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finding that an entry like this has been missing all along (all we seem to have had was this paragraph at enriched category) I have now created it with some minimum content, for completeness
Is “enriched product category” a common term? I think I’ve only seen this construction referred to as the “tensor product of enriched categories” (or similar). I would probably expect “enriched product category” to refer to a cartesian product in the 2-category VCat.
No, I tried to find the most systematic term.
How about “enriched tensor product category”, as a compromise?
My experience matches varkor’s.
My experience is no different, but I make decisions based on internal consultation :-).
Feel free to rename the entry, but please keep the current title as redirect, not to break links.
(I’ll be offline in a minute.)
Coming back online…
Just for the heck of it I asked Google who else says “enriched product category” and the answer is:
Bartosz Milewski (here and here) and hackage.haskell
(here).
The close variant “product of enriched categories” (without adjective) is used (apart from Bartosz Milewski) by John Baez (here), Forcey et al. (here and here) and more.
A thesis here even says “Kelly’s product of enriched categories”.
In any case, in practice the key advantage of “enriched product category” is the systematics, for instance it allows one to say (as Bartosz does here) things like “… the (enriched) product category…”.
But I have added more alternative redirects and won’t care if the entry gets renamed.
In any case, in practice the key advantage of “enriched product category” is the systematics, for instance it allows one to say (as Bartosz does here) things like “… the (enriched) product category…”.
I suppose my opinion is that “tensor product of (enriched) categories” works just as well, though I grant it’s slightly more of a mouthful. One could probably elide “product” instead and simply say “tensor of (enriched) categories”, which is roughly as as the current name.
added pointer to:
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