Topological construct

Question

I just started working with some category theory and I would like to understand the link between what I am studying now and what I know about topological spaces.

By definition, a construct (in our case, ($\textbf{Top}$,$U$) with U the forgetful functor) is topological if ever $U$-structured source has a unique initial lift.

How does this translate in Topology terminology? These are the definition I am using the my thoughts

A $U$-structured source, for the forgetful functor, is just a source in $\textbf{Set}$ of the form ($f_i : X \rightarrow UA_i)_{i \in I}$ and an initial lift is an initial source $(\bar f_i : A \rightarrow A_i)_{i \in I}$ in $\textbf{Top}$ such that $U \bar f_i = f_i$ and $UA=X$. In $\textbf{Top}$, the initial source has to be equipped with the initial topology to be an initial source.

But what does it mean to have an unique lift? Is there another way to show that $\textbf{Top}$ is a topological construct?

Thanks everyone!

Answer 1

It seems to be a simple consequence of the fact that $U\colon \bf Top\to Set$ admits a left adjoint $F$ which sends a set $X$ in the discrete topological space $FX$ (it must be what you mean by "initial topology").

So a simple way to characterize the lift of $f_i\colon X\to UA_i$ is via the bijection $$ {\bf Set}(X, UA)\cong {\bf Top}(FX,A) $$ sending $f_i$ to $\tilde{f}_i\colon FX\xrightarrow{Ff_i}FUA_i\xrightarrow{\epsilon_{A_i}}A_i$.

Both functors on the adjunction $F\dashv U$ act like the identity on the objects, so it's evident that $UFX=X$ and that $U\tilde f_i$ can be identified with $f_i$.

But what does it mean to have an unique lift?

I think that the above argument can be exported verbatim to a similar context (this is one of the best features of Category Theory :) ). You need a pair of adjoint functors $F\dashv U$ such that...