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In order to accompany the nCafe discussion I have started to add some content to the entry Euler characteristic
added the notion of “homotopical Euler characteristic” for “$\infty$-groupoid cardinality” and mentioned how Leinster cardinality interpolates between that and “homological” Euler characteristic.
Added stub sections for Euler char of enriched and higher categories and for Gauss-Bonnet theorem
I have added some more discussion of the symmetric-monoidal Euler characteristic, and rearranged a little bit. I think it would be better to group the various definitions by style, rather than by input data – the “homological” Euler characteristic of a topological space fits into the same picture as the Euler characteristic of a chain complex and the symmetric monoidal notion, whereas the “homotopical” one is a different sort of beast.
I’m also not sold on the phrase “homotopical Euler characteristic”. The notion of “Euler characteristic of a topological space” is ancient and well-established, and I don’t think we should try to bifurcate its meaning. Especially since we have a perfectly good alternate name: “$\infty$-groupoid cardinality”. I know that the “Euler characteristic of a category” reduces to both in special cases, but I would rather just say that “the Euler characteristic of a groupoid, considered as a category, is its groupoid cardinality”.
Sure, thanks. I have edited a bit more, accordingly. Also added a brief comment on Euler char of higher categories and added plenty of references, using John’s material.
Looks great, thanks!
I added the reference by Efremovich and Rudyak which singles Euler characteristic of finite CW complexes as the only (up to multiplicative constant) additive homotopy invariant and the related 1966 reference by C.T.C. Wall on arithmetic invariants of finite simplicial complexes, with a free pdf link at Canadian J. Math.
added the following statement (here):
Let $X$ and $Y$ be closed manifolds (topological or differentiable) of even dimension. Then the Euler characteristic of their connected sum is the sum of their separate Euler characteristics, minus 2:
$\chi \big( X \sharp Y \big) \;=\; \chi \big(X\big) + \chi \big(Y\big) -2$added this also to connected sum
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