TeXipedia

varinjlim

Represents a direct limit (also known as an inductive or injective limit) in category theory and abstract algebra.

Overview

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

  • Used when working with directed systems of objects and morphisms
  • Frequently appears in the study of filtered colimits and direct systems
  • Important in constructing algebraic objects like direct limits of modules or groups
  • Often paired with morphisms and diagrams in category-theoretic contexts
  • Provides a more visually distinct alternative to standard lim notation for direct limits

Examples

Direct limit of an increasing sequence of vector spaces.

V=lim→⁡n→∞VnV = \varinjlim_{n \to \infty} V_n 
V = \varinjlim_{n \to \infty} V_n

Direct limit notation in a commutative diagram showing morphisms.

lim→⁡Fn→ϕlim→⁡Gn\varinjlim F_n \xrightarrow{\phi} \varinjlim G_n 
\varinjlim F_n \xrightarrow{\phi} \varinjlim G_n

Direct limit of a system of groups indexed by natural numbers.

G=lim→⁡n∈NGnG = \varinjlim_{n \in \mathbb{N}} G_n 
G = \varinjlim_{n \in \mathbb{N}} G_n