What are the projective and the injective objects in the category of spectra (of simplicial sets)?
Does the category of spectra have enough projectives and injectives?
An object $P$ of a category $C$ is called projective, if $\operatorname{hom}_C(P,-)$ preserves epimorphisms.
An object $I$ of a category $C$ is called injective, if $\operatorname{hom}_C(-,I)$ takes monomorphisms to epimorphisms.
The category of spectra (of simplicial sets) in question has as objects $X$ sequences of pointed simplicial sets $X_0, X_1,X_2,\ldots$ together with (pointed) structure maps $\Sigma X_n\to X_{n+1}$ and as morphisms $X\to Y$ sequences of morphisms $X_n\to Y_n$ of pointed simplicial sets making the obvious diagram involving the structure maps commute.