Is there a name for an over-category where the morphisms to the object is required to be a monomorphism?

Question

Let there be a category $C$. I would like to work on the category $C/c$ where the morphism to $c$ is a monomorphism.

I go like this: Let $C_m$ be the subcategory of $C$ where all morphisms are monomorphisms, and work in $C_m/c$.

My question is: is there a name for this construction in category theory?

Answer 1

What you are describing is the first step in constructing the subobject poset of $c$. So let us recall how that goes.

Definition. Given monomorphisms $f: x \to c$ and $g: y \to c$, we say write $f \leq g$ if there is an arrow $h: x \to y$ such that $gh = f$.

Exercise: prove that $h$ itself must be a monomorphism, and that it is unique.

Definition. The above definition defines a pre-order on the monomorphisms into $c$. We say that two such monomorphisms $f: x \to c$ and $g: y \to c$ are equivalent if $f \leq g$ and $g \leq f$.

Exercise: if $f$ and $g$ are equivalent, then the arrows witnessing $f \leq g$ and $g \leq f$ are actually isomorphisms. So in particular, $x$ and $y$ are isomorphic.

Definition. A subobject of $c$ is an equivalence class of monomorphisms (under the previously defined equivalence relation). We denote by $\operatorname{Sub}(c)$ the collection of all subobjects of $c$. The pre-order on monomorphisms then becomes a partial order on subobjects, making $\operatorname{Sub}(c)$ into a partial order.

Exercise: verify that we indeed get a partial order on $\operatorname{Sub}(c)$.

This is the basic construction of the subobject poset. In the original question you basically just do not take the equivalence classes.

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A few interesting notes about these subobjects.

Subobjects often have very natural descriptions. For example, in the category of sets they are just subsets and $\operatorname{Sub}(c)$, for a set $c$, is essentially just the powerset of $c$.

I wrote "collection" on purpose for $\operatorname{Sub}(c)$, and not "set", because it can be too big to be a set. For example, if we consider the class of ordinals with the reverse order as a category, then $0$ (or in fact, any object in that category) has a proper class of subobjects. However, in most categories we consider $\operatorname{Sub}(c)$ is a set, we call such categories well-powered.

Different kind of categories can guarantee a lot of extra structure on $\operatorname{Sub}(c)$. For example, in the category of sets it is always a Boolean algebra. In a topos it is always a Heyting algebra. In the category of groups we get a complete lattice (subobjects are subgroups there).