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  "body": "\n## Summary\n\nA continuous group homomorphism is a map between [[topological groups|92700QAB5Y]] that respects the group law and is continuous.\nThese maps are the morphisms of the category of topological groups.\nBecause translations are homeomorphisms, a homomorphism is continuous as soon as it is continuous at the identity.\nCharacters of locally compact abelian groups and finite-dimensional representations of Lie groups are continuous homomorphisms.\n\n## Explanations\n\n### L1 human\n\nA group homomorphism turns products into products.\nIn topological groups one also wants the map to send nearby elements to nearby elements.\nFor example, the map $t \\mapsto e^{2\\pi i t}$ wraps the real line around the circle.\nIt turns sums of real numbers into products of points on the circle, and it is continuous, so it is a continuous group homomorphism.\n",
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