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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 22nd 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexNo discussion page for [[cosmos]], and now we have $\infty$-cosmos to worry about in [Infinity category theory from scratch](http://arxiv.org/abs/1608.05314).

   No discussion page for cosmos, and now we have $\infty$-cosmos to worry about in Infinity category theory from scratch.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 5th 2018
   • (edited Jun 5th 2018)
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   Author: Urs
   Format: MarkdownItexIs there a reference for the definition of "cosmos" in the sense of [Bénabou’s definition](https://ncatlab.org/nlab/show/cosmos#bnabous_definition)? It is a bit sad that there is so little coherence across authors as to the definition of "cosmos". In Riehl-Verity's "$\infty$-cosmoi" even the default monoidal structure is dropped, it seems.(?) In discussion of elementary enriched category theory, is it still recommendable to say "cosmos" in Bénabou’s sense, or will it just add to the confusion?

   Is there a reference for the definition of “cosmos” in the sense of Bénabou’s definition?

   It is a bit sad that there is so little coherence across authors as to the definition of “cosmos”. In Riehl-Verity’s “$\infty$-cosmoi” even the default monoidal structure is dropped, it seems.(?)

   In discussion of elementary enriched category theory, is it still recommendable to say “cosmos” in Bénabou’s sense, or will it just add to the confusion?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 5th 2018
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   Author: Mike Shulman
   Format: MarkdownItexI think we just have to prefix the word: Benabou-cosmos, Street-fibrational-cosmos, Street-module-cosmos, Riehl-Verity-$\infty$-cosmos. In [enriched indexed categories](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html) I introduced "indexed Benabou-cosmoi", perhaps I should add those to the page. Riehl-Verity $\infty$-cosmoi are more like Street's fibrational cosmoi, so the monoidal structure is cartesian.

   I think we just have to prefix the word: Benabou-cosmos, Street-fibrational-cosmos, Street-module-cosmos, Riehl-Verity-$\infty$-cosmos. In enriched indexed categories I introduced “indexed Benabou-cosmoi”, perhaps I should add those to the page. Riehl-Verity $\infty$-cosmoi are more like Street’s fibrational cosmoi, so the monoidal structure is cartesian.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 5th 2018
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   Author: Mike Shulman
   Format: MarkdownItexHmm, looks like the latest changes created a new "cosmos" thread since this one was in a different discussion category.

   Hmm, looks like the latest changes created a new “cosmos” thread since this one was in a different discussion category.