Dan wrote:

> This is a minor point, but since Michael Hong brought [into discussion the notation](https://forum.azimuthproject.org/discussion/comment/18388/#Comment_18388), I think the definition:

> > for every two objects \\(x,y\\), one specifies an element \\(\mathcal{X}(x,y)\\) of \\(\mathcal{V}\\).

> should say "an element \\(\mathcal{X}(x,y)\\) of \\(V\\)" rather than "[...] of \\(\mathcal{V}\\)" (which denotes a tuple, not a set).

That's true, given Fong and Spivak's unfortunate decision to use different names for the monoidal preorder \\(\mathcal{V}\\) and its set of elements, \\(V\\).

This is mainly unfortunate because they never did it before when discussing monoidal preorders! Bringing it in now seems more confusing than helpful to me.

Second, I'm never in favor of building such distinctions into the notation. For example, I don't say "a group \\(\mathcal{G} = (G, \cdot, 1) \\)", distinguishing the group \\(\mathcal{G}\\) and its underlying set \\(G\\). It's a waste of brain cells. There are times when this distinction is very important, but there are other ways to convey it.

> This is a minor point, but since Michael Hong brought [into discussion the notation](https://forum.azimuthproject.org/discussion/comment/18388/#Comment_18388), I think the definition:

> > for every two objects \\(x,y\\), one specifies an element \\(\mathcal{X}(x,y)\\) of \\(\mathcal{V}\\).

> should say "an element \\(\mathcal{X}(x,y)\\) of \\(V\\)" rather than "[...] of \\(\mathcal{V}\\)" (which denotes a tuple, not a set).

That's true, given Fong and Spivak's unfortunate decision to use different names for the monoidal preorder \\(\mathcal{V}\\) and its set of elements, \\(V\\).

This is mainly unfortunate because they never did it before when discussing monoidal preorders! Bringing it in now seems more confusing than helpful to me.

Second, I'm never in favor of building such distinctions into the notation. For example, I don't say "a group \\(\mathcal{G} = (G, \cdot, 1) \\)", distinguishing the group \\(\mathcal{G}\\) and its underlying set \\(G\\). It's a waste of brain cells. There are times when this distinction is very important, but there are other ways to convey it.