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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeNov 14th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexLet $C$ be the monoidal category whose underlying category is $Set \times Set$ and whose tensor product is $(a, b) \otimes (c, d) \coloneqq (a \times c \uplus b \times d, a \times d \uplus b \times c)$ (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|$\mathcal{M}$-category]], but of a rather special kind.

   Let $C$ be the monoidal category whose underlying category is $Set \times Set$ and whose tensor product is $(a, b) \otimes (c, d) \coloneqq (a \times c \uplus b \times d, a \times d \uplus b \times c)$ (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 $\mathcal{M}$-category, but of a rather special kind.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 14th 2011
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   Author: Mike Shulman
   Format: MarkdownItexA $\mathbb{Z}/2$-graded category?

   A $\mathbb{Z}/2$-graded category?

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeNov 14th 2011
   • (edited Nov 14th 2011)
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   Author: Tim_Porter
   Format: MarkdownItexA 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](http://en.wikipedia.org/wiki/Lorenz_cipher)))

   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))

4. • CommentRowNumber4.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 14th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexHaving 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 $\mathbb{R}^2$ 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?

   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 $\mathbb{R}^2$ 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?

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 14th 2011
   • (edited Nov 14th 2011)
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   Author: Todd_Trimble
   Format: MarkdownItexYou're right, Andrew, but I expect what was meant was "$\mathbb{R}^2$ as a direct product $\mathbb{R} \times \mathbb{R}$ of rings". Edit: I went ahead and changed it.

   You’re right, Andrew, but I expect what was meant was “$\mathbb{R}^2$ as a direct product $\mathbb{R} \times \mathbb{R}$ of rings”.

   Edit: I went ahead and changed it.