Properties of $\pi_n$ from a category theoretical point of view

Question

This will be a more open question.

I would like to have a better understanding about how the homotopy group functors behave with different constructions such as limits or colimitis.

Some examples:

1) In the case $n=0$ we have $\pi_0$ is a left-adjoint for $Set \rightarrow Top$, $X \mapsto X_d$ with the discrete topology.

2) $\pi_n$ commutes with (finite ?) products. If I remember correctly this fact derives from a more general fact about the behavior of $\pi_n$ with regard to fibrations.

3) $\pi_n(\vee_{i \in I} S^n) = \oplus_{i \in I} \pi_n({S^n})$, if $n \geq 2$. Is it a coincidence that in this case $\pi_n$ commutes with the direct sum? I guess this won't hold in arbitrary spaces, but maybe in a sufficiently nice category of spaces?

4) According to this post on stack.overflow, $\pi_1$ seems to at least be a left-adjoint in the homotopy category of connected $CW$-complexes.

I am neither very good in topology nor in category theory and this is exactly why I am asking. I would like to know more properties similar to above to get on the one hand a better understanding of how to apply category theory to topology, on the other hand to get some practical tools for avoiding explicit computations when dealing with the $\pi_{\ast}$ functors. Do you know any?

Answer 1

It's not quite true that $\pi_0$ is a left adjoint for the discrete spaces functor – one should restrict to, say, locally connected spaces (if by $\pi_0$ you mean connected components) or locally path connected spaces (if by $\pi_0$ you mean path components). This difficulty goes away if you work with simplicial sets instead.

For $n \ge 1$, some of the good properties of $\pi_n$ come from the fact that $\pi_n$ is homotopy-representable. More precisely, let $\mathbf{H}_*$ be the homotopy category of pointed CW complexes (or Kan complexes). Then $\pi_n : \mathbf{H}_* \to \mathbf{Grp}$ is representable: there exists a cogroup structure on the (pointed) $n$-sphere $(S^n, *)$ such that $\pi_n (X, x)$ is naturally isomorphic to the set of homotopy classes of maps $(S^n, *) \to (X, x)$ equipped with the group structure induced by the cogroup structure of $(S^n, *)$. Of course, this is by definition of $\pi_n$, but it implies that $\pi_n$ preserves all limits as a functor $\mathbf{H}_* \to \mathbf{Grp}$. Since products in $\mathbf{H}_*$ coincide with products in the usual sense (at least if we work with Kan complexes), it follows that $\pi_n$ preserves products.

On the other hand, there are some things going on here that are not just abstract nonsense. For instance, the van Kampen theorem says that $\pi_1$ preserves certain pushouts (as computed in $\mathbf{Top}_*$) – from the point of view of pure category theory, this is not something we expect. However, we can define the fundamental groupoid functor $\pi_{\le 1} : \mathbf{Kan} \to \mathbf{Grpd}$ as a left adjoint, so perhaps it's not so surprising after all. More generally one expects that there should be a fundamental $n$-groupoid functor $\pi_{\le n}$ that is a left adjoint up to coherent homotopy.

Answer 2

In homotopy theory one has to note that identifications in low dimensions can change homotopy information in high dimensions. One first sees this in dimension $1$, where the circle $S^1$ is obtained from the unit interval $[0,1]$ by identifying $0$ and $1$, i.e. by an identification in dimension $0$. The $2$-sphere is obtained from the disk $D^2$ by shrinkng the bounding $S^1$ to a point.

All this suggests that to obtain useful colimit theorems in homotopy theory you need algebraic structures which have structure in a range of dimensions, to model the gluing of spaces.

In dimension $1$ this is obtained by using the fundamental groupoid $\pi_1(X,A)$ on a set of base points, chosen according to the geometry of the situation. Groupoids may be seen as having structure in dimensions $0$ and $1$. The use of this structure is developed in my book Topology and Groupoids (third edition of a book published in 1968, so this idea is not at all new). In this way one gets useful information on homotopy $1$-types.

Going further with this plan, to dimensions $2$, can be done by using the information on a space at levels $0,1,2$, i.e. by using triples $X_*:= X_0 \subseteq X_1 \subseteq X_2$ and involving relative homotopy groupoids $\pi_2(X_2,X_1,X_0)$ and their relation with $\pi_1(X_1,X_0)$. So one does not get direct information on a homotopy group $\pi_2(X_2,x)$. There are also connectivity conditions needed. Nonetheless, one does get some useful information, see for example my answer to this stack exchange question on homotopical excision in dimension 2.

One thing to note is that even in dimension $2$, and even with the structure of module over $\pi_1$, the second homotopy group $\pi_2$ is but a pale shadow of the homotopy $2$-type. For colimit theorems in homotopy theory, one needs better models of homotopy $n$-types. Such models reflect the idea of more complicated, and "more nonabelian", structures than groups, to capture some features of higher homotopy theory. Currently these seem to be best defined on certain "structured spaces", not just on "bare" topological spaces.