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In Quillen's original text on model categories (homotopical algebra) he uses $Q$ and $R$ to denote cofibrant and fibrant replacement respectively.

This notation has been used by several other authors including Hovey in Model Categories and Dwyer and Spalinski in Homotopy Theories and model categories.

Personally I find this notation confusing, especially when taking derived functors and having to work with the derived functor of fibrant replacement - usually denoted $RR$ - hence the confusion sometimes appears.

I have seen other authors use $\hat{c}$ and $\hat{f}$ for cofibrant and fibrant replacement respectively, which personally I prefer. This notation runs into problems if you take the fibrant replacement of a map $f$ - obtains $\hat{f}f$.

So my question is,

is there a "standard" notation for cofibrant and fibrant replacement that everyone can agree on, and that has minimal ambiguity?

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  • $\begingroup$ I think, another reasonable and common notation for a fibrant replacement of an object $X$ is $X^f$. I find this particularly convenient if you did not choose once and for all a functorial fibrant replacement as the notation $RX$ for a fibrant replacement might suggest that $R$ is a functor. $\endgroup$ – Lennart Meier Jul 25 '18 at 11:30
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Quillen's notations $Q$ and $R$ are by far the most commonly used in my experience (apart from ad hoc constructions like "let $A' \to A$ be a cofibrant replacement"). In general it is a good idea to follow standard notation unless you have a very compelling reason to deviate. The target audience for your paper will often have spent 20+ years reading the literature which uses this standard notation and you'll make it easier for them if you do so too.

Regarding the ugliness of $RR$ (how often do you use that?), you could use different R's for the two different meanings: of course you can choose between $R$, $\mathrm R$, $\mathbf R$, $\mathsf R$, $\mathbb R$,...

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I find convenient a version of the notation used by Durov in "New Approach to Arakelov Geometry".

Let $A\xrightarrow{(c)} B$, $A\xrightarrow{(w)}B$, $A\xrightarrow{(f)} B$ denote that the map $A\longrightarrow B$ is a fibration, a weak equivalence, and a cofibration, respectivelly. You can combine labels, thus $A\xrightarrow{(cw)} B$, $A\xrightarrow{(wf)}B$ denote acyclic cofibration and acyclic fibrations.

Cofibrant replacement is then denoted $\perp\xrightarrow{(c)} B_{(c)}\xrightarrow{} B$; similarly, fibrant replacement is denoted $A\xrightarrow{(c)} A_{(wf)}\xrightarrow{(wf)} \top$.

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