I'm trying to better understand the relationship between monicity and injectivity as properties of a morphism. My question has two parts.
what are the necessary and sufficient conditions in a category for every monic morphism to be injective?
(I'm guessing that this is not an open question!)
I am also interested in the reverse implication (i.e. every injection is monic). In this case, the standard definition of injective (i.e. $f(x) = f(y) \Rightarrow x = y$) already implies that every injective morphism is monic, but this definition is "pointwise", which is not a very satisfying as a categorical criterion. Therefore, the second part of my question is
is there a way to define injective that does not make any reference to individual elements of sets, but yet is equivalent to the "standard" definition?
(My hope is that from such a point-free definition I will be able to come up with a point-free, purely categorical proof of "injective $\Rightarrow$ monic".)
P.S. This question was motivated by some remarks on p. 32 of Sets for Mathematics by Lawvere and Rosebrugh (2003), where they define what a monomorphism is. Basically their definition says that a morphism $i:S \rightarrow A$ is monomorphic iff, for every pair of morphisms $s_1, s_2:T \rightarrow S, (i \circ s_1 = i \circ s_2) \Rightarrow (s_1 = s_2)$, which they call the "cancellation property with respect to composition on [the morphism's] right". Then they go on to say that
The difference between monomorphic and injective is that, for the "mono" property, we require cancellation for all $T$ [whereas for the 'injective' property, we require cancellation only when composing with morphisms with domain $T = 1$]. This does not matter in the case of abstract sets, where cancellation with the general $T$ or with just $T > = 1$ means the same thing ... . *That "mono" implies injective is * *tautologous because a general statement always implies any of its special * cases. The converse statement is $not$ tautologous; it depends on the existence of sufficiently many elements. (emphasis added)
The sentence I emphasized above puzzles me, because I know of examples where a monic morphism is not injective. The following statement is also a bit puzzling, because the implication that they don't consider tautologous ("injective $\Rightarrow$ monic") is the one that looks entirely tautologous to me. And I am also puzzled by what exactly they meant by $T = 1$. Did they mean "$T =$ the category's terminal object"? Or "$T =$ the category's separator object?" Or "$T =$ a/the singleton object"? In $Sets$ all these descriptions apply to 1, but it is not clear to me which of them is the basis for the authors' comments. Furthermore, it means that the authors are either (1) implying that the term injective can be meaningfully applied only to morphisms in categories that have a "1" object (with 1 = terminal/separator/singleton, depending on what the authors meant), or else (2) implying that every concrete category has one such object.