Q:

Exact Constructions of Homotopy Fiber and Cofiber of Spectra

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?
Thanks

A:

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

