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The cardinality of a set is just the number of elements.

To make sense of the cardinality of a category, one has to account for the morphisms. The usual definition is the sum over the isomorphism classes of 1/#automorphisms.

What's the cardinality of a higher category?

I.e., should I somehow take into account the automorphisms between the automorphisms, etcetera?

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    $\begingroup$ Everywhere that you say "category" you should say "groupoid." There is a notion of the cardinality of a category which takes into account non-invertible morphisms; it appears e.g. in work of Tom Leinster (arxiv.org/abs/math/0610260) and produces a classical invariant for posets. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 6, 2015 at 23:27

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Note that the formula "sum 1/#automorphisms" only depends on the (maximal sub-)groupoid of the category.

So your question becomes: what's a good notion of cardinality for $n$-groupoids (a.k.a. spaces with vanishing $\pi_k$ for all $k>n$)?

One possible answer is to take the sum over connected components of the alternating product of the cardinality of the homotopy groups:

$$ \sum_{components} \frac1{|\pi_1|}\cdot |\pi_2|\cdot \frac1{|\pi_3|}\cdot |\pi_4|\cdot\ldots $$

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    $\begingroup$ This is the unique answer which 1) returns $1$ for a point, 2) is additive with respect to disjoint union, and 3) is multiplicative with respect to fiber sequences (with connected base). It appears e.g. when doing Dijkgraaf-Witten theory with gauge group a "finite $n$-group." $\endgroup$
    – Qiaochu Yuan
    Commented Apr 6, 2015 at 23:23
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    $\begingroup$ This number can also be called the "Euler characteristic", as these properties are shared by the Euler characteristic of finite cell complexes. John Baez (among others) has thought a lot about this connection and has a nice page on it here. $\endgroup$
    – Eric Wofsey
    Commented Apr 7, 2015 at 1:10
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    $\begingroup$ By calling this number "Euler characteristic", you also learn what to do (in some cases) when the homotopy groups are infinite or there are infinitely many of them (if you work with $\infty$-groupoids aka homotopy types). $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 7, 2015 at 2:26
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    $\begingroup$ Qiaochu Yuan : this is the unique answer which satisfies this properties when we restrict to groupoid such that $\pi_i = \{1\}$ for $i$ large enough. One can imagine class of higher groupoid which does not satisfy this condition and for which there is a notion of "cardinal" satisfying the condition you state and which cannot be expressed under this form, at least not without some sort of renormalized product. (for example, using the Euler carateristic) $\endgroup$
    – Simon Henry
    Commented Apr 7, 2015 at 8:00
  • $\begingroup$ @Simon: yes, of course I want the obvious finiteness conditions. I would hesitate to call this number "Euler characteristic," though; to my mind the Euler characteristic is set up to have nice behavior with respect to cofiber sequences rather than fiber sequences. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 9, 2015 at 1:19

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