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My professor recently said that the final topology induced by a family of functions can be thought of as a colimit.

Note that I am aware of the normal categorical description of the final topology as an initial morphism in a coslice category $((X,f)\downarrow U')$ for the forgetful functor $U':Cone(Y)\rightarrow Cone(UY)$, where $Y:\mathcal{J}\rightarrow \mathbf{Top}$ is a diagram picking out the family of spaces $(Y_i)_{i\in \mathcal{J}}$ and $U:\mathbf{Top}\rightarrow \mathbf{Set}$ is the usual forgetful functor.

The specific example we were working with is following: let $X$ be a topological space, and take as the family of functions all continuous functions $\varphi:V\rightarrow X$ where $V$ is a open subspace of $\mathbb{R}^n$ for some $n$. We want the final topology induced by this family of functions, which produces a finer topology whose open sets are exactly those sets whose pre-images are open under every such map $\varphi$ above.

He quoted the following diagram as that which the final topology is a colimit of: $(\mathbf{Test}\downarrow X)\rightarrow (\mathbf{Top}\downarrow X)\rightarrow \mathbf{Top}$, where $\mathbf{Test}$ is the full subcategory of $\mathbf{Top}$ with objects as open subspaces of Euclidean spaces and morphisms continuous functions. The notation $(\mathcal{C}\downarrow X)$ is the slice category.

My questions are as follows:

  1. What exactly are the functors between these categories creating the functor? The first I would imagine is just an inclusion functor, but I'm blanking on the second.
  2. Is this expandable to any final topology? At least as the above goes, the family of spaces may not be a category, which could bring up issues.

Additionally, in the answer to this question: Categorification of the (co-)induced topology the author writes the the final topology can be seen as a colimit of the coslice category $(f\downarrow (X\downarrow \mathbf{Top}))$. Can this be extended in such a way that instead of the final topology being induced by just $f$, we instead have it by a family of functions? I would expect that just replacing the constant functor here with a functor from a discrete category to the family would work, but I'm a little uncertain.

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All initial objects are colimits of course (of empty diagrams), but that's a trivial remark. –  Marc Feb 18 at 2:21
The second map is a forgetful functor, mapping the function in the slice category to the domain of the function. See ncatlab.org/nlab/show/overcategory . –  Marc Feb 18 at 2:24

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