The Whitehead Theorem and Model Categories

Consider the category of topological spaces where morphisms are continuous maps. There are distinguished morphisms called homotopy equivalences as well as morphisms which induce isomorphisms on homotopy groups; these are often called weak equivalences. There are also distinguished objects called CW complexes which are constructed from gluing topological disks together. One important result from algebraic topology is that for any topological space $X$, there exists a CW complex $Z$ and a weak equivalence $Z \to X$. We call these maps CW approximations because, as far as homotopy groups are concerned, $Z$ and $X$ are equivalent.

There are situations in which a weak equivalence is in fact, a homotopy equivalence.

Theorem (Whitehead) Let $X,Y$ both be CW complexes. If $f:X \to Y$ is a weak equivalence, then it is in fact, a homotopy equivalence.

This is a very useful theorem. Since in general, homotopy equivalences are weak equivalences, then Whitehead’s theorem shows that in the context of CW complexes, weak and homotopy equivalences coincide. Also, we can combine it with the Hurewicz theorem. For any path-connected space $X$ and positive integer $n$, there is a group homomorphism $h_*:\pi_n(X) \to H_n(X,\mathbb{Z})$.

Theorem (Hurewicz): For $n \geq 2$, if $X$ is $(n-1)$-connected; i.e. its homotopy groups up to degree $n-1$ are all trivial, then the reduced homology groups $\widetilde{H}_i(X,\mathbb{Z})=0$ for all $i<n$ and the Hurewicz map $h_*:\pi_n(X) \to H_n(Z,\mathbb{Z})$ is an isomorphism. For $n=1$, the Hurewicz map induces an isomorphism between the abelianization of $\pi_1(X)$ and $H_1(X,\mathbb{Z})$.

Corollary: Combining Whitehead and Hurewicz, a map $f:X \to Y$ between simply connected CW complexes that induces an isomorphism on all integral homology groups is a homotopy equivalence.

For example, suppose $M$ is a simply connected closed complex manifold of complex dim 3 (so a real dim 6 manifold) and $f:M \to S^6$ is a weak equivalence. If $M$ only has nontrivial integral homology in degrees 0 and 6, then $M$ and $S^6$ are homotopy equivalent. The proof of the generalized Poincaré conjecture establishes that $M$ is diffeomorphic to $S^6$ and thus, $S^6$ would admit an integrable complex structure.

On the other hand, it is very important for Whitehead’s theorem that the isomorphisms between homotopy groups is induced by a map.

Example: Manifolds are CW complexes and $S^3 \times \mathbb{RP}^2$ and $S^2 \times \mathbb{RP}^3$ have the same universal cover. Thus, we know they have the same homotopy groups $\pi_k$ for $k \geq 2$. They are also both simply connected so their homotopy groups are all isomorphic. But there is no map which induces this weak equivalence because, if there were, the two manifolds would be homotopy equivalent. However, we can see from their different homologies that they are not homotopy equivalent.

Model Categories

As it turns out, Whitehead’s theorem has manifestations in other categories as well. We would need a proper notion of weak equivalence and homotopy equivalence in those categories. The proper unifying framework are model categories which were established by Daniel Quillen.

To keep this post less technical, I will skim over many details but you can find more details in Noam Kantor’s note. A model category $M$ consists of a category with three distinguished classes of morphisms: weak equivalences, fibrations, and cofibrations. The classes must include identities and be closed under composition. There are additional axioms they need to satisfy. As it turns out, within a model category, any 2 of the 3 classes of morphisms determines the third.

One of the axioms is that $M$ needs to be complete and cocomplete; i.e. contain all its limits and colimits. In particular, $M$ must admit initial and terminal objects. An object $A$ is fibrant if the map from $A$ to the terminal object $A \to *$ is a fibration and cofibrant if the map from the initial object $\varnothing \to A$ is a cofibration. If an object is both, we say it is bifibrant.

Here are some examples of model categories; the chart is from Kantor’s notes.

Examples include categories of topological spaces but also a more combinatorial category like simplicial sets and also chain complexes and DGAs. We can add to this list the stable homotopy category as well, the homotopy category of spectra. To define a notion of homotopy in this more generalized setting takes more work since we don’t have available to us the unit inverval $[0,1]$ in non-topological categories.

Model Category Whitehead Theorem: Suppose $X,Y$ are bifibrant objects in a model category $M$. Then $X$ and $Y$ are weakly equivalent if and only if they are homotopy equivalent.

Moreover, in the category of topological spaces, CW complexes are bifibrant objects and we can “approximate” objects by bifibrant objects. Any object $X$ in a model category has a fibrant replacement $FX$ by factoring the map $\varnothing \to X$ as a fibration followed by an acyclic cofibration. $FX \to X$ is a weak equivalence and $FX$ is fibrant. SImilarly, there is a cofibrant replacement $QX$ and $QFX$ is in fact bifibrant. For a given map $g:X \to Y$, these constructions induce maps $Fg,Qg$ as well. The model category axioms don’t require functoriality for factorizing maps so $F,Q$ might not be functors. Despite this, we can still define the homotopy category of a model category $M$.

Definition: Given a model category $M$, the homotopy category $\text{Ho}(M)$ has the same objects as $M$ but morphisms defined by $\text{Ho}(M)(X,Y) := M(QFX,QFY)$.

Since weak equivalences between bifibrant objects actually have “real” homotopy inverses by the Whitehead theorem for model categories, we have successfully localized/inverted all the weak equivalences in this homotopy category. To make this a more precise statement:

Theorem: The functor $L:M \to \text{Ho}(M)$ given by the identity on objects and $QF$ on morphisms is a localization of $M$ with respect to its weak equivalences. In other words, weak equivalences in $M$ turn into isomorphisms in $\text{Ho}(M)$. Moreover, any functor $K:M \to C$ in which weak equivalences go to isomorphisms naturally factor through $L$.

Example: Beginning with some familiar abelian category like modules over a commutative ring $R$, one can produce a category of chain complexes and specify quasi-isomorphisms as the weak equivalences; i.e. they induce isomorphisms when we take homology. The (co)fibrations are injective and projective resolutions. Then the derived category is equivalent to the homotopy category defined above.

In the category of modules over $R$, for a fixed module $B$, we have a tensoring functor $T(A) = A \otimes_R B$. This is a right exact functor and we can define a left derived functor where we take a projective resolution $… \to P_2 \to P_1 \to P_0 \to A \to 0$, remove $A$ and then tensor by $B$ to get $… \to P_2 \otimes_R B \to P_1 \otimes_R B \to P_0 \otimes_R B \to 0$. The Tor groups are the homology of this complex. Note that we did a fibrant replacement (in fact, it is a bifibrant replacement).