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For a while I've been trying to come up with a satisfying way of defining what kind of object the Euler characteristic is. Naively it is a function $\chi:Top\to \mathbb{Z}$, but the LHS, $Top$, is not a set. I thought that it could maybe be considered a functor somehow, although I wouldn't know how?

I am aware that one can consider the Euler characteristic for complex varieties as a realisation map $\chi:K_0(Var)\to \mathbb{Z}$ where now the LHS is actually a set. However, this does not work for the topological Euler characteristic, since even the class of all homotopy-equivalence classes of topological spaces are too big to be a set (I think).

Also more generally, if I have a way of associating to objects in a category an element of a specific object of some category, then what is such an association called?

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    You will need monoidal categories formalism to have a satisfactory answer. See https://ncatlab.org/nlab/show/Euler+characteristic for a brief explanation. – 2017-01-09
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    @Caligula Thank you. If I understand this correctly the Euler characteristic is actually a morphism (in the this abstract setting)? How would you describe other invariants like for example something simple as the cardinality? So the association $X\mapsto |X|$. Is it too much to ask for a single unifying framework to capture all these things? – 2017-01-09

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