Editing Initial topology (section)

==Categorical description==

In the language of [[category theory]], the initial topology construction can be described as follows. Let <math>Y</math> be the [[functor]] from a [[discrete category]] <math>J</math> to the [[category of topological spaces]] <math>\mathrm{Top}</math> which maps <math>j\mapsto Y_j</math>. Let <math>U</math> be the usual [[forgetful functor]] from <math>\mathrm{Top}</math> to <math>\mathrm{Set}</math>. The maps <math>f_j : X \to Y_j</math> can then be thought of as a [[cone (category theory)|cone]] from <math>X</math> to <math>UY.</math> That is, <math>(X,f)</math> is an object of <math>\mathrm{Cone}(UY) := (\Delta\downarrow{UY})</math>&mdash;the [[category of cones]] to <math>UY.</math> More precisely, this cone <math>(X,f)</math> defines a <math>U</math>-structured cosink in <math>\mathrm{Set}.</math>

The forgetful functor <math>U : \mathrm{Top} \to \mathrm{Set}</math> induces a functor <math>\bar{U} : \mathrm{Cone}(Y) \to \mathrm{Cone}(UY)</math>. The characteristic property of the initial topology is equivalent to the statement that there exists a [[universal morphism]] from <math>\bar{U}</math> to <math>(X,f);</math> that is, a [[terminal object]] in the category <math>\left(\bar{U}\downarrow(X,f)\right).</math><br/>
Explicitly, this consists of an object <math>I(X,f)</math> in <math>\mathrm{Cone}(Y)</math> together with a morphism <math>\varepsilon : \bar{U} I(X,f) \to (X,f)</math> such that for any object <math>(Z,g)</math> in <math>\mathrm{Cone}(Y)</math> and morphism <math>\varphi : \bar{U}(Z,g) \to (X,f)</math> there exists a unique morphism <math>\zeta : (Z,g) \to I(X,f)</math> such that the following diagram commutes:
[[File:UniversalPropInitialTop.jpg|300px|center]]
The assignment <math>(X,f) \mapsto I(X,f)</math> placing the initial topology on <math>X</math> extends to a functor
<math>I : \mathrm{Cone}(UY) \to \mathrm{Cone}(Y)</math>
which is [[adjoint functor|right adjoint]] to the forgetful functor <math>\bar{U}.</math> In fact, <math>I</math> is a right-inverse to <math>\bar{U}</math>; since <math>\bar{U}I</math> is the identity functor on <math>\mathrm{Cone}(UY).</math>