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Created hypervirtual double category.
If I may be so bold, I think it is a bad idea to use ’virtual’ as part of the name.
As I read it, the only nod to ’virtual’ in Virtual double category is :
The relevant transformations are a “virtual” version of vertical transformations between ordinary double category
and
It is a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad T T on graphs-internal-to-Cat whose algebras are double categories. This is the origin of the name “virtual double category,” in line with the general terminology “virtual T T-algebra” of Cruttwell-Shulman for such generalized multicategories.
Thus if the only use of virtual here, as I read it, is that two things are “virtually the same” and if so, IMO, this will lead to confusion.
Personally, as a physicist, ’virtual’ carries too much of the original “virtual particle” connotation: not ’real’ (i.e., virtual) but convenient for calculations.
On top of that, from my programming days, “virtual reality”’ again emphasizes that this reality is not the ’real’ one but a virtual one.
If this is the sense of virtual in these categories, that they are not ’real’ in some context, I apologize and I’ll read more carefully.
But otherwise, based on these quotes…
A virtual double category or fc fc-multicategory is a common generalization of a monoidal category, a bicategory, a double category, and a multicategory. […] Virtual double categories are related to double categories precisely as ordinary multicategories are related to monoidal categories (see generalized multicategory and tensor product).
…something like “generalized double category” would be less confusing IMO.
Here again, for example:
The profunctors between [virtual] double categories are a similar [“virtualization” of] the notion of double profunctor between double categories.
If you change “virtualiztion” to generalization, you don’t need to use air-quotes and, as I read it, you preserve (my perceived) intent behind “virtualization”:
The profunctors between [generalized] double categories are a similar [generalization on] the notion of double profunctor between double categories.
The general principle of the name is that a “virtual $T$-algebra” is close to, but not quite, a “real” $T$-algebra. In other words, exactly what you want.
The adjective “generalized” doesn’t convey any indication of in exactly what way something has been “generalized”, whereas by using a particular term of art we can be specific.
I understand your intent and this is a good lesson for me on using the world “generalized”; thanks for that.
Just keep in mind that if a person familiar with Computer Science and/or Physics comes across this term, they may conflate among the various different definitions already in usage for ’virtual’, especially the existential context where ’virtual’ means “isn’t real” and “doesn’t exist”.
Since virtual double categories can be described as a generalized multicategory, can these augmented virtual double categories be described as kind of generalized polycategory?
Yes! Except that generalized polycategories haven’t been defined yet…
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