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Let C be the monoidal category whose underlying category is Set×Set and whose tensor product is (a,b)⊗(c,d)≔(a×c⊎b×d,a×d⊎b×c) (the multiplication rule for perplex numbers). Surely there is some name for a category enriched over C?
In more elementary terms, a C-category has both morphisms and antimorphisms, with a composite of such being anti iff oddly many of the originals are anti. Examples: groups, homomorphisms, and antihomomorphisms; (strict) categories, functors, and contravariant functors; etc.
A C-category is of course an ℳ-category, but of a rather special kind.
A ℤ/2-graded category?
A Tunny category. (I happen to watched a programme recently on Bill Tutte and Tommy Flowers describing the work, in Bletchley Park, of the ’Tunny’ code breakers. (see here))
Having never heard of perplex numbers before, I wandered to a nearby wiki and took a look. I don’t understand this description:
the direct product of ℝ2 as a ring;
I expect “the direct product of” to be “the direct product of X with Y” but that doesn’t fit in here. What am I missing?
You’re right, Andrew, but I expect what was meant was “ℝ2 as a direct product ℝ×ℝ of rings”.
Edit: I went ahead and changed it.
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