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Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the first fundamental group of $X$, and $K(G,1)$ is the first Eilenberg-MacLane space of $G$. I guess he is describing an adjunction of functors here, between the category of homotopy classes of maps between connected pointed CW complexes and groups.

This surprised me. If $\pi_1$ is a left-adjoint functor, then we should conclude that it is cocontinuous, i.e. takes pushouts to pushouts. But I had understood the van Kampen theorem to say something like "$\pi_1$ takes certain pushouts in $\text{hTop}_*$ to pushouts in groups". For example, van Kampen requires the morphisms to be inclusions, among other things. Presumably then not all pushouts are preserved under $\pi_1$, for example if the maps are not inclusions.

I tried to come up with a pushout of non-injective pointed topological spaces which would give a counterexample to van Kampen, but I could not. Is there one? Can you give one?

And if there is one, why doesn't that contradict the status of $\pi_1$ as a left-adjoint? And if there isn't one, then why can't the hypotheses of the van Kampen theorem be weakened?

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I always thought that the conditions of van Kampen theorem were about making sure that the diagram we start with is actually a pushout in the category of pointed spaces. But maybe I am wrong. –  Andrej Bauer Oct 16 '12 at 4:25
Question: What about etale (and other) analogues of this? –  David Corwin Oct 16 '12 at 21:43
@Davidac: The étale fundamental pro-groupoid functor is also a left adjoint functor from the étale $\infty$-topos to pro-groupoids, and it therefore preserves all homotopy colimits. In particular, if you have any étale cover of a scheme $X$, then the étale fundamental pro-groupoid of $X$ is the 2-colimit of the diagram formed by the pro-groupoids of all the finite "intersections" of the schemes in the cover. –  Marc Hoyois Oct 17 '12 at 0:46
The OP seems to be under a misapprehension. The hypotheses can be (and usually are) weakened: the inclusion maps are not required to be $\pi_1$-injective. See, for instance, Wikipedia: en.wikipedia.org/wiki/Seifert–van_Kampen_theorem . –  HJRW Oct 17 '12 at 5:22
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2 Answers

The problem is that there are not a lot of actual colimits in the homotopy category of (connected) CW complexes, so knowing that $\pi_1$ preserves them (which is true) is pretty much useless. The pushouts appearing in the van Kampen theorem are pushouts in $Top$ but not in the homotopy category, so the van Kampen theorem does not follow from this adjunction. On the other hand, the functor $\pi_1$ preserves all homotopy colimits, and the hypotheses in the van Kampen theorem guarantee that the pushout in Top is a homotopy pushout.

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One should replace the fundamental group by the fundamental groupoid. only then it is "homotopy cocontinuous". Else there are issues if the spaces are non-connected. –  Lennart Meier Oct 16 '12 at 8:48
So I think the categories Top, Top*, CWComp, and CWComp* are all cocomplete. When you say "not a lot of colimits", I guess that you mean that some of the corresponding homotopy categories are not cocomplete? Is there an easy way to see this? I would have guessed that since we can fill out the pushout diagram in Top, we can fill it out in hTop using the homotopy classes of the arrows in Top. –  Joe Hannon Oct 16 '12 at 13:46
hTop isn't just not cocomplete, it has extremely few colimits--essentially the only colimits that exist are coproducts. The problem with your guess is that a diagram that commutes in hTop need not lift to a commuting diagram in Top, since it only commutes up to homotopy. –  Eric Wofsey Oct 16 '12 at 23:39
For a specific counterexample that hTop* is not cocomplete, see mathoverflow.net/questions/10364 –  Martin Brandenburg Oct 22 '12 at 8:41
Thanks Marc. Google led me to this blog post: schommerpries.wordpress.com/2012/02/20/… with an explanation and link to lecture notes for an example of a colimit that doesn't exist in hTop. I note that the blog post is by Chris Sommer-Preis, who is commenting on the answer below. –  Joe Hannon Oct 25 '12 at 20:44
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This question is related to the paper of P. Olum, Non-abelian cohomology and van Kampen's theorem. Ann. of Math. (2) {68} (1958) 658--668. He defines nonabelian singular cohomology $H^1(X,A;G)$ of a pair of spaces with coefficients in an in general nonabelian group, and verifies that $H^1(X,x;G) \cong Hom(\pi_1(X,x),G)$ if $X$ is pathconnected and $x \in X$.

If $A,B$ are subspaces of $X$, then under the assumption that $H^1(A \cup B; G) \cong H^1(S(A) \cup S(B);G) $ he obtains a Mayer-Vietoris type sequence $$\matrix{H^0(A\cap B,x;G)& \to & H^1(A \cup B,x;G)& \to & H^1(A,x;G) \cr &&\downarrow&&\downarrow\cr &&H^1(B,x;G) & \to &H^1(A \cap B,x;G) }$$ and proves exactness conditions which imply that if $A,B,A \cap B$ are pathconnected and $x \in A \cap B$ then we obtain the usual pushout diagram of the standard Seifert-van Kampen Theorem. This result is put in the context of groupoids in R. Brown, P.R. Heath, K.H. Kamps, ``Groupoids and the Mayer-Vietoris sequence'', J. Pure Appl. Alg. 30 (1983) 109-129.

Later: with regard to the question of $\pi_1$ as a left adjoint, one can say that $\pi_1$ as a functor from Simplicial Sets to Groupoids is a left adjoint to the nerve functor, and some have assumed that all van Kampen type theorems are of this kind of depth, i.e. not much.

However more work is needed to formulate and prove the case of the fundamental groupoid with a set of base points, which require connectivity conditions on intersections of the sets of the cover; and such simple adjointness arguments have not touched the higher homotopy Seifert-van Kampen Theorems, which require more complex connectivity assumptions, and so imply of course that they solve only some problems.

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What is the easiest illustrative example of the phenomena described in the last paragraph that goes beyond the fundamental groupoid? –  Chris Schommer-Pries Oct 16 '12 at 20:24
@Chris: In calculating $\pi_1 S^1$ it is good to use 2 base points. In a van Kampen situation $A$ needs to meet each path component of each $1$-, $2$-, $3$-fold intersrsection of the sets of the cover. See my paper with Razak, [41] in my publication list. See also "Topology and Groupoids" Section 9.1. See also arXiv:math/0111073 for a wider application. In higher dimensions, the theory that really works well is for filtered spaces, which is against tradition; the approach yields a new exposition of basic algebraic topology without using singular homology! See EMS Tract vol 15. –  Ronnie Brown Nov 3 '12 at 20:58
@Chris: are you asking about the higher Seifert van Kampen theorems? If so, a simple example is to give a precise description of $\pi_2(X \cup_f CA,X,x)$ in terms of the morphism of fundamental groups of $A$ and $X$ induced by the attaching map $f$ of the cone. This generalises a classic theorem of JHC Whitehead on free crossed modules. The proof uses a homotopy double groupoid of a pair. –  Ronnie Brown Nov 3 '12 at 23:14
See also my answer to math.stackexchange.com/questions/198348 –  Ronnie Brown Nov 4 '12 at 10:47
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