Is the fundamental group functor a left-adjoint?

Question

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?

Answer 1

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.

Answer 2

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.