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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeMay 9th 2011

I did not classify this under latest changes as I expect some discussion after some questions I will ask later this week in this thread, related to thread.

Before that let me archive an old query from center here:

Mike Shulman: It seems to me that the monoid of endofunctions of a set would be the decategorification of $[C,C]$, not $[C,C](Id_C,Id_C)$. The center of a set should be the endotransformations of the identity endofunction (of which there is only one, the identity). Moreover, since the center of a category is a commutative monoid, and the center of a bicategory is a braided monoidal category (horizontally categorifying the center of a monoidal category), the center construction acts like a knights-move on the periodic table; thus it makes sense that the center of a set should be a symmetric monoidal $(-1)$-category, i.e. “True.”

Toby: I didn't look closely enough at your centre of a category then! What you say here contradicts what you wrote below —that the centre of a $k$-tuply monoidal $n$-category is a $(k+1)$-tuply monoidal $n$-category, which is my understanding— and contradicts what John Baez writes in Section 1.1 (page 5) of HDA1.

Mike Shulman: I expanded it a lot; let me know if this is any better. It’s even more confusing than I realized at first.

Toby: I understand it, but it still doesn't actually include the centre of a set (or more generally of an $n$-category) that I learnt about from HDA1. Now, maybe that's not a very useful concept … except that it fits in so well with the centre of a $k$-tuply monoidal $n$-category for $k \gt 0$! How many of these $k + 1$ different centres that a $k$-tuply monoidal $n$-category has are used?

Mike Shulman: Hmm, that appears to be a different notion of center than the one I was used to. (I didn’t make this one up, but I don’t remember where I learned it; has anyone else seen it?) Perhaps that one is better; it also has the advantage that it gives automatically that the center of a $k$-tuply monoidal $n$-category is $(k+1)$-tuply monoidal.

Toby: I'll try to get John's attention.

Mike Shulman: The HDA1 definition also leaves me wondering: why make only that particular choice of what level to stop at? Is there anything interesting to say about other choices?

Toby: H'm, John is ’too busy’ until September 22. (’_‘)

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeMay 9th 2011
• (edited May 11th 2011)

OK, first question.

There is also sometimes used another notion of a center of an object $X$ in an arbitrary monoidal category $C$. It is by the definition simply the hom-set $C(1,X)$. In important cases, one can consider the right derived functor of $X\mapsto C(1,X)$ which corresponds to Hochschild cohomology. Now, an object $X$ is said to be central if the center in this sense, $C(1,X)$, has the special property that for every object $Y$ and for every pair $f,g: X\to Y$ such that $f\neq g$ there is $c\in C(1,X)$ such that $f\circ c \neq g\circ c$. If $C$ is the category of $R$-bimodules and $R$ commutative, then we get that the subcategory $ZC$ of central objects is the subcategory of central bimodules. There are many other special cases. I do not see how this corresponds to the treatment in center.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeMay 11th 2011

It could be that this is just “yet another” notion of center which should be added to the page center alongside the existing ones. I suspect that the center of a monoidal n-category may be related to its center, in your sense, in the ((n+1)-)category of monoidal n-categories, but that’s just a guess.