Let $R$ be a commutative ring (with identity) and let $R\mathbf{Alg}$ denote the category of $R$-algebras.
My question:
Is there a suitable notation for the full subcategory of commutative $R$-algebras?
I would like to know this in order to classify the polynomial ring $R\left[S\right]$ - where $S$ is a set - as an object of category ? free over set $S$.
In the special case $R=\mathbb{Z}$ the notation $\mathbf{CRing}$ will do, since rings can be recognized as $\mathbb{Z}$-algebras. But what in the general case?
I've seen the notation $\mathsf{CAlg}_R$ in many places. I prefer $\mathsf{CAlg}(R)$ in order to stress the functoriality. This is also used in Yves Diers' work. Many papers restrict to commutative rings and algebras in the first place and therefore just write $\mathsf{Alg}_R$ (which might be confusing - but this is just a local notation).
More generally, if $C$ is a symmetric monoidal category, then I've seen $\mathsf{CAlg}(C)$, $\mathsf{CMon}(C)$ and $\mathsf{Comm}(C)$ for the category of commutative algebras in $C$. For $C=\mathsf{Mod}(R)$ one often abbreviates $?(\mathsf{Mod}(R))$ with $?(R)$, so that you would write $\mathsf{CAlg}(R)$ or $\mathsf{Comm}(R)$ for the category of commutative $R$-algebras.
As for the font, instead of $\mathsf{CAlg}$ of course you can also write $\mathrm{CAlg}$ or $\mathbf{CAlg}$. This is not standard (as for module categories).