# A set of objects classically generates the full subcategory of compact objects iff it generates the whole category

Sorry in advance if my question doesn't have the level of this community.

I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated categories.

As long as I was figuring out the definitions of "classically generated", "generated" and "compactly generated", I came up with the Theorem 2.1.2 of this paper which says that in a compactly generated category $$\mathcal{C}$$, a set of objects $$\mathcal{E}\subset \mathcal{C}^{c}$$ classically generates $$\mathcal{C}^{c}$$ if and only if it generates $$\mathcal{C}$$. For the proof I started to read the reffering paper by Neeman and successfully finished and understood it. But there are some problems of connecting the information of the 2 papers.

1. First of all, while theorem 2.1.2 uses the notion of "generates" that Bondal and Van Den Bergh define, in Neeman's paper there is nowhere such a notion and maybe instead of this, he uses the notion of the smallest localising triangulated subcategory that contains a set of objects. I think that there is the following connection which I unfortunately can't prove $$\mathcal{R}\ \text {is the smallest localising triangulated subcategory that contains a set}\ R\subset \mathcal{C}^{c}\ \Leftrightarrow$$$$R^{\perp}=0$$ where $$R^{\perp}$$ is defined as in Bondal's paper as a subcategory of $$\mathcal{R}$$ and $$\mathcal{R}$$ is the smallest localising triangulated subcategory that contains a set $$R\subset \mathcal{C}^{c}$$ as in Neeman's paper. I strongly think that this is true under the hypothesis that $$\mathcal{R}\subset \mathcal{C}$$ where $$\mathcal{C}$$ is compactly generated.
2. If the 1. is true then I construct the proof of Theorem 2.1.2 in the following way: by lemma 2.2 in Neeman's Paper(in the main proof of theorem 2.1) we clearly have the implication $$\mathcal{E}$$ generates $$\mathcal{C}$$ $$\implies$$ $$\mathcal{E}\subset \mathcal{C}^{c}$$ classically generates $$\mathcal{C}^{c}$$. For the other direction we note that if the smallest localising triangulated subcategory that contains $$\mathcal{C}^{c}$$ is the whole $$C$$, then $$\mathcal{C}$$ consists of coproducts of objects in $$\mathcal{C}^{c}$$. Then it is easy, since $$Hom(M,\bigoplus E_{i})\cong \bigoplus Hom(M,E_{i})$$.

As it seems there is a problem of combining these two papers and any help will be accepted.

Edit: To be honest I hadn't found this post that has an answer from Leonid Positselski, so I reform my question: as the proof is for well-generated categories (a generalised notion of compact objects) and uses the Brown representability theorem for triangulated categories I was wondering if there is a simpler proof for compact objects without using it. Maybe the first proof before the introduction of well-generatedness by Krause and others.

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• Cross-posted on math.stackexchange – Jeremy Rickard 2 days ago
• Thank you very much for the comment Profesor @JeremyRickard.I also edited the post above. – T.Karawolf yesterday

I agree that the various notions of 'generates' can be confusing. I think the following result may clarify what you are after (this can be found in Lemma 2.2.1 of 'Stable model categories are categories of modules' by Schwede and Shipley).

Let $$\mathcal{C}$$ be a triangulated category with infinite coproducts and let $$\mathcal{P}$$ be a set of compact objects. Then the following are equivalent:

(i) The smallest localizing subcategory of $$\mathcal{C}$$ containing $$\mathcal{P}$$ is $$\mathcal{C}$$.

(ii) An object $$X \in \mathcal{C}$$ is trivial if and only if $$[P,X]_* = 0$$ for all $$P \in \mathcal{P}$$ (in the language of the question, $$\mathcal{P}^{\perp} = 0$$).

Finally, we also have the following, due to Thomason:

Suppose $$\mathcal{C}$$ is a compactly generated triangulated category, and $$\mathcal{A}$$ is a set of compact objects, then $$\mathop{Loc(A)} \cap \mathcal{C}^{c} = \mathop{Thick(A)}$$.

Here $$\mathop{Loc}(\mathcal{A})$$ denotes the smallest localizing subcategory of $$\mathcal{C}$$ containing $$\mathcal{A}$$, and $$\mathop{Thick}(\mathcal{A})$$ denotes the smallest thick (epaisse in Bondal--Van Den Bergh) subcategory of $$\mathcal{C}^c$$ containing $$\mathcal{A}$$.

• Thank you very much @Drew Heard for the answer.Actually,after your response and some reading of Lemma 2.2.1 of 'Stable model categories are categories of modules' by Schwede and Shipley i figured out that in particular lemma 1.7 of Neeman's paper gives what i wanted (as Schwede and Shipley explains but it was unclear to me in Neeman's paper) – T.Karawolf yesterday