Quantalification

Extending Functors to Quantale Categories

For now, assuming the quantale to be commutative, we refer to Adriana Balan ; Alexander Kurz ; Jiří Velebil - Extending set functors to generalised metric spaces for details. We only extract some information for reference.

Examples

Constant Functor

Let \(T:\Set\to\Set\), be a constant functor at a set \(S\). Then \(\overline T\) acts as follows: For any \(\Qu\)-category \(X\), \(\overline TX\) is the constant \(\QuCat\)-functor to the \(\Qu\)-category with \(S\) as set of objects, with \(\Qu\)-distances

\[\begin{split} \overline TX(x',x) = \begin{cases} \, \top \ \ , \ \ x'=x \\ \bot \ \ , \ \ \mbox{otherwise} \end{cases} \end{split}\]

That is, \(\overline TX = S \cdot \One_\top\) is the coproduct in \(\QuCat\) of \(S\) copies of the terminal \(\Qu\)-category \(\One_\top\). In case \(\Qu\) is integral, we have \(\overline T X=D S\).

Power

Let \(T:\Set\to\Set\) be the functor \(TX= X^n\), for \(n\) a natural number. Then \(\overline T\) maps a quantale category \(X\) to its \(n\)-th power \(X^n\), where

\[ \begin{equation} X^n((x'_0,\dots,x'_{n-1}),(x_0,\dots,x_{n-1})) = X(x'_0,x_0)\wedge\dots\wedge X(x'_{n-1},x_{n-1}). \end{equation} \]

If \(n\) is an {\em arbitrary/} cardinal number, the quantalification of \(T:\Set\to \Set\), \(TX=X^n\) also exists and \(\overline TX((x'_i),(x_i))=\bigsqcap_i X (x'_i,x_i)\). That is, \(\overline TX=X^n\).

Polynomial Functor

The quantalification of a finitary polynomial functor \( X\mapsto \coprod_n X^n\coprod \Sigma n \) is the ``strongly polynomial’’ \(\Qu\)-functor \( X\mapsto \coprod_n X^n \otimes D\Sigma n \), where \(n\) ranges over finite sets.

In particular, the quantalification of the list functor \(LX = X^* = \coprod_n X^n\) maps a \(\Qu\)-category \(X\) to the \(\Qu\)-category having as objects tuples of objects of \(X\), with non-trivial \(\Qu\)-distances only between tuples of same order

Powerset Functor

…

Distribution Functor

…

References

…