In my reading of Hovey’s “Model Categories,” Hovey talks a bit about ordinals, cardinals, transfinite compositions, small objects and ordinals being $\gamma$ filtered.  I found that I didn’t have a very good intuition for such situations, and thought it might be helpful to write a short article on it, since such issues, while often ignored or downplayed, become relevant in the category theoretic facet of homotopy theory. One note: some of this early stuff I’m doing from memory, so excuse any notational discrepancies, I think the basic idea is still there. And also, we will, without even mentioning it, be assuming the axiom of choice here.

(Editor’s Note: This first part just contains foundational stuff on ordinals and cardinals, i.e. what are they?  The next piece of this article will actually get into Hovey’s stuff.)

First of all, I’d like to briefly discuss the construction of the ordinals.  We can consider an ordinal “number” $\alpha$ to be the set of all “smaller” ordinals.  Of course, without any background, the above statement doesn’t make any sense.  So, let’s start at the beginning, i.e. the empty set $\emptyset$.  This is our first ordinal.  Next we take the set containing the empty set, which we might write as $\emptyset+1$ or, if we’re feeling adventurous, just $1$. The next ordinal will be the set containing all smaller ordinals, i.e. $\{\emptyset, 1\}$ or maybe just $2$.  We can continue this process ad infinitum.  One thing to note here is that when we look at the cardinality of these ordinals (i.e. how many elements are in each, since each is a set) we’re not counting the cardinality of the sets the ordinal contains.  We’re counting each contained set as one object in itself.  Thus while the next ordinal could be written as $\{\emptyset, 1, 2\}$ it would be $3$ since it contains three elements.

Now, unless we’re just doing our taxes, we’d like a way to enumerate things that aren’t finite (i.e. infinite…).  To do this we take a “limit ordinal,” (up till now we’ve been taking what are called “successor ordinals”).  We let $\omega$ just be the set of all the finite ordinals, which we note is in bijection with $\mathbb{N},\mathbb{Z}$ and $\mathbb{Q}$. This ordinal is, as you probably know, the first of the countable (and infinite) ordinals.  Of course, continuing the process from the finite ordinals, we might now take the set of all finite ordinals AND $\omega$, which we denote by $\omega+1$.  Note that $\omega+1$ is a successor ordinal again, since we got it by just bumping up a step, instead of taking the union of some infinite sequence of things (some might say “But we are taking the union of an infinite number of things, because $\omega+1$ has both $\omega$ in it and all the finite ordinals,” but the idea is that we’ve already dealt with that process by jumping up to $\omega$ and can just start taking successor ordinals again). Similarly to what we did before, we can now just keep taking successor ordinals, getting $\omega+k$ for every $k\in\mathbb{N}$.  However, this is not enough for us now.  So again we take a limit ordinal, this time taking the union of all the finite ordinals, $\omega$, $\omega+1$…. and call this thing $\omega+\omega$ or $2\omega$.  Of course we can now repeat, getting $\omega\cdot\omega$, $\omega^\omega$ and so on and so forth.

The thing to note here is that all of the things we’ve gotten above, and infinity of them in addition to the ones we’ve talked about, have one thing in common.  As sets, they are all in bijection with $\omega$.  That is, they’re all countable (there is lots of interesting stuff going on with ordinals, and lots of interesting definitions; check out wikipedia to follow that particular path, specifically Cantor-Normal form, Church Kleene Ordinal and recursive ordinals are rather interesting).  So, for any one of those ordinals, say $\alpha$, we know that $\vert\alpha\vert=\aleph_0$ (except for finite ordinals, which have cardinality themselves, and so are in fact also the finite cardinals), which is by definition, the cardinality of something which is “countable.” In general, the cardinality of an ordinal $\alpha$ is defined to be the smallest ordinal which is in bijection with $\alpha$.  Hence $\aleph_0=\omega$ from our point of view.

So, like we’ve done before, let’s take the union of all the countable ordinals! We do, and what we get is what we denote as the set $\omega_1$.  This is the smallest uncountable ordinal, and so we also call it $\aleph_1$.  The point is, this process just keeps going, forever. I think that what we’ve done so far however will be enough to serve intuition.

As a side note, the content of the Continuum Hypothesis, which is something I struggled with for a time, is that $2^{\aleph_0}=\aleph_1$.  That is, the cardinality of the power set of $\omega$ is in fact $\omega_1$.  It is necessarily true that the cardinality of $\aleph_1$ is less than or equal to $2^{\aleph_0}$ since in some sense $\aleph_1$ is the next biggest cardinal after $\aleph_0$ and that’s the way we constructed it, but showing that they are equal is independent of ZFC.

-JB

We’d like to discuss the homotopy groups of spectra, i.e. what they are, and some very basic things about them.  In the process we will make some vague categorical statements. Hopefully, along the way it will be clear why the category of spectra is in some sense the natural place to do stable homotopy theory.

First of all, we’re going to make some assumptions:

1. All spaces are pointed CW complexes, all maps take base point to point, and all homotopies are homotopies of pointed maps (that is, given a homotopy $H:X\times I\to Y$ such that $H(0,x)=f(x)$ and $H(1,x)=g(x)$, we have that $H(t,x):X\to Y$ is a pointed map.

2. As clearly follows from above, we’ll be working with pointed CW spectra.  We could just as easily work with spectra of simplicial sets.

Now towards defining homotopy groups of spectra, we note that there are always group homomorphisms $\pi_{n+r}(E_n)\to\pi_{n+r+1}(E_{n+1})$.  How can you be so sure, you say?  This is how we can be so sure:  Say we have some map $f:S^{n+r}\to E_n$ which defines a homotopy class.  Then we can define $\Sigma f:\Sigma S^{n+r}\cong S^{n+r+1}\to\Sigma E_n$, where in general, since $\Sigma X=X\wedge S^1$, we can just takse $\Sigma f=f\wedge 1$ (we assume knowledge of the smash product here, especially its construction, but more information can be found at http://en.wikipedia.org/wiki/Smash_product).  This last map $\Sigma f$ clearly defines an element of $\pi_{n+r+1}(\Sigma E_n)$ so by composition with the structure map $\sigma_{n+r}:\Sigma E_{n+r}\to E_{n+r+1}$ we obtain a representative of some class in $\pi_{n+r+1}(E_{n+1})$. Thus, for each fixed $r$ we have a diagram in the category of groups indexed by $\mathbb{N}$, the natural numbers (corresponding to $n$, $n+1,\ldots$).  Since we are in the category of groups, a cocomplete category, we know that the colimit of this diagram exists.  So for fixed $r$ and spectrum $E$ we define $\pi_{r}(E)$ to be $\mathrm{colim}_n\pi_{n+r}(E_n)$.  Note, this colimit exists even for negative $r$ even if we ignore lower degrees where the groups may not exist.

Now, what do we mean when we ask about the stable homotopy groups of a space $X$?  Well, we mean precisely what is discussed above, that is, we mean the elements of homotopy group that remain after an arbitrary number of suspensions.  For instance, the $r$th stable homotopy group of spheres is defined to be $colim_n\pi_{n+r}(S^n)$ but that is precisely the $r$th homotopy group of the sphere spectrum $S$ where $S_n=\Sigma^nS_0=S^n$.

We wish also to have relative homotopy groups of spectra.  This follows directly from our knowledge of relative homotopy groups of spaces.  As we’ve seen in earlier posts, we can just define subspectra as spectra which are level-wise contained in another spectra with natural structure (suspension) maps.

We go ahead and define relative homotopy as

$\pi_{r}(E,A)=\mathrm{colim}_n\pi_{n+r}(E_n,A_n)$

where $A$ is some subspectrum of $E$. The details work out similarly to the above.  Note that for each fixed $r$ and  we have a long exact sequence in homotopy, the one for homotopy groups of spaces.   The diagram we are taking the colimit of is called “filtered,” and filtered colimits are always exact (this is not hard to prove), so we can take the colimit of the LES along $n$ again, which yields the desired LES in the homotopy of spectra.

Before we go too far into the world of spectra and generalized cohomology, I want to catalog some nice theorems that I have used but haven’t studied much in the past.  Let’s start with the Freudenthal suspension theorem, which nicely falls out of the Serre spectral sequence.  It will follow from the following proposition.

Proposition: If $X$ is an $n$-connected CW-complex, then the canonical map $j: X \rightarrow \Omega \Sigma X$ is a $(2n+1)$-equivalence.

Proof: Consider the path-loop fibration $\Omega \Sigma X \rightarrow P \Sigma X \rightarrow \Sigma X$.   By the Hurewicz theorem, $H_k(X)=0$ for $k \leq n$.  Since suspension induces an isomorphism on homology, we have that $H_k(\Sigma X)=0$ for $k \leq n+1$.  By the Hurewicz theorem again, we have $\pi_k(\Sigma X)=0$ for $k \leq n+1$.  By the adjointness of loops and suspension, we have $\pi_k(\Omega \Sigma X)=\pi_{k+1}(\Sigma X)$, which is zero for $k \leq n$.  So in the above fibration, the fiber is $(n+1)$-connected and the base is $n$-connected.  Looking at the Serre spectral sequence, we see that for $k \leq 2n+1$, the first nonzero differential from $H_k(\Sigma X)$ and the first nonzero differential into $H_{k-1}(\Omega \Sigma X$ must be the transgression, $\tau$.  Since $P\Sigma X$ is contractible, $\tau$ is an isomorphism for $k \leq 2n+1$.  Now we have the following diagram

(diagram coming soon–I need to figure out how to draw commutative diagrams on this blog and in TeX in general)

where $J: CX \rightarrow P\Sigma X$ is given by $J(u, x)(t)=(tu, x)$ where $CX$ is the cone on $X$ and $q$ is the map that collapses $X$ to a point.  Notice that the composition $X \rightarrow CX \rightarrow \Sigma X$ is exactly the suspension.  Going to the long exact sequences in homology, we have

(diagram coming soon)

where the composition of $latex$\delta$and $q_*$ is exactly the suspension isomorphism on homology. TO BE COMPLETED… Freudenthal Suspension Theorem: Let $X$ be an $n$-connected CW-complex. Then $S: \pi_k(X) \rightarrow \pi_{k+1}(\Sigma X)$ is an isomorphism for $k \leq 2n$. Proof: We have $[S^k, X] \rightarrow [S^k, \Omega \Sigma X] \rightarrow [\Sigma S^{k+1}, \Sigma X]$ for $k \leq 2n$$\Box$ More theorems will be added soon: Blakers-Massey, how to get long exact sequences from fibrations, etc. Next week I plan on writing up some notes on complex oriented cohomology theories and the connection to formal group laws. I will also continue studying spectra on the way to the Adams spectral sequence. So far, I’ve been getting most of my notes from the book Bordism, Stable Homotopy, and Adams Spectral Sequences by S.O. Kochman. I have been trying to work out the details of what spectra are and the various constructions one can make with them. Today I’m going to define spectra and say a little bit about why they are useful. In the future, I’d like to continue this series of posts to talk about Brown representability, Eilenberg-MacLane spectra, the Adams spectral sequence, and maybe some other topics. Today’s notes are mostly from chapter 2 of Hatcher’s book on spectral sequences. Let’s start with a few definitions. Definition: A CW spectrum is a collection of CW spaces $\{X_n\}$ for integers $n$ together with a collection of maps of CW complexes $\Sigma X_n \rightarrow X_{n+1}$ that are inclusions of subcomplexes. A spectrum $X'$ is a subspectrum of $X$ if for each $n$ we have that $X_n' \subseteq X_n$, and $\Sigma X'_n \rightarrow X'_{n+1}$ is the restriction of the map $\Sigma X_n \rightarrow X_{n+1}$ to $\Sigma X'_n$. Notice that a non-basepoint $k$-cell of $X_n$ suspends to a $(k+1)$-cell of $X_{n+1}$. Now there is sort of a subtle question of how to define maps between CW spectra. Our first guess would be to define what we’ll call a strict map $f: X \rightarrow Y$ as a collection of cellular maps $f_n: X_n \rightarrow Y_n$ which fit into commutative squares with the suspension maps and the maps $\Sigma f_{n-1}: \Sigma X_{n-1} \rightarrow \Sigma Y_{n-1}$. But it turns out that a nicer and weaker condition will let us do the things that we want (for example, maps of CW spectra should induce maps on the homology, cohomology, and homotopy groups of spectra, when we define them). First, we define a subspectrum $X'$ of $X$ to be cofinal in $X$ if for every $n$, every cell $e_{\alpha}^i$ of $X_n$ is such that there is some large enough $k$ such that $\Sigma^k e_{\alpha}^i$ is a cell in $X'_{n+k}$. In other words, every cell of a space in $X$ eventually suspends to a cell of a space in $X'$. Now we define a map of CW spectra $f: X \rightarrow Y$ to be an equivalence class of strict maps $X' \rightarrow Y$ for cofinal subspectra $X'$ of $X$, where we regard two maps $f_1: X' \rightarrow Y$ and $f_2: X'' \rightarrow Y$ as equivalent if there is a subspectrum $X'''$ cofinal in both $X'$ and $X''$ such that $f_1$ and $f_2$ agree on $X'''$. It is a useful exercise to check that composing two maps of CW spectra $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ gives us a map of CW spectra. The idea is to choose a cofinal subspectrum $X''$ of the cofinal subspectrum $X'$ on which $f$ is defined, such that $X''$ has the property that each of the cells in each of its complexes $X''_n$ maps into $Y'_n$, where $Y'$ is the cofinal subspectrum on which $g$ is defined. Then we may compose the restriction of $f$ to $X''$ with $g$, and it just remains to check that $X''$ is cofinal in $X$ (it is because it is cofinal in $X'$). We define a homotopy of maps of CW spectra, $f_1: X \rightarrow Y$ and $f_2: X \rightarrow Y$, to be a map of spectra $F: X \times I \rightarrow Y$ which is the map $f_i$ on $X \times \{i\}$, $i=0, 1$. Here we regard $X \times I$ as a spectrum with $(X \times I)_n=X_n \times I$ and we mean the reduced product, with the interval above the basepoint collapsed to a point so that we have that the reduced suspension of $X_n \times I$ is just the reduced product of the reduced suspension of $X_n$ with $I$. We denote the homotopy classes of maps as $[X, Y]$. Now we come to a proposition which explains what the homotopy category of spectra is good for and how it is different from the category of CW spaces. This proposition says that the suspension functor is invertible. Proposition: The map $[X, Y] \rightarrow [\Sigma X, \Sigma Y]$ is an isomorphism. Proof: To see that the above map is surjective, let $f: \Sigma X \rightarrow \Sigma Y$ be a map of CW spectra. If $f$ is a strict map on a cofinal subspectrum $Z'$ of $\Sigma X$, we set $X'_n=Z_{n-1}$. Then we have that the spectrum $X'$ is cofinal in $X$, the spectrum $\Sigma X'$ is cofinal in $\Sigma X$, and the map $f$ is strict on $\Sigma X'$. In other words, we may assume that $f: \Sigma X \rightarrow \Sigma Y$ is strict in the first place. Now we rewrite $\Sigma X=X \wedge S^1$ where $(X \wedge S^1)_n=X_n \wedge S^1$ and the map from $\Sigma(X_n \wedge S^1)$ to $X_{n+1} \wedge S^1$ is $\Sigma \wedge 1$. So we have $f: X \wedge S^1 \rightarrow Y \wedge S^1$. We may replace $f_n$ by its restriction $\Sigma f_{n-1}: \Sigma(X_{n-1} \wedge S^1) \rightarrow \Sigma(Y_{n-1} \wedge S^1)$ As we noted above, this map is independent of the coordinate “in”$\latex \Sigma$. We want to homotope it to a map that is also independent of the coordinate in $S^1$. Thus, each $f_n$ will be replaced by a map $h_{n} \wedge 1$. Then this map $h$ will be sent to $f$ via the homomorphism above. So how can we homotope $\Sigma f_{n-1}$ to a map which is independent of the coordinate in $S^1$? Well, we write $\Sigma(X_{n-1} \wedge S^1)= X_{n-1} \wedge \Sigma S^1$. Now the map may depend on the coordinate in $S^1$, identified with the equator in $\Sigma S^1$. We homotope the sphere $\Sigma S^1=S^2$ by rotating it by 90 degrees. Now the new map, also called $f_n$, on this sphere is independent of the new equator, and we identify $X_{n-1} \wedge \Sigma S^1$ with $\Sigma(X_{n-1} \wedge S^1)=\Sigma X_{n-1} \wedge S^1$. This new map is independent of both $\Sigma$ and $S^1$ coordinates (since $S^1$ is now the new equator inside $\Sigma S^1$, so it is of the form $h_n \wedge 1: X_{n-1} \wedge \Sigma S^1=\Sigma(X_{n-1} \wedge S^1) \rightarrow \Sigma(Y_{n-1} \wedge S^1)$ The proof of injectivity of the above homomorphism is similar. Let $f$ and $g$ be maps $X \rightarrow Y$ such that $\Sigma f$ and $\Sigma g$ are homotopic. That is, there is a map of CW spectra $F: \Sigma X \times I \rightarrow \Sigma Y$ restricting to $\Sigma f$ and $\Sigma g$ on the two endpoints. As above, we may assume that $F$ is strict. We need to come up with a homotopy $X \times I \rightarrow Y$. We can find $X_n \times I$ inside $\Sigma X_n \times I$ by choosing the copy of $X_n$ that lies in the middle of each suspension. This gives us a map$\latex X \times I \rightarrow \Sigma Y\$.

But since the homotopy $F$ above is independent of the $\Sigma$ coordinate (that is, it sends each horizontal cross section of the suspension of $X_n$ into the horizontal cross section of $\Sigma Y_n$ with the same $\Sigma$ coordinate), it maps the copy of $X_n \times I$ in the middle of all of the suspensions into the copy of $Y_n$ in the middle of $\Sigma Y$.  This is a map of CW spectra restricts to $f$ and $g$ at the endpoints. $\Box$

The invertibility of the suspension functor essentially says that the homotopy category of spectra is the place in which to study stable phenomena.  As we will see later, the homology/cohomology/homotopy groups of spectra will be the stable homology/cohomology/homotopy groups of the underlying spaces.