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Fixed the anchor for the reference. It needs to be
* {#Shulman13} [[Mike Shulman]], *Enriched indexed categories*, [TAC](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html) 2013
instead of
* [[Mike Shulman]], *Enriched indexed categories*, [TAC](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html) 2013
{#Shulman13}
(anchor at the beginning) otherwise the system gets confused.
added pointer to an online version of the relevant first page of Street 74, which seems to be the (only?) source for the statement that Jean Bénabou made “cosmos” a definition
(all other sources that Google can find cite the nLab for this, if they care to cite anything)
also added the quote:
to J. Benabou the word means “bicomplete symmetric monoidal category”, such categories $\mathcal{V}$ being rich enough so that the theory of categories enriched in $\mathcal{V}$ develops to a large extent just as the theory of ordinary categories.
Re #2, having it at the end works fine for me.
The way this entry is written (or was, I have adjusted wording slightly for more clarity) it’s no good as a quick reference for the simple concrete notion of bicompleted symmetric closed categories. I am giving Bénabou cosmos its stand-alone entry now and am redirecting the redirects to there.
touched the wording of the Idea-section,
in particular added hyperlink to base of enrichment
also touched the section layout
Does (Benabou) cosmos encompass the case where $V$ can be a bicategory?
According to Bénabou’s notion (according to Street), a cosmos is a kind of monoidal category. However, it doesn’t seem particularly helpful to me to list various different concepts known by “cosmos” on this page, as is the current status. Instead, it would appear to be more helpful to me to list the kinds of constructions one might want to consider in enriched category theory, and explain the necessary assumptions to perform these constructions. Then one can easily see what assumptions on $V$ are useful to capture the concepts one wishes to capture (whether that $V$ be monoidal, bicategorical or something else).
Fine with me, but notice that you were objecting to using “cosmos” in this general sense in recent discussion here. As a result of that comment of yours, Todd created a new entry base of enrichment for the purpose of having the entry “cosmos” be more restrictive in meaning.
I do think it’s fine to let this page disambiguate between Benabou cosmos and cosmos in that sense of Street.
As long as we’re clear on how the term ’cosmos’ occurs in the literature (I’m only aware of the two, and Benabou cosmos to me has only the meaning of bicomplete symmetric monoidal closed category), I don’t mind if there is a section with some general discussion on what assumptions occur where.
Fine with me, but notice that you were objecting to using “cosmos” in this general sense in recent discussion here.
I realise what I wrote was a little confusing.
At the moment, the page treats two distinct, but related, concepts. One is “a cosmos is a good base of enrichment”; the other is “a cosmos is a good place to do category theory”. A cosmos in the first sense induces the equipment $V\text{-}Cat$, which is a cosmos in the second sense. However, I think it’s helpful to identify the two meanings as distinct. In my previous post, I had in mind the former meaning. It’s true that perhaps the information I mentioned in my previous post about useful structure on $V$ could go on base of enrichment instead, but it could also be helpful to view “cosmos” as a concept with an attitude, and give a list of special cases of bases of enrichment that are particularly useful/comment in category theory (the most common being complete and cocomplete symmetric closed monoidal). Alternatively, perhaps making cosmos more clearly a disambiguation page is better (i.e. making it clearer that there are at least two different meanings of “cosmos”), and defer the details of the Bénabou cosmos case to base of enrichment.
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