I'm trying to understand the terminology for reflective subcategories.

On nLab we find the statement that the left adjoint is "called the reflector, and a functor which is a reflector is called a reflection".

However, in this paper by Adamek & Rosicky the left adjoint is only called a "reflector", while the word "reflection" is reserved for components of the unit of adjunction.

I have two questions:

  1. Are "reflections" and "reflectors" usually synonyms in the literature, or is there some variety in terminology?
  2. In the paper linked above, on top of p. 1308, where the authors describe the component of the unit of the reflective adjunction, shouldn't this be $$r_L:L\to EFL$$ rather than $$r_L:L\to FL$$? One of the adjoint functors appears to be missing on the right-hand side.
  • 2
    $\begingroup$ For your second question, you are right $E$ should be included, but since $E$ is the inclusion of a subcategory the authors are suppressing the $E$ functor in the notation since $FL$ is literally an object of the larger category. For example, an Abelian group is a group, and the inclusion of the category of Abelian groups into the category of groups doesn't actually do anything. That said, I dislike these "implicit conversions" especially in a categorical context where a lot of interesting things are adjoints to "trivial" seeming functors including reflectors. $\endgroup$ – Derek Elkins Feb 16 '18 at 17:54

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