definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Question

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while trying to read Lecture 4 of Lurie's notes here on complex oriented cohomology theories:

1. What is the definition of the spectrum $E^X$? I wanted to know what it is as a sequence of spaces, but perhaps it is best to think of it via its defining property of being a path space, which is what? My guesses were it an isomorphism between morphism spaces in the stable homotopy category of spectra $$[F, E^X]=[F \wedge \Sigma^\infty X ,E]$$ for every spectrum $F$? Or perhaps it is $$[\Sigma^\infty Y, E^X]=[Y \wedge X, \Omega^\infty E]$$ for every CW-complex $Y$, where $[Y \wedge X, \Omega^\infty E]$ is homotopy classes of based maps between based topological spaces.

2. What is the nth-space of $E \wedge \Sigma^\infty X$? Is it different from the spectrum whose $n$-th space is $E_n \wedge X$?

3. There is a function $S^1 \wedge E \to \Sigma E$ given by the structure maps of $E$. Is it an isomorphism in the stable homotopy category? (I don't see why it should be).

Answer 1

Thanks for the compliments on the notes. You're right that when you first get introduced to spectra, it's a good idea to focus on the stable homotopy category and its formal properties. Then you can play around a bit, see how stability is really useful for calculating things. I think I remember Ravi Vakil saying something like this: you don't need to know what's under the hood to drive the car.

Your questions 1) and 3) are about the stable homotopy category, so they don't depend on the choice of model. Both your equations in 1) are correct. For 3), there is indeed an isomorphism between $S^1 \wedge E$ and $\Sigma E$ in the stable homotopy category.

Unfortunately, you can't do everything with just the homotopy category. Eventually you need some kind of model to work with. I prefer prespectra and orthogonal spectra, but there are lots of other models too.

In the category of prespectra, your question 2) is correct. $E \wedge X$ is a spectrum which at level $n$ is $E_n \wedge X$. The mapping spectrum $E^X$ is a spectrum which at level $n$ is $Map_*(X,E_n)$. (It's considered good etiquette make $E$ an $\Omega$-spectrum first before taking maps in from $X$.)

For 3), when defining this map explicitly, it depends on what you mean by $\Sigma E$. I usually take that to mean $S^1 \wedge E$, so these two spectra are already isomorphic by definition.

Instead, you might mean $sh E$, a spectrum with levels shifted: $(sh E)_n = E_{n+1}$. Now, this is isomorphic to $\Sigma E$ in the stable homotopy category, but it's harder to make that isomorphism explicit at the level of prespectra. If you try to use the structure maps of $E$ to define a map $S^1 \wedge E \rightarrow sh E$, you won't get a map of prespectra!

(This can be fixed, but it's a bit painful. You can flip the suspension coordinate of your map on all the levels of odd degree, then everything commutes up to homotopy, giving a "weak map" of prespectra. Then you add in a ton of cylinders to keep track of those homotopies. The result is a zig-zag of equivalences of prespectra between $S^1 \wedge E$ and $sh E$.)