Do topological spaces form a full subcategory of spectra?

Question

Let $\Sigma^\infty: Top_* \to Spectra$ be a functor sending a pointed topological space $X$ to its suspension spectrum, that is $(\Sigma^\infty X)_n=\Sigma^nX$ with isomorphisms $\Sigma(\Sigma^\infty X)_n \to (\Sigma^\infty X)_{n+1}$. Is it true that this map make $Top_*$ a full subcategory of $Spectra$, that is $$Hom_{Top_*}(X, Y)=Hom_{Spectra}(\Sigma^\infty X, \Sigma^\infty Y)?$$ Maybe, it is true in homotopy categories: $Hom_{hTop_*}(X, Y)=Hom_{hSpectra}(\Sigma^\infty X, \Sigma^\infty Y)$?

Answer 1

As noted in the comments, this is most definitely false in homotopy categories: the set $[S^1, S^0]$ of homotopy classes of based maps is trivial, but the set $[\Sigma^\infty S^1, \Sigma^\infty S^0]$ of maps in the homotopy category of spectra is $\Bbb Z/2$.

The statement about the suspension spectrum functor being fully faithful may or may not be true in the category of spectra and it strongly depends on which model you use for spectra.

• In the category of "sequential" spectra where a map of spectra is a collection of maps $X_n \to Y_n$ commuting with the structure maps (sometimes called Bousfield-Friedlander spectra), then the answer is yes: this is a full embedding. I suspect this is the definition that you are using.

• In the category of spectra described in Adams' Stable homotopy and generalized homology, he distinguishes the above (a "function" of spectra) from a "map" of spectra. Maps of CW-complexes do not embed fully faithfully into this definition of maps of spectra.

• In the category of symmetric spectra, this is also a full embedding.

• In the category of orthogonal spectra, this is also a full embedding.

• I remember seeing a (draft?) document at some point in the past proving that the suspension spectrum functor produces a fully faithful embedding into Elmendorf-Kriz-Mandell-May's category of S-modules. However, I can't track this down, and it's a bit unexpected (I believe the result is false for the closely related category of spectra developed by Lewis-May-Steinberger).