category theory

# Contents

## 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”. ( Projective limit is an older term for limit.) It is in contrast to “ind” in the dual notion of ind-object, standing for “inductive”, (and corresponding to inductive limit, the old term for colimit). In some (often older) sources, the term ‘projective system’ is used more or less synonymously for pro-object.

## Definition

The objects of the category $\mathrm{pro}$-$C$ are diagrams $F:D\to C$ where $D$ is a small cofiltered category. The hom set of morphisms between $F:D\to C$ and $G:E\to C$ is

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

The limit and colimit is taken in the category Set of sets. Cofiltered limits there are threads and filtered colimits are germs (classes of equivalences). Thus a representative of $s\in \mathrm{pro}C\left(F,G\right)$ is a thread whose each component is a germ:
$s=\left({\mathrm{germ}}_{e}\left(s\right){\right)}_{e\in E}$ which can be more concretely written as $\left(\left[{s}_{{d}_{e},e}\right]{\right)}_{e}$; thus $\left[{s}_{{d}_{e},e}\right]\in {\mathrm{colim}}_{d\in D}C\left(Fd,Ge\right)$ where ${s}_{{d}_{e},e}\in C\left(F{d}_{e},Ge\right)$ is some representative of the class; there is at least one ${d}_{e}$ for each $e$; if the domain $E$ is infinite, we seem to need an axiom of choice in general to find a function $e↦{d}_{e}$ which will choose one representative in each class ${\mathrm{germ}}_{e}\left(s\right)$. Thus $s$ is given by the (equivalence class) of the following data

• function $e↦{d}_{e}$

• correspondence $e↦{s}_{{d}_{e},e}\in C\left(F{d}_{e},Ge\right)$

such that $\left(\left[{s}_{{d}_{e},e}\right]{\right)}_{e}$ is a thread, i.e. for any $\gamma :e\to e\prime$ we have an equality of classes (germs) $\left[G\left(\gamma \right)\circ {s}_{{d}_{e},e}\right]=\left[{s}_{{d}_{e\prime },e\prime }\right]$. This equality holds if there is a $d\prime$ and morphisms ${\delta }_{e}:d\prime \to {d}_{e}$, ${\delta }_{e\prime }:d\prime \to {d}_{e\prime }$ such that $G\left(\gamma \right)\circ {s}_{{d}_{e},e}\circ F{\delta }_{e}={s}_{{d}_{e\prime },e\prime }\circ F{\delta }_{e\prime }$. (Usually in fact people consider the dual of $D$ and the dual of $C$ as filtered domains). Now if we chose a different function $e↦{\stackrel{˜}{d}}_{e}$ instead then, $\left(\left[{s}_{{d}_{e},e}\right]{\right)}_{e}=\left(\left[{s}_{{\stackrel{˜}{d}}_{e},e}\right]{\right)}_{e}$, hence by the definition od classes, for every $e$ there is a $d″\in D$ and morphisms ${\sigma }_{e}:d″\to {d}_{e}$, ${\stackrel{˜}{\sigma }}_{e}:d″\to {\stackrel{˜}{d}}_{e}$ such that ${s}_{{\stackrel{˜}{d}}_{e},e}\circ F\left({\stackrel{˜}{\sigma }}_{e}\right)={s}_{{d}_{e},e}\circ F\left({\sigma }_{e}\right)$.

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 $\mathrm{ind}$-$C$.

#### Definition of $\mathrm{pro}C$ as a subcategory of functors

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

## References

• (SGA4-1) Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics 269, Springer 1972.

• Michael Artin and Barry Mazur, Étale homotopy theory, 1969, No. 100 in Lecture Notes in Maths., Springer-Verlag, Berlin.

• J.-M. Cordier, Tim Porter, Shape Theory , categorical methods of approximation, Dover (2008) (It is a reprint of the 1989 edition without amendments.)

• S. Mardešić, J. Segal, Shape theory, North Holland 1982

Revised on May 15, 2012 19:11:00 by Stephan Alexander Spahn (79.227.166.2)