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Given a map of spectra (pick whatever category you want), $f:X\to Y$, what are the exact constructions of the fiber and cofibers of this map? Does this depend in any deep way upon the category or spectra you're working with?

For the cofiber, does it suffice to simply mimic the mapping cylinder construction from topological spaces in every degree, or must we make some other adjustments for the structure maps?


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up vote 3 down vote accepted

Nice question. I'll admit I am not 100% sure on what follows - there should be something in Adams' book though, I would have thought?

I believe we get that it all commutes with the structure maps automatically.

Recall that a map of spectra $f:E \to F$ is a collection of maps that are compatible with the structure maps. For two maps $f_0,f_1$ a homotopy is then a map $g:E \wedge I_+ \to F$ such that the restrictions to $E \wedge \{ 0 \}$ and $E \wedge\{1 \}$ are $f_0$ and $f_1$. The map $f$ will be a cofibration if it has the HEP for all spectra, and then we can form the cofiber of $f$ as the pushout of the following diagram

$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} E & \ra{} & CE\\ \da{f} & & \da{} \\ F & \ra{} & C_f=F \cup_f CE \\ \end{array} $$ where $CE$ is the 'cone' $E \wedge I$

Similarly the fiber can be formed by a similar type of pull back diagram.

It should be true, I believe, that (since everything commutes with our maps. Or in Adams' terminology we can use strict maps on cofinal subspectra)

$$(C_f)_n = C_{f_n}$$ and that $$\Sigma C_{f_n} = C_{\Sigma f_n}$$

Ok, the diagram for the fibre should look like

$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} F_f{} & & \ra{} & PF\\ \da{} & & & \da{} \\ E & & \ra{f} & F \\ \end{array} $$

where $PF=\text{Map}(I,E)$

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Thankyou, I believe you're correct regarding the cofiber situation. Would you mind actually showing that pullback diagram for the fiber? Or at least, what is the thing that maps to $F$ that we pullback over, some kind of contractible mapping space or something? And perhaps there is something in Adams' book. There is no index, and I'm moving slowly, so perhaps I have not found it yet. – Jon Beardsley Nov 5 '11 at 20:29
Something like that - I've update with what I think it is (and hopefully one of the experts can chime in and verify all this stuff!) – Juan S Nov 6 '11 at 11:38
Yeah that seems right. I can never remember all these constructions. Thanks Juan! – Jon Beardsley Nov 6 '11 at 15:53

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