Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] Euler characteristic of a type
@ 2018-09-17 19:11 Ali Caglayan
  2018-09-17 22:36 ` Floris van Doorn
  0 siblings, 1 reply; 9+ messages in thread
From: Ali Caglayan @ 2018-09-17 19:11 UTC (permalink / raw)
  To: Homotopy Type Theory


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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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^ permalink raw reply	[flat|nested] 9+ messages in thread

end of thread, other threads:[~2018-09-19  3:52 UTC | newest]

Thread overview: 9+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2018-09-17 19:11 [HoTT] Euler characteristic of a type Ali Caglayan
2018-09-17 22:36 ` Floris van Doorn
2018-09-17 23:07   ` Ali Caglayan
2018-09-18  6:25     ` Michael Shulman
2018-09-18 10:54       ` Ali Caglayan
2018-09-18 16:13         ` Michael Shulman
2018-09-18 19:11           ` Ali Caglayan
2018-09-19  0:04           ` Ali Caglayan
2018-09-19  3:52             ` Michael Shulman

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