nLab
pro-object

Idea

A pro-object of a category $C$ is a “formal cofiltered limit” of objects of $C$. The category of pro-objects of $C$ is written $\mathrm{pro}$-$C$. Such a category is sometimes called a ‘pro-category’, but notice that that is not the same thing as a pro-object in Cat.

“Pro” is short for “projective limit,” an old term for a limit, as contrasted with “ind” in the dual notion for “inductive limit,” the old term for colimit.

Definition

There are many ways to make this notion precise. One is to define the objects of $\mathrm{pro}$-$C$ to be diagrams $F:D\to C$ where $D$ is a small cofiltered category. The set of morphisms between $F:D\to C$ and $G:E\to C$ is then defined to be

(1)$$\mathrm{pro}\text{-}C(F,G)={\lim }_{e\in E}{\mathrm{colim}}_{d\in D}C(Fd,Ge)$$pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)

This definition is perhaps more intuitive in the dual case of ind-objects (pro-objects in $C^{\mathrm{op}}$), where it can be seen as stipulating that the objects of $C$ are finitely presentable in $\mathrm{ind}$-$C$.

Another, equivalent, definition is to let $\mathrm{pro}$-$C$ be the full subcategory of $[C,\mathrm{Set}{]}^{\mathrm{op}}$ determined by those functors which are cofiltered limits of representables. This is reasonable since $[C,\mathrm{Set}{]}^{\mathrm{op}}$ is the free completion of $C$, so $\mathrm{pro}$-$C$ is the “free completion of $C$ under cofiltered limits.”

References

One source for the theory of pro-objects is

• J.-M. Cordier and T. Porter, 2008, Shape Theory, categorical methods of approximation, Dover.

(I might as well get a plug in!! It is a reprint of the 1989 edition without amendments.)

Another good reference is

• P. T. Johnstone, Stone Spaces.