# Categorification of the (co-)induced topology

In second semester analysis we learned about the product topology which is quite easy to categorify using limits. However, we also learned about the coinduced topology $\mathfrak{V}$ induced by $f: X → Y$ and $(X, \mathfrak{U})$ on $Y$. It is the strongest topology on Y such that $f: (X, \mathfrak{U}) → (Y, \mathfrak{V})$ is continuous.

I would love to to express this coinduced topology using the categories Top and Set. Sadly I can't realy figure out how to do it as I have few experience with Category Theory.

Could someone please explain this to me?

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This is an example of a final topology. –  Chris Eagle Apr 27 '12 at 8:22

The final topology $\mathfrak{V}$ induced by a map $f : X \to Y$ satisfies the following universal property: for all spaces $(Z, \mathfrak{W})$ and all continuous maps $h : (X, \mathfrak{U}) \to (Z, \mathfrak{V})$, if $h : X \to Z$ factors through $f : X \to Y$ for some $g : Y \to Z$, then the map $g : (Y, \mathfrak{V}) \to (Z, \mathfrak{W})$ is continuous.

Another way to think of the final topology is as a colimit. If I'm not mistaken, it is the colimit of a diagram induced by the coslice category $(f \downarrow ((X, \mathfrak{U}) \downarrow \textbf{Top}))$, but I don't think this is helpful...

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those coslices and colimits might not sound helpful, but a lot more funny. I'll look at those and the link provided by Chris Eagle and work out the rest myself. –  Rolf Sievers Apr 27 '12 at 18:29