Re: [HoTT] Euler characteristic of a type

[Floris van Doorn <fpvdoorn@gmail.com>]

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Clearly we cannot define E on the whole universe, but only on a
subuniverse.
For example, we could define it on the subuniverse of types with finitely
generated homology groups. For the Euler characteristic we will also need
that the betti numbers are eventually 0. Other than that, I agree that
these properties should hold in HoTT.

On Mon, 17 Sep 2018 at 21:11, Ali Caglayan <alizter@gmail.com> wrote:

> We currently have enough machinary to (kind of) define Betti numbers (for
> homology see Floris van Doorn's thesis
> <http://florisvandoorn.com/papers/dissertation.pdf>). I am confident that
> soon we can start compting Betti numbers of some types. This would allow us
> to define the euler characterstic E : U --> N of a type. If classical
> algebraic topology tells us anything this will satisfy a lot of neat
> identities.
>
> In fact consider U as a semiring with + and * as the operations. E is a
> semiring homomorphism to N (the initial semiring (is this relavent?)). In
> other words we should have
>
> E(X + Y) = E(X) + E(Y)
> E(X * Y) = E(X)  E(Y)
>
> and even maybe, subject to some conditions, a given type family P : X -->
> U would satisfy E( (x : X) * P(x) ) = E(X) * E(P(x_0))
>
> This would be a cool invariant to have. Unfortunately as it stands,
> homology is a bit unwieldy. Perhaps rationalising spaces would help?
>
> Any thoughts or suggestions?
>
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