What are the subobjects of a manifold? Categorically a subobject of an object $a$ of some category $A$ is an object $a'$ with a monic morphism to $a$, ie $a'\to a$, upto isomorphism.
When $A$ is either a Topological or Differential manifold, what are the subobjects of a manifold, and are they the same as submanifolds?
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My own thinking on this is that in the smooth category one ought to work with manifolds together with the tangent bundle structure and then epics turn out to be surjective submersions and monics, injective immersions. These being very useful notions in the wider context of manifold theory.
 A: Let $\mathcal{C}$ be the category of toplogical(resp. differential) manifolds.
The objects of $\mathcal{C}$ are topological(resp. differential) manifolds and the morphisms of $\mathcal{C}$ are continuous(resp. smooth) maps.
Let $f\colon X \rightarrow Y$ be a morphism in $\mathcal{C}$.
We claim $f$ is a monomorphism if and only if $f$ is injective.
Suppose $f$ is a monomorphism.
Let $x, y$ be distinct points of $X$.
Let $p$ be a $0$-dimensional object in $\mathcal{C}$.
There exists the unique morphism $g\colon p \rightarrow X$ such that $g(p) = x$.
Similarly there exists the unique morphism $h\colon p \rightarrow X$ such that $h(p) = y$.
Since $g \neq h$, $fg \neq fh$.
Hence $f(x) \neq f(y)$.
Hence $f$ is an injective map.
Conversly suppose $f$ is injective.
Clearly $f$ is a monomorphism.

"Are they the same as submanifolds?"
Generally no.
Counter-example:
Let $f\colon \mathbb{R} \rightarrow \mathbb{R}^2$ be the map defined by $f(x) = (x^3, 0)$.
$f$ is smooth and injective, but is not an immersion($f'(0) = 0$).
Hence $\mathbb{R}$ cannot be identified with a submanifold of $\mathbb{R}^2$ by $f$.
A: Monic morphisms are injections in concrete categories like the category of manifolds, so a subobject of a manifold $M$ is another manifold with an injection into $M$.  This isn't quite the same thing as a submanifold, as usually submanifolds are required to be embedded, meaning that they inherit their topology from the larger manifold.
For an example of a manifold that is injected into a larger manifold but isn't embedded, let $M$ be a torus (considered as a quotient space of $\mathbb{R}^2$ under integer translations), and let $L$ be the real line, mapped to a line of irrational slope in $\mathbb{R}^2$ and then projected to the torus.  This causes the line to "wrap around" infinitely without touching itself.  This map is an injection, thus $L$ with this map is a subobject, but the image isn't a manifold, since any neighborhood of a point contains infinitely many "nearby" lines.
Edit: As pointed out below, I was erroneous when I said monic morphisms are injective in all concrete categories.  It is true for manifolds, however, as proved in Makoto Kato's answer.
