edition of the 1958 book. So that's one of the first appearances of monad, sending a sheaf to a product of its germs, it seems. Is it important that germs appear? We were wondering about them rather than jets elsewhere.

   If you recall, I was interested a while ago in which modalities are induced by the units and counits of the (differential) cohesive modalities. So we noticed that the jet comonad behaves rather like necessity since both are compositions of adjoints induced by counits, $X \to 1$ and $X \to \Im(X)$. I was wondering what else might arise in this way, and observed that Awodey and Kishida’s work on models for first-order modal logic relied on modalities induced over the unit $\flat(X) \to X$.

   But this latter is surely very close to an instance of the first appearance of the monad concept as “standard construction” by Godement in 1958. At wikipedia: Godement resolution you can see $Gode = p_{\ast} \circ p^{-1}$, where $p$ maps into a space from its discretized version. They mention a 1973 book, but this is the 3rd edition of the 1958 book. So that’s one of the first appearances of monad, sending a sheaf to a product of its germs, it seems.

   Is it important that germs appear? We were wondering about them rather than jets elsewhere.