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Is there a categorification of $\pi$?

I have to admit that this is a very vague question. Somehow it is motivated by this recent MO question, which made me stare at some digits and somehow forgot my animosity about this branch of mathematics, wondering if there is a connection to the branches I love so much.

$\pi$ is the area of the unit circle, so perhaps we have to categorify the unit circle (using the projective line?) and the area of such an object. Areas are values of integrals, and there are some kind of integrals in category theory (ends), but this is really just a wild guess.

For some great examples of categorification see this list on MO, and for the meaning of categorification see this MO question or that article by Baez/Dolan. Inspired by the answers of Todd Trimble, we may consider a categorified sine function

$$\sin(X) = \bigoplus_{n \geq 0} (-1)^{\otimes n} X^{\otimes (2n+1)} / (2n+1)!$$

in any complete symmetric monoidal category, at least if we can make sense of $(-1)$.

Perhaps $(-1)$ should be the universal invertible object $\mathcal{L}$ such that the symmetry on $\mathcal{L} \otimes \mathcal{L}$ equals $-1$. In the theory of $k$-linear cocomplete symmetric monoidal categories, this is the category of super vector spaces over $k$, i.e. $\mathbb{Z}/2$-graded vector spaces, with a twisted symmetry. Here, $\mathcal{L}$ is $1$-dimensional concentrated in degree $1$. Thus, we have a sine function for super vector spaces, namely $$\sin(V) = \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n} \otimes V^{\otimes (2n+1)} / \Sigma_{2n+1}.$$ How does it look like, and can we extract something which resembles $\pi$?

Maybe we can also attack $\pi$ more directly with something like $\pi/4 = \sum_{k \geq 0} (-1)^k / (2k+1)$ which may be categorified to $$\bigoplus_{k \geq 0} \mathcal{L}^{\otimes k} / C_{2k+1},$$ but again this is just a wild guess.

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    $\begingroup$ This question got a bounty because knowledgeable users endorsed it in the Pearl Dive. $\endgroup$ Feb 13, 2020 at 14:38
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    $\begingroup$ I don't know if this is a good idea actually. I doubt that there will be an answer in the next days, which means that the bounty goes to Qiaochu's answer, which doesn't answer the question. $\endgroup$ Feb 14, 2020 at 21:29
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    $\begingroup$ Don't worry about it too much, Martin. The main point of the bounty was to draw attention. I'm not going to give it to Qiaochu's post (given the comments here). My reading of the rules is that only a new post earning at least 2 upvotes is eligible for automatic bounty (=half the bounty amount). So from Qiaochu's point of view the benefit of the bounty has been the exposure. $\endgroup$ Feb 19, 2020 at 10:13
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    $\begingroup$ Anyway, if (now or later) an upgraded answer emerges, we can renew the bounty. $\endgroup$ Feb 19, 2020 at 10:15
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    $\begingroup$ This paper of Street and Janelidze seems closely related (as is this $n$-Category Café post about it ). They construct a category $\mathbf{RSet}_g$ of "real sets", together with a "cardinality" functor $\sharp\colon\mathbf{RSet}_g\to [0,\infty]$, and exhibits an object $\Pi$ such that $\sharp \Pi=\pi$. This doesn't really give much insight though, since the construction is based on the binary expansion of $\pi$ so you could do it for any number. $\endgroup$
    – Arnaud D.
    Feb 19, 2020 at 19:30

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One approach to questions of this general nature is to find a groupoid whose groupoid cardinality is equal to the number in question; this is a form of groupoidification. For example, the groupoid cardinality of the groupoid $\text{FinSet}_0$ of finite sets and bijections is $\sum \frac{1}{n!} = e$, so this is a reasonable categorification of $e$; in fact arguably this is "the" categorification of $e$ and goes a long way towards explaining its prevalence in mathematics. Similarly, for any finite set $X$, the groupoid cardinality of the groupoid of $X$-colored finite sets and color-preserving bijections is $\sum \frac{|X|^n}{n!} = e^{|X|}$. Note that this groupoid is the coproduct of $|X|$ copies of $\text{FinSet}_0$; see also this math.SE question.

In a TWF188 John Baez mentions that he looked into the problem of finding a "natural" groupoid whose cardinality is $\pi$, but that he (and possibly some collaborators) weren't able to come up with any nice examples. So possibly this is the wrong direction to go in the particular case of $\pi$.

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  • $\begingroup$ Can you make sense of the «grupoid of $(-1)$-colored finite sets and finite color-presrving bijections» so as to categorify $e^{-1}$ (which comes up in a sensible combinatorial way, of course) $\endgroup$ Jan 13, 2011 at 3:30
  • $\begingroup$ @Mariano: there are a couple of ways to think about negative numbers in this setting, some of which are described at math.ucr.edu/home/baez/counting . I guess one can replace groupoid cardinality with a form of orbifold Euler characteristic. For example, in Schanuel's approach the interval (0, 1) has Euler characteristic -1, so a (-1)-colored finite set is just a function from that set to (0, 1), and the orbifold Euler characteristic of the n-element (-1)-colored sets, appropriately defined, should be (-1)^n/n!. Note that in this setting it's natural to think of 1/n! as the measure $\endgroup$ Jan 13, 2011 at 3:35
  • $\begingroup$ ... of a fundamental domain for the action of S_n on (0, 1)^n, which is one way this number shows up in probability. $\endgroup$ Jan 13, 2011 at 3:37
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    $\begingroup$ Nice. Now do $e\cdot e^{-1}=1$ :) $\endgroup$ Jan 13, 2011 at 3:47
  • $\begingroup$ @Mariano: e * e^{-1} is the Euler characteristic of the groupoid of (0, 1]-colored sets, and in Schanuel's approach the Euler characteristic of (0, 1] is 0. :) $\endgroup$ Jan 13, 2011 at 3:51

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