TeXipedia

injlim

Represents the injective limit (inductive limit) in category theory and advanced mathematics.

Overview

Serves as a specialized mathematical operator primarily used in advanced algebra and category theory to denote the injective (direct) limit of a directed system.

  • Essential in homological algebra for describing direct limit constructions
  • Commonly appears in research papers and advanced textbooks on category theory
  • Often used alongside projective limits in discussions of categorical limits and colimits
  • Particularly relevant when working with directed systems of modules or algebraic structures

Examples

Expressing the injective limit of a directed system of modules.

inj lim⁡n→∞Mn\injlim_{n \to \infty} M_n 
\injlim_{n \to \infty} M_n

Showing the injective limit in a sequence of homomorphisms.

inj lim⁡i∈IAi≅B\injlim_{i \in I} A_i \cong B 
\injlim_{i \in I} A_i \cong B

Representing the injective limit of a family of abelian groups.

Hn(X)≅inj lim⁡U⊃KHn(U)H^n(X) \cong \injlim_{U \supset K} H^n(U) 
H^n(X) \cong \injlim_{U \supset K} H^n(U)