In category theory, a monomorphism (also called a monic morphism or a mono) is defined to be a left-cancellative morphism. It seems that this definition generalizes the definition of injections. However, even in a concrete category, a monomorphism may not be an injection. Why could this happen? I know examples of monomorphisms which are not injections but what is the reason behind the existence of such examples? Isn't monomorphism a generalised concept of injection? Similar questions can be asked about epimorphisms and surjections.
[Math] Why the underlying function of a monomorphism may not be an injection
ct.category-theory
Related Solutions
Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related question was discussed here.) I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.
Your third question was asked, closed, and deleted once; I started a blog discussion about it here.
I'll attempt to answer some of your questions. First off, 1) and 3) are equivalent. This is because the bicategory of fractions in 3) is the Morita bicategory of topological groupoids, which is equivalent to the bicategory of topological stacks. What maps are you inverting in 3)? Well, if you have an internal functor $F:G \to H$ of topological groupoids, you basically want to know when is this functor morally an equivalence, but I can be more precise. If you have a functor of categories, it is an equivalence if and only if it is essentially surjective and full and faithful. Note that this uses the axiom of choices, namely, that every epimorphism splits.
An internal functor $F:G \to H$ is a Morita equivalence if
1) it is essentially surjective in the following sense: The canonical map $$t \circ pr_1:H_1 \times_{H_0} G_0 \to H_0$$
which sends a pair $(h,x),$ such that $h$ is an arrow with source $F(x),$ to the target of $h,$ is an surjective local homoemorphism 'etale surjection)
2) $G_1$ is homeomorphic to the pullback $(G_0 \times G_0) \times_{H_0 \times H_0} H_1,$ which is literally a diagramatic way of saying full and faithful.
You asked in 1), why 'etale surjection? Because this is what makes the map, when viewed as a map in in the topos $Sh(Top)$ an epimorphism, since the Grothendieck topology on topological spaces can be generated by surjective local homeomorphisms. If you want to use another Grothendieck topology (e.g. the compacty generated one, as I do in one of my papers), you must adjust accordingly.
Anyway, $F$ satisfies 1) and 2) if and only if the induced map between the associated topological stacks is an equivalence. Note though, that surjective local homeomorphisms don't always split (if they did, then every Morita equivalence would have an inverse internal functor). Hence, we have to use spans to represent morphisms, where one leg "ought to be" invertible.
Finally, you say that $G$ and $H$ are Morita equivalent if there is a diagram $G \leftarrow K \to H$ of Morita equivalences. This is the standard definition of Morita equivalent. Since the bicategory of fractions with respect to Morita equivalences is equivalent to topological stacks, one can also say that $G$ and $H$ are Morita equivalent of the have equivalent topological stacks.
Now, I'll respond to some of the comments:
@Zhen: If $G$ and $H$ are 'etale (or more generally 'etale complete) then they have equivalent classifying topoi if and only if they have equivalent stacks. In fact, there is an equivalence of bicategories between 'etale topological stacks, and the topoi which are classifying topoi of 'etale topological groupoids ('etendue). For more general topological groupoids however, there is information lost when passing to their classifying topoi.
@Ben: I believe this is related to Zhen's question. The definition in the way you stated it, usually appears in topos literature, and is related to the fact that open surjections of topoi are of effective descent. This is not a good concept when the groupoids in question are not etale. To deal with torsors one really wants to have some version of local sections.
Best Answer
There are several issues here. Let $\mathcal C$ be a category.
(1) The question only makes sense when $\mathcal C$ is concrete, as otherwise "injective" has no meaning. When $\mathcal C$ is concrete, we can choose a faithful functor $\pi : \mathcal C\to Set$, and regard a morphism $f:X\to Y$ as being injective if $\pi(f)$ is injective (i.e. a monomorphism in $Set$).
(2) However, this notion of "injective" depends on the choice of concrete structure. Indeed, if we consider one of the simplest possible categories, which has two objects $A,B$ and one nonidentity morphism $f: A\to B$, then we can concretize this category by mapping $A,B,$ to an $n$-element set and an $m$-element set, respectively, and mapping the morphism to any function between those two sets--the faithfulness condition is essentially vacuous. But now $f$ was a monomorphism (and an epimorphism, although not an isomorphism) and its concretization will not typically be any of these things, depending on how the concretization was chosen.
(3) This counterexample highlights what can go wrong. First of all the notion of monomorphism depends solely on the category, and not on a choice of concretization, so it is a better "invariant" generalization of the notion of injection from $Set$. Furthermore, it can be discussed whether or not a concrete structure exists. Finally, the phenomenon that a monomorphism can fail to be injective relative to some concrete structure is roughly attributable to a lack of "enough" morphisms and objects in the original category, so that in $Set$ one sees the map is not a monomorphism, but in the original category there are not enough objects and morphisms to find a "preimage" of any diagram exhibiting the failure of injectivity.