Category theory; homological algebra Definition

Pro-object

Builds on Ind-completion Opposite category Filtered category Limit Profinite group

Definition

Let $\mathcal{C}$ be a category. A small category $\mathcal{I}$ is cofiltered if its opposite category $\mathcal{I}^{\mathrm{op}}$ is filtered. A pro-object in $\mathcal{C}$ is a functor $X \colon \mathcal{I} \to \mathcal{C}$ with $\mathcal{I}$ small and cofiltered, written $(X_i)_{i \in \mathcal{I}}$ . For pro-objects $X \colon \mathcal{I} \to \mathcal{C}$ and $Y \colon \mathcal{J} \to \mathcal{C}$ the morphisms are $\mathrm{Pro}(\mathcal{C})(X, Y) = \lim_{j} \operatorname{colim}_{i} \mathcal{C}(X_i, Y_j).$ The pro-objects with these morphisms form the category $\mathrm{Pro}(\mathcal{C})$ .

Remarks

There is an equivalence $\mathrm{Pro}(\mathcal{C}) \simeq \mathrm{Ind}(\mathcal{C}^{\mathrm{op}})^{\mathrm{op}}$ with the Ind-completion of the opposite category. An object $c$ gives the constant pro-object indexed by the one-point category, and this gives a fully faithful functor $\mathcal{C} \to \mathrm{Pro}(\mathcal{C})$ . Every pro-object is the limit in $\mathrm{Pro}(\mathcal{C})$ of its stages.

Example: in the category of finite groups, the groups $\Z / p^n \Z$ for $n \ge 1$ , with the reduction maps, form a pro-object. Its index category is the cofiltered poset $\N$ with an arrow $m \to n$ whenever $m \ge n$ . More generally, every profinite group is the limit of the pro-object of its finite quotients.

Non-example: a functor from the discrete category with two objects to $\mathcal{C}$ is not a pro-object. The discrete category with two objects is not cofiltered, since its two objects have no common object mapping to both.

Reading path

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Filtered category concept
Ind-completion concept
Limit concept
Opposite category concept
Profinite group concept
Pro-object definition

Tag	`PQRJ30QTYD`
URI	`m2://PQRJ30QTYD`
Kind	definition
Writing	W0 stub
MSC	18A30
Defines	Pro-object
Uses	Ind-completion, Opposite category, Filtered category, Limit, Profinite group

Pro-object

Definition

Remarks

Reading path

Further reading

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