# Why is full- & faithful- functor defined in terms of Set properties?

Wikipedia entry or Roman's "Lattices and Ordered Sets" p.286, or Bergman's General Algebra and Universal Constructions, p.177 and in fact every definition of full and/or faithful functor is defined in terms of the Set-theoretical properties: surjective and injective on (compatible) arrows.

Why aren't full-/faithful- defined in terms of epic and monic, in other words, in terms of algebraic invertibility or cancellation properties, eg if it is required to consider not a set of arrows but a topology or order (or any other category) of them.

Is this an historical accident awaiting suitable generalization, or is there some fundamental reason why Set seems to always lurk in the background?

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Because being a fully faithful functor is not a property internal to a category but rather a 2-category. Moreover ordinary categories are enriched over $\textbf{Set}$ so it is only natural to describe the conditions in terms of monomorphisms and epimorphisms in $\textbf{Set}$, i.e. injections and surjections. –  Zhen Lin Jul 3 '12 at 20:35
@ZhenLin, how do you interpret Lawevere: "...natural generalization of category theory within itself...whose hom-functors take values in a given closed category (not necessarily in the category of abstract sets), it's possible to regard a metric space as a category..." -- FW Lawvere "Metric spaces, generalized logic and closed categories" TAC 2002 –  alancalvitti Jul 3 '12 at 21:51
That's just some fluff about enriched categories. As long as you are enriching over a category (and it need not be closed!) then it makes sense to define a fully faithful functor as one that induces isomorphisms on hom-objects. (This isn't something that generalises too well, though.) –  Zhen Lin Jul 4 '12 at 5:21
That fluff was written by the mathematician who invented the concept of enriched categories. –  alancalvitti Jul 4 '12 at 16:09
Lawvere invented many things, but enriched category theory is not one of them. The credit should go to Eilenberg and Kelly. –  Zhen Lin Jul 4 '12 at 16:20

Full and Faithful can be easily defined in general with no reference to Set. Just state:

Faithful functor F

$\forall (f,g: A \to B)$: $Ff = Fg$ implies $f = g$

Full functor F

$\forall (h: FA\to FB)$ $\exists (f: A \to B):Ff = h$

Just FOL, no set theory or category of sets.

You can find this in CWM chapter 1 section Functors

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Thanks magma, so it was there from the start... I'm trying to find out from Prof Bergman what compels reference to Set all over what should be Universal Algebra. –  alancalvitti Aug 4 '12 at 17:35

I think requiring injective/surjective suffices, since these concepts are equivalent to epic/monic in the category of sets, and the categories are locally small. So the point is that we've already assumed that the categories themselves have sets of morphisms. In an abelian (more generally, preadditive) category, for example, we might be interested in requiring a more stringent condition on the hom-sets.

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The obvious analogue for abelian/additive categories asks for isomorphisms of hom-groups... but this amounts to asking for an additive functor that is fully faithful in the usual sense, because an isomorphism of abelian groups is the same thing as a bijective homomorphism. –  Zhen Lin Jul 3 '12 at 20:42
Yes, thanks. I didn't really know what a $2$-category is, I just wanted to emphasize that the definition of fully faithful seems to rely on the structure of the hom-objects. –  Andrew Jul 3 '12 at 20:53
Set is one category among many. Suppose you want to consider a Topos of morphisms, or order them or topologize them or do anything else that you can do to spaces of objects...? –  alancalvitti Jul 4 '12 at 16:11
Exactly. My point was that injective/surjective suffice when we only consider the collections of morphisms as sets. –  Andrew Jul 4 '12 at 16:26
@user34377, why is it the case that a collection of morphisms are considered as a set, as opposed to a category? - In particular, why should categories of functors refer to Set at all? –  alancalvitti Jul 4 '12 at 17:01