Timeline for Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?

6 events

when toggle format	what		by	license	comment
Jul 12, 2016 at 10:05	comment	added	Ronnie Brown		A groupoid proof of the Kurosh subgroup theorem is in Philip Higgins' 1971 book "Categories and Groupoids" available at tac.mta.ca/tac/reprints/articles/7/tr7abs.html . Philip told me he thought of the use of groupoids when reading Hilton and Wylie's account of covering spaces.
Jul 12, 2016 at 5:46	history	edited	Qiaochu Yuan	CC BY-SA 3.0	deleted 6 characters in body
Jul 11, 2016 at 20:25	comment	added	Tyler Lawson		In fact we always have $\pi_{2n-1}(K(A,n) \vee K(B,n)) = A \otimes B$ for $n > 1$, and so this is often not an $n$-type. Something genuinely special is happening in the case $n=1$.
Jul 11, 2016 at 19:37	comment	added	Qiaochu Yuan		@Omar: I know an example not covered by that statement; the inclusion of $1$-types (not $\tau_{\le 1}$, which is its left adjoint) also preserves homotopy quotients by (discrete) group actions.
Jul 11, 2016 at 19:36	comment	added	Omar Antolín-Camarena		The most general homotopy colimit I know that $\tau_{\le 1}$ preserves is given by Theorem 1B.11 in Hatcher's book. It gives you that $\mathrm{hocolim} BF$ is a 1-type if $F : \mathcal{C} \to \mathrm{Groups}$ is a diagram of groups and injective group homomorphisms and $\mathcal{C}$ is the free category on a directed graph.
Jul 11, 2016 at 19:05	history	answered	Qiaochu Yuan	CC BY-SA 3.0