# What are the projective and the injective objects in the category of spectra?

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.

• 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. – Qiaochu Yuan Feb 9 '15 at 21:11
• 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. – user8463524 Feb 9 '15 at 21:15
• There should be plenty of projectives, as with any other category of essentially algebraic structures. – Zhen Lin Feb 9 '15 at 21:57