varprojlim

Denotes a projective limit (inverse limit) in category theory and advanced algebra.

Overview

Essential notation in advanced mathematics for describing the limit of an inverse system, particularly common in algebraic topology and homological algebra.

Used when working with inverse systems of algebraic objects like groups, rings, or topological spaces
Frequently appears alongside direct limits (colimits) in categorical constructions
Important in studying pro-finite groups and completion theory
Provides a way to construct new mathematical objects as limits of inverse systems of simpler objects

Examples

Expressing the projective limit of an inverse system of groups.

G = \varprojlim_{n \in \mathbb{N}} G_n

G = \varprojlim_{n \in \mathbb{N}} G_n

Defining the p-adic integers as a projective limit.

\mathbb{Z}_p = \varprojlim_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z}

\mathbb{Z}_p = \varprojlim_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z}

Representing the inverse limit of a sequence of rings.

R = \varprojlim_{k} R/I^k

R = \varprojlim_{k} R/I^k