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.

Definition

Source and sink

A source $(X,(Y_i{)}_{i\in I},(f_i:X\to Y_i{)}_{i\in I})$ in a category ${ℰ}$ consists of the following data:

• an object $X\in {ℰ}$,
• a (possibly proper) class of objects $(Y_i{)}_{i\in I}\subseteq {ℰ}$
• and a class of morphisms $(f_i:X\to Y_i{)}_{i\in I}$.

Dually, a sink $(X,(Y_i{)}_{i\in I},(f_i:Y_i\to X{)}_{i\in I})$ in ${ℰ}$ consists of

• an object $X\in {ℰ}$,
• a class of objects $(Y_i{)}_{i\in I}\subseteq {ℰ}$
• and a class of morphisms $(f_i:Y_i\to X{)}_{i\in I}$.

In particular, a source $(f_i:X\to Y_i{)}_{i\in I}$ 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 $(f_i:X\to Y_i{)}_{i\in I}$ be a source in a category ${ℰ}$ and let $\mathrm{\Pi }:{ℰ}\to {ℬ}$ be a functor. The source $(f_i{)}_{i\in I}$ is said to be a $\mathrm{\Pi }$-initial source if it satisfies the following universal property.

• For every object $X^{\prime }\in {ℰ}$, a morphism $\hat{g}:\mathrm{\Pi }(X^{\prime })\to \mathrm{\Pi }(X)$ and a family of morphisms $(f{\text{'}}_i:X^{\prime }\to Y_i{)}_{i\in I}$ such that $\mathrm{\Pi }(f_i)\circ \hat{g}=\mathrm{\Pi }(f{\text{'}}_i)$ for each $i\in I$, there exists a unique ${ℰ}$-morphism $g:X^{\prime }\to X$ such that $\hat{g}=\mathrm{\Pi }(g)$ and $\mathrm{\forall }i\in I:f_i\circ g=f{\text{'}}_i$.

  $\begin{array}{ccc}{ℰ}& \stackrel{\mathrm{\Pi }}{\to }& {ℬ}\\ \begin{array}{c}{X}^{\prime }\\ \mathrm{\exists }!g\downarrow \mathrm{\exists }!g& \searrow {}^{f{\text{'}}_i}\\ X& \underset{f_i}{\to }& Y_i\end{array}& \stackrel{\mathrm{\Pi }}{\mapsto }& \begin{array}{c}\mathrm{\Pi }{X}^{\prime }\\ \hat{g}\downarrow \hat{g}& \searrow {}^{\mathrm{\Pi }f{\text{'}}_i}\\ \mathrm{\Pi }X& \underset{\mathrm{\Pi }{f}_i}{\to }& \mathrm{\Pi }{Y}_i\end{array}\end{array}$

Similarly one defines the dual notion of $\mathrm{\Pi }$-final sink. When $I$ is a set of a single element, the initial source is called a Cartesian morphism.

Lift

Let ${ℰ}$, ${ℬ}$ be two categories. Let $\mathrm{\Pi }:{ℰ}\to {ℬ}$ be a functor. A source $({\hat{f}}_i:\hat{X}\to {\hat{Y}}_i{)}_{i\in I}$ in ${ℬ}$ is a $\mathrm{\Pi }$-structured source if for each $i$ we have ${\hat{Y}}_i=\mathrm{\Pi }(Y_i)$ for some $Y_i\in {ℰ}$. One similarly defines a $\mathrm{\Pi }$-structured sink. A lift of a $\mathrm{\Pi }$-structured source $({\hat{f}}_i:\hat{X}\to \mathrm{\Pi }(Y_i){)}_{i\in I}$ is a source $(f_i:\hat{X}\to Y_i{)}_{i\in I}$ in ${ℰ}$ such that $\mathrm{\Pi }(X)=\hat{X}$ and $\mathrm{\Pi }(f_i)={\hat{f}}_i$ for each $i\in I$

$\begin{array}{ccc}{ℰ}& \stackrel{\mathrm{\Pi }}{\to }& {ℬ}\\ \begin{array}{c}\mathrm{\exists }X\\ \mathrm{\exists }{f}_i\downarrow \mathrm{\exists }{f}_i\\ Y_i\end{array}& \stackrel{\mathrm{\Pi }}{\mapsto }& \begin{array}{c}\hat{X}\\ {\hat{f}}_i\downarrow {\hat{f}}_i\\ \mathrm{\Pi }{Y}_i\end{array}\end{array}$

A lift of a $\mathrm{\Pi }$-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 $\mathrm{\Pi }$-structured source $(\hat{X}\to \mathrm{\Pi }(Y_i){)}_{i\in I}$ has an initial lift $(X\to Y_i{)}_{i\in I}$, we say that $X$ is an initial ${ℰ}$-structure on $\hat{X}$ with respect to $(\hat{X}\to \mathrm{\Pi }(Y_i){)}_{i\in I}$. Similarly for a final ${ℰ}$-structure with respect to a $\mathrm{\Pi }$-structured sink.

Topological functor

Let $\mathrm{\Pi }:{ℰ}\to {ℬ}$ be a functor. Then the following two conditions are equivalent.

• Every $\mathrm{\Pi }$-structured source has an initial lift. That is, an initial structure always exists.
• Every $\mathrm{\Pi }$-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 $({ℰ},F)$ is called a topological (concrete) category if the forgetful functor $F:{ℰ}\to \mathrm{Set}$ is topological. (A topological category can also mean an enriched category enriced over the category $\mathrm{Top}$ of topological spaces.) Some require a topological category to satisfy two additional conditions.

• Constant functions in $\mathbf{S}\mathbf{e}\mathbf{t}$ lift to ${ℰ}$-morphisms.
• Fibers ${\mathrm{\Pi }}^{-1}(\hat{X})$ ($\hat{X}\in \mathbf{S}\mathbf{e}\mathbf{t}$) are small (they are sets and not proper classes).

Properties

Every topological functor is faithful. Let $\mathsf{P}$ be one of the following four properties of categories:

• complete category
• cocomplete category
• well-powered category
• co-well-powered category

If $\mathrm{\Pi }:{ℰ}\to {ℬ}$ is topological and ${ℬ}$ has property $\mathsf{P}$, then ${ℰ}$ also has property $\mathsf{P}$. Let ${ℰ}$ be a category. Then the topological functors ${ℰ}\to \mathrm{Set}$ are unique up to natural isomorphism.

Examples

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