Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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. (’_‘)
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.
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.
1 to 3 of 3