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.

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    $\begingroup$ I don't like calling this "the category of spectra." Spectra naturally form an $\infty$-category, not a category. There are model categories of spectra which present this $\infty$-category, and there is a homotopy category of spectra. Whatever this is, it doesn't yet have enough data attached to it to even count as a presentation of the $\infty$-category of spectra. $\endgroup$ – Qiaochu Yuan Feb 9 '15 at 21:11
  • $\begingroup$ Dear Qiaochu Yuan, I am sorry for the probably badly-chosen name. Anyway, I neither want to consider the stable homotopy category (as a triangulated category, all its objects are injective and projective) nor the $\infty$-category of spectra, I am talking about an ordinary 1-category. $\endgroup$ – user8463524 Feb 9 '15 at 21:15
  • $\begingroup$ There should be plenty of projectives, as with any other category of essentially algebraic structures. $\endgroup$ – Zhen Lin Feb 9 '15 at 21:57

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