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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeOct 20th 2017
• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeOct 20th 2017

I’d like to write something about non-symmetric $\ast$-autonomous categories, but I can’t decide whether it should go at star-autonomous category (which is presently only about the symmetric version) or on its own page. Any thoughts?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeOct 20th 2017

Is it that in the literature people generally say $\ast$-autonomous category and intend the symmetric case? Then there’s annoying ’not necessarily symmetric’ kind of qualification. Isn’t it better to let the general term win out, then specify ’symmetric’?

If the latter, is there enough material to split pages, or could the symmetric case just feature in a subsection?

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeOct 20th 2017

Well, I think it’s more that the original notion was symmetric, with the non-symmetric case introduced rather later. My impression (although I’m not an expert in this field) is that recently people have been pushing to make the general term win out, as you say. But whoever wrote our page chose to talk only about the symmetric case, so I didn’t want to unilaterally make the change in case they had a good reason for that choice.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 20th 2017

I’m not sure it was really a ’choice’, in the sense that the author held the symmetric notion in one hand and the non-symmetric one in the other, and then weighed in favor of the symmetric notion. As far as I know, the symmetric notion has been studied much more (for example, from the perspective of studying coherence problems, I’m much more at home with the symmetric notion).

I for one have no objection to making adjustments to the page to redress this. But once one has the non-symmetric notion, it seems one is invited to ’oidify’ (horizontally categorify); has an appropriate 2-dimensional notion been considered in the literature?

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 20th 2017

it seems one is invited to ’oidify’ (horizontally categorify)

That’s the name of this thread: linear bicategory. (-:

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeOct 20th 2017

(Or, more precisely, a linear bicategory with linear adjoints for all morphisms.)

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeOct 20th 2017

Re #6: oh, so it is! Helps to look at the thread title now and then. :-)

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeOct 20th 2017

Well, it’s partly my fault for taking the thread in a different (though related) direction in #2.