Spooky categories

Suppose $C$ is an abelian category and $B\subseteq C$ is a subcategory which is also abelian. Is $B$ a sub-(abelian category) of $C$?

Categorical noetherian-ness

A ring $A$ is noetherian iff the category $_A\mathrm{mod}$ of finitely generated $A$-modules is a full abelian subcategory of the category $_A\mathrm{Mod}$ of all $A$-modules.

Compact objects

Let $\mathcal{C}$ be a category. An object $C\in\mathcal{C}$ is small or compact if the representable functor $[C,-]$ preserves filtered colimits: Notice that for any diagram $\{A_i\}$ there is always a natural arrow $colim [C,A_i] \to [C, colim A_i]$. The colimit is preserved if this arrow is an isomorphism.

To get used to this definition prove that the compact objects in the category of sets are the finite sets. If you like algebra you can try to characterize the compact objects the category of modules over some ring. In the vein of the following problems you can try to characterize the compact objects in the category of (linear) functors $mod_A \to mod_K$. Finally, if you like topology you can study the relationship of categorical compactness and topological compactness. For example see what happens in the category of CW-complexes and cellular maps.

Now to the probems.

Problem 1: Let $\mathcal{A}$ be a category and take $\mathcal{C} := A^{op} \to Set$. Show that the representable functors are compact objects in $\mathcal{C}$.

Problem 2: Show that finite colimits of compact objects are compact.

This implies that finite colimits of representable functors are compact. Well, there’s more:

Problem 3: In the setting of Problem 1 show that an object is compact if and only if it is a finite colimit of representable functors.