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”. ( Projective limit is an older term for limit.) It is in contrast to “ind” in the dual notion, 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)$$\mathrm{pro}\text{-}C(F,G)={\lim }_{e\in E}{\mathrm{colim}}_{d\in D}C(Fd,Ge)$$pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)

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 $s\in \mathrm{pro}C(F,G)$ is a thread whose each component is a germ:
$s=({\mathrm{germ}}_e(s){)}_{e\in E}$ which can be more concretely written as $([s_{d_e,e}]{)}_e$; thus $[s_{d_e,e}]\in {\mathrm{colim}}_{d\in D}C(Fd,Ge)$ where $s_{d_e,e}\in C(F{d}_e,Ge)$ 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 $e\mapsto d_e$ which will choose one representative in each class ${\mathrm{germ}}_e(s)$. Thus $s$ is given by the (equivalence class) of the following data

• function $e\mapsto d_e$

• correspondence $e\mapsto s_{d_e,e}\in C(F{d}_e,Ge)$

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

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

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 $[C,\mathrm{Set}{]}^{\mathrm{op}}$ determined by those functors which are cofiltered limits of representables. This is reasonable since the copresheaf category $[C,\mathrm{Set}{]}^{\mathrm{op}}$ is the free completion of $C$, so $\mathrm{pro}$-$C$ is the “free completion of $C$ under cofiltered limits.”

Examples

• profinite group

• progroup

• profinite space

Related concepts

• ind-object / ind-object in an (∞,1)-category

• pro-object / pro-object in an (∞,1)-category

References

• A. Grothendieck,Technique de descente … II

• (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.)

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves

• Peter Johnstone, Stone Spaces.

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