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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeNov 2nd 2017
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   Author: Mike Shulman
   Format: MarkdownItexI just noticed that the [[Isbell envelope]] $E(\mathcal{T})$ embeds fully-faithfully into the [[Chu construction]] $Chu(Set^{\mathcal{T}^{op}\times\mathcal{T}},\hom_{\mathcal{T}})$. An object of the latter consists of two endo-profunctors $P,F : \mathcal{T}^{op}\times\mathcal{T}\to Set$ together with a natural transformation $P\times F \to \hom_{\mathcal{T}}$, and the image of $E (\mathcal{T})$ consists of those triples such that $P$ depends only on $\mathcal{T}^{op}$ and $F$ depends only on $\mathcal{T}$. Does this mean anything? Is it good for anything?

   I just noticed that the Isbell envelope $E(\mathcal{T})$ embeds fully-faithfully into the Chu construction $Chu(Set^{\mathcal{T}^{op}\times\mathcal{T}},\hom_{\mathcal{T}})$. An object of the latter consists of two endo-profunctors $P,F : \mathcal{T}^{op}\times\mathcal{T}\to Set$ together with a natural transformation $P\times F \to \hom_{\mathcal{T}}$, and the image of $E (\mathcal{T})$ consists of those triples such that $P$ depends only on $\mathcal{T}^{op}$ and $F$ depends only on $\mathcal{T}$. Does this mean anything? Is it good for anything?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 27th 2018
   • (edited May 27th 2018)
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   Author: David_Corfield
   Format: MarkdownItexStumbling upon this on a Sunday morning wander, we talk about duality in an Isbell context ([Isbell envelope](https://ncatlab.org/nlab/show/Isbell%20envelope#isbell_duality), [[Isbell duality]]) and in a Chu context ([Chu construction](https://ncatlab.org/nlab/show/Chu+construction#chu_spaces_general_case)), but without relating them. Isbell pages don't mention 'Chu', and vice versa. It seems ([Linear Logic for Constructive Mathematics](https://golem.ph.utexas.edu/category/2018/05/linear_logic_for_constructive.html)) that the Chu outlook is good for setting constructions in terms of linear logic, so might we expect that it could be used to formulate that work on linear HoTT? If Chu constructions act interestingly on toposes of various kinds (Heyting algebras, etc), then why not on $(\infty, 1)$-toposes? How about a $Chu((\infty, 1)$-topos, object classifier)? That would seem to take us in a [[integral transforms on sheaves]] direction, with a fibration over a product as an integral kernel.

   Stumbling upon this on a Sunday morning wander, we talk about duality in an Isbell context (Isbell envelope, Isbell duality) and in a Chu context (Chu construction), but without relating them. Isbell pages don’t mention ’Chu’, and vice versa.

   It seems (Linear Logic for Constructive Mathematics) that the Chu outlook is good for setting constructions in terms of linear logic, so might we expect that it could be used to formulate that work on linear HoTT? If Chu constructions act interestingly on toposes of various kinds (Heyting algebras, etc), then why not on $(\infty, 1)$-toposes?

   How about a $Chu((\infty, 1)$-topos, object classifier)? That would seem to take us in a integral transforms on sheaves direction, with a fibration over a product as an integral kernel.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 28th 2018
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   Author: David_Corfield
   Format: MarkdownItexI see that Vaughan Pratt found a common generalization of the Isbell envelope and Chu spaces in the form of his _communes_, [talk](http://boole.stanford.edu/pub/sortprop.pdf) and [paper](https://pdfs.semanticscholar.org/c496/5eb1b0ce45befddfe9da964c81adbfd16028.pdf).

   I see that Vaughan Pratt found a common generalization of the Isbell envelope and Chu spaces in the form of his communes, talk and paper.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMay 31st 2018
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   Author: Mike Shulman
   Format: MarkdownItexIt's a bit funny: the "prototypical" Chu construction on a 1-category is $Chu(Set,Prop)$ ([[Chu spaces]]), which as you say categorifies to $Chu(\infty Gpd, \infty gpd)$. But since (-2)-categories are trivial, its obvious decategorification would be $Chu(Prop,1)$, whereas in LLCM I used instead $Chu(Prop,0)$. The latter also categorifies to $Chu(Set,0)$ and $Chu(\infty Gpd,0)$. And any of these can be done "fibered" over the base DTT for $Set$ or $\infty Gpd$, producing a model of [[dependent linear type theory]] with linear types depending on nonlinear ones; the fibered $Chu(Set,0)$ appears briefly in Remark 9.12 of LLCM.

   It’s a bit funny: the “prototypical” Chu construction on a 1-category is $Chu(Set,Prop)$ (Chu spaces), which as you say categorifies to $Chu(\infty Gpd, \infty gpd)$. But since (-2)-categories are trivial, its obvious decategorification would be $Chu(Prop,1)$, whereas in LLCM I used instead $Chu(Prop,0)$. The latter also categorifies to $Chu(Set,0)$ and $Chu(\infty Gpd,0)$. And any of these can be done “fibered” over the base DTT for $Set$ or $\infty Gpd$, producing a model of dependent linear type theory with linear types depending on nonlinear ones; the fibered $Chu(Set,0)$ appears briefly in Remark 9.12 of LLCM.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 1st 2018
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   Author: Mike Shulman
   Format: MarkdownItexThere's another construction that "combines the idea of a category of presheaves with that of the Chu construction" in Hyland's [Proof theory in the abstract](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2002/pta02.pdf), section 5, called the "envelope of a polycategory".

   There’s another construction that “combines the idea of a category of presheaves with that of the Chu construction” in Hyland’s Proof theory in the abstract, section 5, called the “envelope of a polycategory”.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 2nd 2018
   • (edited Jun 2nd 2018)
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   Author: David_Corfield
   Format: MarkdownItexSo if the $Chu(n Gpd, 0)$ family corresponds to a linear type theory, how would one describe the object classifier family? It would seem to include a groupoidified linear algebra.

   So if the $Chu(n Gpd, 0)$ family corresponds to a linear type theory, how would one describe the object classifier family? It would seem to include a groupoidified linear algebra.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJun 2nd 2018
   • (edited Jun 2nd 2018)
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   Author: Mike Shulman
   Format: MarkdownItex*All* Chu constructions are $\ast$-autonomous and hence have a linear type theory.

   All Chu constructions are $\ast$-autonomous and hence have a linear type theory.