I am self-learning category theory (though, at this point, I no longer remember what got me started), and I have encountered a troubling definition on wikipedia.
The formal definitions of (full) subcategory seem straightforward enough. However, there is a key bit of confusion in the definition of reflective subcategory, which is (at present) given as follows (with only changes in format):
A full subcategory $\mathsf A$ of a category $\mathsf B$ is said to be reflective in $\mathsf B$ if for each $\mathsf B$-object $B$ there exists an $\mathsf A$-object $A_B$ and a $\mathsf B$-morphism $r_B:B\to A_B$ such that for each $\mathsf B$-morphism $f:B\to A$ there exists a unique $\mathsf A$-morphism $\overline f:A_B\to A$ with $\overline f\circ r_B=f.$
It likewise states (as does every other source that I've checked) the following definition (again, up to formatting):
A full subcategory $\mathsf A$ of a category $\mathsf B$ is said to be reflective in $\mathsf B$ when the inclusion functor from $\mathsf A$ to $\mathsf B$ has a left adjoint.
Now, I suspect that the first definition entails the second if and only if the Axiom of Global Choice holds, but that is a question for another time (i.e.: I'd like to play with it some more, first). I am currently trying to prove that the second definition entails the first, using the counit-unit definition of adjunction.
My initial interpretation of the first definition was that $A$ was a $\mathsf B$-object for which a $\mathsf B$-morphism $f:B\to A$ exists, but as I delved into the proving process, I have come to suspect that $A$ is intended to be an $\mathsf A$-object, instead. I am nearing a proof (I hope) in the case that $A$ is intended to be and $\mathsf A$-object, but I haven't the foggiest notion how to proceed if it may be any $\mathsf B$-object.
I am aware of the often-used convention of alphabetical order (though it is not yet second nature for me, being a relative newbie to cat-theory), which leads me to suspect that my latter interpretation was correct. With that in mind, my question is as follows:
- Is this simply an instance of said convention?
- If so, can you provide a counterexample (that is, a reflective subcategory $\mathsf A$ in the sense of the second definition which does not satisfy my initial interpretation of the first definition?
- If not, then it seems I'm missing something fairly deep about one or more of the definitions mentioned (and possibly some that weren't mentioned), so I'd appreciate an outline/hint of how one proves existence of the morphism $\overline f$ described in the first definition.
Yes, $A$ is supposed to be an $\mathsf{A}$-object, not just a $\mathsf{B}$-object. However, I would describe this as simply an error on Wikipedia, not an adherance to an unstated convention. The "convention" that you refer to is not at all any sort of universally understood thing, and especially in a context like the definition of a reflective subcategory, you should be explicit about what categories your objects live in (unless you are in an informal discussion and there is established context that makes it clear what you mean).
For a counterexample to your initial interpretation, just take any reflective subcategory such that not every object of $\mathsf{B}$ is isomorphic to an object of $\mathsf{A}$ (i.e., such that the inclusion is not an equivalence). For if $A_B$ exists by your first interpretation, then taking $f=1_B$ gives a map $\bar{f}$ which is an inverse of $r_B$ (to see that it is a right inverse, note that $1_{A_B}r_B=r_B1_B=r_B(\bar{f}r_B)=(r_B\bar{f})r_B$ and use the uniqueness part of the universal property of $r_B$ in the case $f=r_B$).