I think this is an issue of different conventions in different communities. I don’t see a problem with sticking with $\ast$, which is common in some communities, and people who are familiar with it would probably be just as confused by $\top$ as you are by $\ast$. The important thing is to define notation when there is any potential for confusion, and the entry does that in the first bullet point where it said “the tensor unit is a terminal object $\ast$”. If this is too far away from the use of $\ast$ in the previous line, we could just remove it from the previous line and write “let $\mathcal{V}$ be a cartesian monoidal category” and only introduce the notation $\ast$ for the terminal object in the bullet point that states explicitly that it’s a terminal object.

]]>Thanks for saying, now I understand your first comment. I am regularly using “$\ast$” for terminal objects, it’s all over the nLab.

]]>Agreed on the other point. ]]>

Have to admit that I am not sure what you are getting at with either of these comments.

We need a cartesian monoidal base of enrichment in order to state the enriched existence of inverses. The entry is clear about this, and the notation seemed just fine.

Whether this encompasses “ordered groups” is something that the entry on ordered groups needs to deal with. The notion of enriched groupoids is what it is.

]]>Using * for the terminal object, makes it look like you have a cartesian AND monoidal category.

]]>added the example of strict 2-groupoids (here)

]]>Fixed a typo. $id_Y$ should have benn $id_X$.

]]>I know, that’s why I suggested the reformulation “serving as…”.

Now *base of enrichment* is redirecting to *cosmos*. Feel invited to split it off as a separate entry.

I think varkor is suggesting that the cosmos (or cartesian cosmos) assumptions are rather more than what is needed. In that case, I wouldn’t say “cosmos for enrichment”, but “base for enrichment” or something similar. Not sure we have a page that accommodates that extra generality, but if not, I think we should.

]]>how about formulating it this way:

]]>…a

cartesian monoidal category(serving as an cosmos for enrichment)…

Presumably the assumption that $V$ be a cosmos can be relaxed to simply asking for $V$ to be a cartesian monoidal category?

]]>added also pointer to

- John F. Jardine, §9.3 in:
*Local homotopy theory*, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]

(who does finally switch terminology from “simplicial groupoid” to “groupoid enriched over simplicial sets” but still does not state the definition of enriched groupoids)

together with more comments on the history of the notion.

]]>added this pointer:

- Jacopo Emmenegger, Fabio Pasquali, Giuseppe Rosolini, §3 in:
*Elementary fibrations of enriched groupoids*, Mathematical Structures in Computer Science**31**9 (2021) 958-978 [doi:10.1017/S096012952100030X]

for completeness

]]>