Fibrations and Cofibrations of spectra are "the same"

Question

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:

"For spectra every cofibration is equivalent to a fibration" (e.g. in the accepted answer here https://mathoverflow.net/a/56575/18744),

Another one is:

"For spectra, fibrations and cofibrations sequences are the same" (which is stronger than the statement above, because it works both ways)

At this point I am already quite happy with a suitable reference. However I do have some follow-up questions:

To what extent can I switch between spaces and spectra? The suspension functor does not preserve fibrations, correct? So what can be said about the relation of the homotopy fibre (cofibre) of spaces and the homotopy fibre (cofibre) of their suspension spectra? What if the spaces involved are infinite loop-spaces? The resulting $\Omega$-spectra don't carry much different information from the spaces itself, do they?

Concretely:

I have a sequence $X\to Y\to Z$ of group-like H-spaces, and know that they form a homotopy fibration. I would like to make statements about the homotopy cofibre of $X\to Y$. I got the hint to 'work in spectra', where the two are "the same", but don't know what to make of it.

Edit: Thank you all for your answered. Together they cover quie a number of different points of view. Initially I hoped to be able to return back to spaces after doing an excursion through spectra (having fairly explicit $\Omega$-spectra for group like $H$-spaces). It seems that there is no totally generic way to do this and I think I have now enough material to think about the particularities.

Answer 1

One specific statement that people are likely referring to when they say things about fibrations and cofibrations being "the same" in spectra is that a homotopy pushout square of spectra is also a homotopy pullback square (considering squares with one corner trivial gives homotopy fibration and cofibration sequences). A brief explanation of this is given by Goodwillie here: homotopy pullback/pushout.

If you are interested in a formal statement in one of the model categories of spectra, then you need to look at a proof that these model categories are stable, as defined for instance in Hovey's book Model Categories (Chapter 7). As mentioned here Homotopy limit-colimit diagrams in stable model categories, Hovey explains (Remark 7.1.12) that homotopy pullback squares and homotopy pushout squares coincide in any stable model category. Proofs that the standard model categories of spectra are stable can be found in the basic references for these categories: for instance, for the category of symmetric spectra, Hovey-Shipley-Smith prove this in Theorem 3.1.14 of their paper (Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208, available here http://www.ams.org/journals/jams/2000-13-01/S0894-0347-99-00320-3/S0894-0347-99-00320-3.pdf).

Answer 2

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with $j$ copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case the homotopy fiber of $B_j \to B$ $(B = BG)$ has the homotopy type of $j$-fold join of copies of $G$. The space $B_j$ is the orbits of $G$ on this iterated join, and we get the standard $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

Remark on fiber/cofiber sequences: The following is my take on Tillman's answer. IMO, Algebraic Topologists have been historically a bit sloppy with the definitions (I have also been guilty of such sloppiness). For me, a cofiber sequence $A \to X \to C$ is really shorthand notation for a commutative homotopy cocartesian square $$ \require{AMScd} \begin{CD} A @>>> P \\ @VVV @VVV \\ X @>>> C \end{CD} $$ in which $P$ is some (possibly weakly) contractible space. Likewise a fiber sequence $F \to E \to B$ is shorthand notation for a commutative homotopy cartesian square $$ \require{AMScd} \begin{CD} F @>>> E \\ @VVV @VVV \\ P @>>> B \end{CD} $$ for some (possibly weakly) contractible space $P$. In the spectrum case, these two notions agree.

Answer 3

I'd like to expand on Matthias's answer a little bit. There is some unfortunate terminology going around. Being a cofibration resp. fibration is a particular property a map can have; for spaces Hurewicz cofibrations and Serre cofibrations (relative CW-complexes) are ones that get most use, and they have associated notions of fibrations; the concept of a model category mentined by Dan Ramras is in the background of this but I don't think your question warrants, or requires, getting out the big guns. For spectra, in the usual model structure, a cofibration will be a relative CW-spectrum, and $X → *$ is a fibration iff $X$ is an $\Omega$-spectrum.

Every map can be turned into a cofibration or into a fibration if you allow yourself to change your target resp. source by a weak equivalence. For spaces, the mapping cylinder construction will turn $f\colon A → X$ into a Hurewicz cofibration $A → (A \times I) \sqcup X/\sim$, and the path space construction will turn $f\colon E → B$ into a Serre fibration. Similar constructions exist for Serre cofibrations and Hurewicz fibrations, and in fact it's an axiom of model structures that such replacements must always exist. Also in spectra.

In other words, in the homotopy category, it doesn't make sense to say that a map is a cofibration or fibration.

A cofibration sequence, or fibration sequence, is a completely different animal, although there's of course a relation. The usual definition of a cofibration sequence is as follows: if $A → X$ is a cofibration, then $A → X → X/A$ is a cofibration sequence, and any $K → L → M$ weakly equivalent to it is a cofibration sequence, too. So the first map need not be a cofibration. In fact, any $A → X$ can be extended to a cofibration sequence (by the mapping cone, to be explicit in spaces). Analogously for fibration sequences.

It is true in spectra that if $A → B → C$ is a cofibration sequence then it is also a fibration sequence, and vice versa. As a special case (choose $B$ to be a point), the loop functor is inverse to the suspension functor. Morally, in spectra, you've inverted the suspension functor $\Sigma$, so both $\Sigma^{-1}$ and $\Omega$ want to be right adjoint to $\Sigma$, so they have to agree. That's not a proof, though, but you'll find explicit proofs in any textbook on stable homotopy theory, e.g. Adams's "Stable homotopy and generalised homology", part III. The most conceptual reason of this is the Blakers-Massey theorem, which is a theorem about pushouts/pullbacks of spaces, which roughly says that a pushout diagram with highly connected maps is also a pullback diagram in a range of dimensions. The Freudenthal suspension theorem mentioned by Matthias is a corollary of this.

To address your more specific questions: $\Sigma^\infty$ preserves cofibration sequences, $\Omega^\infty$ preserves fibration sequences, but not vice versa. In fact, $\Omega^\infty$ turn cofibration sequences of spectra into fibration sequences of spaces, which is not surprising since cofibration sequences of spectra are fibration sequences after all.

And for your concrete question: you have a natural map from the homotopy cofiber of $X → Y$ to $Z$ whose connectivity depends on the connectivity of $X → Y$; the Blakers-Massey theorem will tell you the details.

Answer 4

I am not exactly sure if this is the sort of answer you are looking for, but here it goes. It seems that the actual question you are asking is about the unstable comparison of homotopy fiber and cofiber, and I am not convinced that working in spectra really solves the problem.

The classical examples concerning the interplay of homotopy fiber and homotopy cofiber come from the loop space fibration resp. the suspension cofibration. For a space $X$ we can consider the loop space fibration $\Omega X\to \ast\to X$, and then the cofiber of $\Omega X\to \ast$ is $\Sigma\Omega X$. If $X$ is a group-like $H$-space, this may be an example of your situation. Similarly, the homotopy fiber of $\ast \to \Sigma X$ is $\Omega\Sigma X$. The precise relation between $X$ and $\Omega\Sigma X$ is given by the Freudenthal suspension theorem - you get isomorphisms on homotopy in some range, and outside that range you might still be able to say something using the James model.

More generally, you may be able to use the relative Hurewicz theorem to get a relation between the homotopy fiber (controlling the relative homotopy groups) and the homotopy cofiber (controlling the relative homology groups). You might want to have a look at the discussion of the relative Hurewicz theorem in the "Simplicial homotopy theory" book by Goerss and Jardine.

Finally, I am not sure if I would agree to a statement like "for spectra cofibrations and fibrations are the same". They are still different classes of maps in the model structure. Certainly cofiber sequences and fiber sequences are the same.