# Topological functor

In category theory and general topology, a topological functor is one which has similar properties to the forgetful functor from the category of topological spaces. The domain of a topological functor admits construction similar to initial topology (and equivalently the final topology) of a family of functions. The notion of topological functors generalizes (and strengthens) that of fibered categories, for which one considers a single morphism instead of a family.$^{:407,§1}$

## Definition
### Source and sink

A source $(X,(Y_{i})_{i\in I},(f_{i}\colon X\to Y_{i})_{i\in I})$ in a category ${\mathcal {E}}$ consists of the following data:$^{:125,Definition1.1(1)}$
* an object $X\in {\mathcal {E}}$,
* a (possibly proper) class of objects $(Y_{i})_{i\in I}\subseteq {\mathcal {E}}$
* and a class of morphisms $(f_{i}\colon X\to Y_{i})_{i\in I}$.
Dually, a sink $(X,(Y_{i})_{i\in I},(f_{i}\colon Y_{i}\to X)_{i\in I})$ in ${\mathcal {E}}$ consists of
* an object $X\in {\mathcal {E}}$,
* a class of objects $(Y_{i})_{i\in I}\subseteq {\mathcal {E}}$
* and a class of morphisms $(f_{i}\colon Y_{i}\to X)_{i\in I}$.
In particular, a source $(f_{i}\colon X\to Y_{i})_{i\in I}$ is an object $X$ if $I$ is empty, a morphism $X\to Y$ if $I$ is a set of a single element. Similarly for a sink.

### Initial source and final sink

Let $(f_{i}\colon X\to Y_{i})_{i\in I}$ be a source in a category ${\mathcal {E}}$ and let $\Pi \colon {\mathcal {E}}\to {\mathcal {B}}$ be a functor. The source $(f_{i})_{i\in I}$ is said to be a $\Pi$-initial source if it satisfies the following universal property.$^{:Definition2.1(1)}$
* For every object $X'\in {\mathcal {E}}$, a morphism ${\hat {g}}\colon \Pi (X')\to \Pi (X)$ and a family of morphisms $(f'_{i}\colon X'\to Y_{i})_{i\in I}$ such that $\Pi (f_{i})\circ {\hat {g}}=\Pi (f'_{i})$ for each $i\in I$, there exists a unique ${\mathcal {E}}$-morphism $g\colon X'\to X$ such that ${\hat {g}}=\Pi (g)$ and $\forall i\in I\colon f_{i}\circ g=f'_{i}$.
    ${\begin{matrix}{\mathcal {E}}&\qquad {\overset {\Pi }{\to }}\qquad &{\mathcal {B}}\\\hline {\begin{matrix}X'\\{\scriptstyle \exists !g}\downarrow {\color {White}\scriptstyle \exists !g}&\searrow \!\!^{f'_{i}}\!\!\!\!\!\!\\X&{\underset {f_{i}}{\to }}&Y_{i}\end{matrix}}&\qquad {\overset {\Pi }{\mapsto }}\qquad &{\begin{matrix}\Pi X'\\{\scriptstyle {\hat {g}}}\downarrow {\color {White}\scriptstyle {\hat {g}}}&\searrow \!\!^{\Pi f'_{i}}\!\!\!\!\!\!\\\Pi X&{\underset {\Pi f_{i}}{\to }}&\Pi Y_{i}\end{matrix}}\end{matrix}}$
Similarly one defines the dual notion of $\Pi$-final sink.
When $I$ is a set of a single element, the initial source is called a Cartesian morphism.

### Lift

Let ${\mathcal {E}}$, ${\mathcal {B}}$ be two categories. Let $\Pi \colon {\mathcal {E}}\to {\mathcal {B}}$ be a functor. A source $({\hat {f}}_{i}\colon {\hat {X}}\to {\hat {Y}}_{i})_{i\in I}$ in ${\mathcal {B}}$ is a $\Pi$-structured source if for each $i$ we have ${\hat {Y}}_{i}=\Pi (Y_{i})$ for some $Y_{i}\in {\mathcal {E}}$.$^{:128,Definition1.1(2)}$ One similarly defines a $\Pi$-structured sink.
A lift of a $\Pi$-structured source $({\hat {f}}_{i}\colon {\hat {X}}\to \Pi (Y_{i}))_{i\in I}$ is a source $(f_{i}\colon {\hat {X}}\to Y_{i})_{i\in I}$ in ${\mathcal {E}}$ such that $\Pi (X)={\hat {X}}$ and $\Pi (f_{i})={\hat {f}}_{i}$ for each $i\in I$
    ${\begin{matrix}{\mathcal {E}}&\qquad {\overset {\Pi }{\to }}\qquad &{\mathcal {B}}\\\hline {\begin{matrix}\exists X\\{\scriptstyle \exists f_{i}}\downarrow {\color {White}\scriptstyle \exists f_{i}}\\Y_{i}\end{matrix}}&\qquad {\overset {\Pi }{\mapsto }}\qquad &{\begin{matrix}{\hat {X}}\\{\scriptstyle {\hat {f}}_{i}}\downarrow {\color {White}\scriptstyle {\hat {f}}_{i}}\\\Pi Y_{i}\end{matrix}}\end{matrix}}$
A lift of a $\Pi$-structured sink is similarly defined. Since initial and final lifts are defined via universal properties, they are unique up to a unique isomorphism, if they exist.
If a $\Pi$-structured source $({\hat {X}}\to \Pi (Y_{i}))_{i\in I}$ has an initial lift $(X\to Y_{i})_{i\in I}$, we say that $X$ is an initial ${\mathcal {E}}$-structure on ${\hat {X}}$ with respect to $({\hat {X}}\to \Pi (Y_{i}))_{i\in I}$. Similarly for a final ${\mathcal {E}}$-structure with respect to a $\Pi$-structured sink.

### Topological functor

Let $\Pi \colon {\mathcal {E}}\to {\mathcal {B}}$ be a functor. Then the following two conditions are equivalent.$^{:128,Definition2.1(3)}$$^{:29–30,§2}$$^{:2,Example2.1(25)}$$^{:4,Definition2.12}$
* Every $\Pi$-structured source has an initial lift. That is, an initial structure always exists.
* Every $\Pi$-structured sink has a final lift. That is, a final structure always exists.
A functor satisfying this condition is called a topological functor.
One can define topological functors in a different way, using the theory of enriched categories.
A concrete category $({\mathcal {E}},F)$ is called a topological (concrete) category if the forgetful functor $F\colon {\mathcal {E}}\to \operatorname {Set}$ is topological. (A topological category can also mean an enriched category enriced over the category $\operatorname {Top}$ of topological spaces.) Some require a topological category to satisfy two additional conditions.
* Constant functions in $\mathbf {Set}$ lift to ${\mathcal {E}}$-morphisms.
* Fibers $\Pi ^{-1}({\hat {X}})$ (${\hat {X}}\in \mathbf {Set}$) are small (they are sets and not proper classes).

## Properties

Every topological functor is faithful.$^{:129,Theorem3.1}$
Let ${\mathsf {P}}$ be one of the following four properties of categories:
* complete category
* cocomplete category
* well-powered category
* co-well-powered category
If $\Pi \colon {\mathcal {E}}\to {\mathcal {B}}$ is topological and ${\mathcal {B}}$ has property ${\mathsf {P}}$, then ${\mathcal {E}}$ also has property ${\mathsf {P}}$.
Let ${\mathcal {E}}$ be a category. Then the topological functors ${\mathcal {E}}\to \operatorname {Set}$ are unique up to natural isomorphism.$^{:6,Corollary2.2}$

## Examples

An example of a topological category is the category of all topological spaces with continuous maps, where one uses the standard forgetful functor.