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Yes, of course. The coderived and contraderived categories (or more specifically, the absolute derived categories) are defined precisely in such a way that short exact sequences of CDG-modules induce …

The answer to your question is positive. See Theorem 1.20 and subsequent Remark in the book of Adamek and Rosicky.

This short answer is a complement to Simon's detailed answer. In Simon's answer he considers arbitrary $\kappa$-accessible categories $A$ (in the first theorem) and locally $\kappa$-presentable ones …

One reference is H. Krause, "Derived categories, resolutions, and Brown representability", Contemporary Math. vol.436, AMS, 2007, p.101-139 or https://arxiv.org/abs/math/0511047 , Section 1.6. Another …

It appears that this result may go back to a 1984 letter of Joyal to Grothendieck. The reference to this letter, as well as some other early references, can be found in Example 3.2 in the paper Cotor …

The standard result in this direction is the dual Eklof lemma (for your first problem) or the Eklof lemma (for your dual problem). Any version of the Eklof lemma presumes that your direct/inverse sys …

A) It depends on what you are interested in. If you do not impose the finiteness condition, then it means that you are describing a different class of abelian categories. Which class is that, depend …

This question is a reference request for the following result or two results, which I believe are rather easy to prove. Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\subs …

Let $R$ be a ring and $\kappa$ be a strongly compact cardinal such that $|R|<\kappa$. Then the class of all projective $R$-modules is closed under $\kappa$-filtered colimits. This is Theorem 3.3 in …

You are right. There is no such a map as the one you are trying to describe. Here is a map that actually exists. Let $R$ and $S$ be two noncommutative rings with units, and let $P$ be an $R$-$S$-bi …

The abelian categories in which all short exact sequences split I would call "split abelian categories", reserving the term "semisimple abelian category" for a more restrictive condition. Roughly, pe …

Another and probably more natural interpretation of the sentence in the Wikipedia article may be called "localizing an adjunction to an equivalence". Let $\mathcal C$ and $\mathcal D$ be two categori …

Let $\mathcal C$ and $\mathcal D$ be two categories, and let $F\colon\mathcal C\longrightarrow \mathcal D$ and $G\colon\mathcal D\longrightarrow\mathcal C$ be two functors, with $F$ left adjoint to $G …

The following pair of examples follows the idea of Jeremy Rickard suggested in a comment on Math Stack Exchange under the link. Inverting the arrows, it suffices to construct an example of abelian ca …

Yes, it is true. Moreover, neither cocommutativity nor characteristic $0$ are needed as conditions. The tensor product of two conilpotent coassociative dg-coalgebras over any field $k$ is a conilpot …