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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 6th 2012
   • (edited Aug 6th 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOld discussion at [[star-autonomous category]], which I think was addressed in the entry, and which I'm now moving here: > +--{: .query} [[Mike Shulman|Mike]]: Can someone fill in some examples of $*$-autonomous categories that are not compact closed? > [[Finn Lawler|Finn]]: Blute and Scott in 'Category theory for linear logicians' (from [here](http://www.site.uottawa.ca/~phil/papers/)) give an example: reflexive topological vector spaces where the topologies are 'linear', i.e. Hausdorff and with 0 having a neighbourhood basis of open linear subspaces; 'reflexive' meaning that the map $d_V$ as above is an isomorphism. It seems this category is $*$-autonomous but not compact. I don't know enough topology to make much sense of it, though. > [[Todd Trimble|Todd]]: Finn, I expect that example is in Barr's book, which would then probably have a lot of details. But I must admit I have not studied that book carefully. Also, the Chu construction was first given as an appendix to that book. > [[John Baez|John]]: I get the impression that a lot of really important examples of $*$-autonomous categories --- important for logicians, anyway --- are of a more 'syntactical' nature, i.e., defined by generators and relations. =--

   Old discussion at star-autonomous category, which I think was addressed in the entry, and which I’m now moving here:

   +–{: .query} Mike: Can someone fill in some examples of $*$-autonomous categories that are not compact closed?

   Finn: Blute and Scott in ’Category theory for linear logicians’ (from here) give an example: reflexive topological vector spaces where the topologies are ’linear’, i.e. Hausdorff and with 0 having a neighbourhood basis of open linear subspaces; ’reflexive’ meaning that the map $d_V$ as above is an isomorphism. It seems this category is $*$-autonomous but not compact. I don’t know enough topology to make much sense of it, though.

   Todd: Finn, I expect that example is in Barr’s book, which would then probably have a lot of details. But I must admit I have not studied that book carefully. Also, the Chu construction was first given as an appendix to that book.

   John: I get the impression that a lot of really important examples of $*$-autonomous categories — important for logicians, anyway — are of a more ’syntactical’ nature, i.e., defined by generators and relations. =–

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 5th 2014
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   Author: Urs
   Format: MarkdownItexI have edited the Definition-section at _[[star-autonomous category]]_ a bit for (hopefully) formatting clarity (gave definition numbers, separated the argument that the two definitions are equivalent).

   I have edited the Definition-section at star-autonomous category a bit for (hopefully) formatting clarity (gave definition numbers, separated the argument that the two definitions are equivalent).