A retraction of a topological space is an idempotent continuous map from the topological space to itself. In other words, it is a continuous map from the topological space to a subspace, such that the restriction to that subspace is the identity map.
The image of a retraction is termed a retract (this is a subspace property).
Definition with symbols
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Given any functor from the category of topological spaces to the category of groups, or the category of modules over a ring, the topological space notion of retract gets mapped to the notion of retract in the relevant category.
Let be a functor from the category of topological spaces to the category of groups.
Suppose is a topological space, is a retraction and is the image of the retraction. Suppose is the inclusion of in . Since is the identity map on , we get by functorality that is the identity on . This forces that is injective, and is surjective. Identifying with its image subgroup , we can view as a retraction of groups.