Hurewicz theorem in HoTT

[Ali Caglayan]

Is there any progress on proving the Hurewicz theorem in HoTT?

I stumbled across this mathoverflow question: https://mathoverflow.net/questions/283199/an-abstract-nonsense-proof-of-the-hurewicz-theorem

I wonder if we can adapt the following argument:

Define H_n(X; Z) as [S^{n+t}, X /\ K(X, t)] for some large t and pointed space X. The Hurewicz map is induced by a generator g : S^t->K(Z, t) of H_n(S^n). Given by postcomposition with (id_X /\ g).

H : [S^{n+t}, X /\ S^t] ---> [S^{n+k}, X /\ K(Z, t)]

Now since X is (n-1)-connected and it can be shown that g is n-connected (an (n+1)-equivalence in the answer), then it follows that (id_X /\ g)_* is an isomorphism.

The only trouble I see with this argument working is the definition of homology. Instead of having a large enough t floating around we would have to use a colimit and that might get tricky. Showing that g is n-connected is possible I think using some lemmas about modalities I can't name of the top of my head.

Do you think this argument will work? Let me know what you all think.

Thanks,

Ali Caglayan

[Luis Scoccola]

Hello Ali,

Dan Christensen and I have a draft paper that proves the Hurewicz
isomorphism theorem in HoTT, and we plan to make it available sometime
soon. We have not formalized the proof, and it also relies on the
adjunction between pointed mapping spaces and smash products and some
coherences of the associativity of the smash product that have not been

completely formalized.

Our starting point is similar to yours, so to answer your question, I
would say that yes, that idea can be made to work.  However, at least in
our approach, most of the work is in proving that the Hurewicz map is
actually a group homomorphism. In that argument we really need the fact
that homology is defined as a colimit and we can't just use some
sufficiently large t.

Another key ingredient in the approach is the relation between smash
products of types and tensor products of their homotopy groups, which
we had to develop for this purpose.

For completeness, note that Floris van Doorn points out in his thesis
that another possible approach is to use the Serre spectral sequence for
homology.  In this approach one still has to prove the dimension 1 case.

Best,
Luis

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