---
tag: 9J5PQMG3HJ
kind: concept
name: Continuous group homomorphism
msc: [22A05]
specializes: [0JYTG5K9Y4]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Summary

A continuous group homomorphism is a map between [[topological groups|92700QAB5Y]] that respects the group law and is continuous.
These maps are the morphisms of the category of topological groups.
Because translations are homeomorphisms, a homomorphism is continuous as soon as it is continuous at the identity.
Characters of locally compact abelian groups and finite-dimensional representations of Lie groups are continuous homomorphisms.

## Explanations

### L1 human

A group homomorphism turns products into products.
In topological groups one also wants the map to send nearby elements to nearby elements.
For example, the map $t \mapsto e^{2\pi i t}$ wraps the real line around the circle.
It turns sums of real numbers into products of points on the circle, and it is continuous, so it is a continuous group homomorphism.