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