I need the connection between dense(see https://ncatlab.org/nlab/show/dense+subcategory) and reflective (https://ncatlab.org/nlab/show/reflective+subcategory) subcategories, i.e. If $C$ is a dense subcategory when $C$ also is reflective? or if $C$ is reflective then $C$ is dense?
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$\begingroup$ The "pro-reflective" sort of "dense subcategory" is by definition a generalization of a reflective subcategory; thus every reflective subcategory is dense in this sense. For the other (more common in category theory) meaning of "dense subcategory", I don't think there is any relationship to reflectivity one way or the other. $\endgroup$– Mike ShulmanCommented Feb 9, 2017 at 18:17
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There are dense subcategories (in the sense that every object is a colimit of objects in them) which are not reflective. Take R-mod, the finitely presented right R-modules. This is dense in R-Mod (the whole category of modules) as every module is a directed colimit of finitely presented modules. It is not reflective in R-Mod, as R-Mod is complete: A small category (R-mod is essentially small) cannot be complete unless it is a pre-order (R-mod is not a pre-order unless R=0), yet if R-mod was reflective in R-Mod it would be complete (in fact, it would be closed under limits).
0 is reflective in R-Mod but certainly not dense!
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$\begingroup$ In fact, $\{R^2\}$ is dense in $R\text{-Mod}$. $\endgroup$ Commented Mar 10, 2017 at 10:31