Categorical Description
In the language of category theory, the initial topology construction can be described as follows. Let Y be the functor from a discrete category J to the category of topological spaces Top which selects the spaces Yj for j in J. Let U be the usual forgetful functor from Top to Set. The maps {fj} can then be thought of as a cone from X to UY. That is, (X, f) is an object of Cone(UY)—the category of cones to UY.
The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from the forgetful functor
- U′ : Cone(Y) → Cone(UY)
to the cone (X, f). By placing the initial topology on X we therefore obtain a functor
- I : Cone(UY) → Cone(Y)
which is right adjoint to the forgetful functor U′. In fact, I is a right-inverse to U′ since U′I is the identity functor on Cone(UY).
Read more about this topic: Initial Topology
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