[-- Attachment #1.1: Type: text/plain, Size: 1479 bytes --] 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 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 2125 bytes --] <div dir="ltr"><div>Is there any progress on proving the Hurewicz theorem in HoTT?</div><div><br></div><div>I stumbled across this mathoverflow question: <a href="https://mathoverflow.net/questions/283199/an-abstract-nonsense-proof-of-the-hurewicz-theorem">https://mathoverflow.net/questions/283199/an-abstract-nonsense-proof-of-the-hurewicz-theorem</a></div><div><br></div><div>I wonder if we can adapt the following argument:</div><div><br></div><div>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).<br></div><div><br></div><div>H : [S^{n+t}, X /\ S^t] ---> [S^{n+k}, X /\ K(Z, t)]</div><div><br></div><div>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.</div><div><br></div><div>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.</div><div><br></div><div>Do you think this argument will work? Let me know what you all think.<br></div><div><br></div><div>Thanks,</div><div><br></div><div>Ali Caglayan<br></div></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com</a>.<br />

[-- Attachment #1: Type: text/plain, Size: 3331 bytes --] 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 On Sat, Jul 27, 2019 at 2:18 PM Ali Caglayan <alizter@gmail.com> wrote: > 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 > > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com > <https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com?utm_medium=email&utm_source=footer> > . > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAKBukGjsZ6g_D2Xs%3D5V3Ak%2BAQWgXvV%3D%2Bqh%3DScMY2AXgMNJv64w%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 4707 bytes --] <div dir="ltr"><span class="gmail-im">Hello Ali,<br> <br> Dan Christensen and I have a draft paper that proves the Hurewicz<br> isomorphism theorem in HoTT, and we plan to make it available sometime<br> soon. We have not formalized the proof, and it also relies on the<br></span> adjunction between pointed mapping spaces and smash products and some<br> coherences of the associativity of the smash product that have not been<div class="gmail-yj6qo gmail-ajU"><div class="gmail-im"><img class="gmail-ajT" src="https://ssl.gstatic.com/ui/v1/icons/mail/images/cleardot.gif">completely formalized.<br> <br> Our starting point is similar to yours, so to answer your question, I<br> would say that yes, that idea can be made to work. However, at least in<br> our approach, most of the work is in proving that the Hurewicz map is<br> actually a group homomorphism. In that argument we really need the fact<br> that homology is defined as a colimit and we can't just use some<br> sufficiently large t.<br> <br> Another key ingredient in the approach is the relation between smash<br> products of types and tensor products of their homotopy groups, which<br> we had to develop for this purpose.<br> <br> For completeness, note that Floris van Doorn points out in his thesis<br> that another possible approach is to use the Serre spectral sequence for<br> homology. In this approach one still has to prove the dimension 1 case.<br> <br> Best,<br> Luis</div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sat, Jul 27, 2019 at 2:18 PM Ali Caglayan <<a href="mailto:alizter@gmail.com">alizter@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>Is there any progress on proving the Hurewicz theorem in HoTT?</div><div><br></div><div>I stumbled across this mathoverflow question: <a href="https://mathoverflow.net/questions/283199/an-abstract-nonsense-proof-of-the-hurewicz-theorem" target="_blank">https://mathoverflow.net/questions/283199/an-abstract-nonsense-proof-of-the-hurewicz-theorem</a></div><div><br></div><div>I wonder if we can adapt the following argument:</div><div><br></div><div>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).<br></div><div><br></div><div>H : [S^{n+t}, X /\ S^t] ---> [S^{n+k}, X /\ K(Z, t)]</div><div><br></div><div>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.</div><div><br></div><div>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.</div><div><br></div><div>Do you think this argument will work? Let me know what you all think.<br></div><div><br></div><div>Thanks,</div><div><br></div><div>Ali Caglayan<br></div></div> <p></p> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com?utm_medium=email&utm_source=footer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/f50d39d3-0017-4820-9c76-877760449e78%40googlegroups.com</a>.<br> </blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAKBukGjsZ6g_D2Xs%3D5V3Ak%2BAQWgXvV%3D%2Bqh%3DScMY2AXgMNJv64w%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAKBukGjsZ6g_D2Xs%3D5V3Ak%2BAQWgXvV%3D%2Bqh%3DScMY2AXgMNJv64w%40mail.gmail.com</a>.<br />