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Stream: learning: questions

Topic: weighted colimits in cats enriched over weighted sets

fosco (Feb 04 2022 at 18:40):

Let $Q$ be a quantale; define the category ${\bf wSet}[Q]$ of weighted sets over $Q$ to be the category where

• objects are functions $u : X \to Q$
• morphisms $(X,u)\to (Y,v)$ are those functions $f : X \to Y$ that are "nonexpansive", i.e. such that (in $Q$) one has $v(f(x)) \le u(x)$ for every $x\in X$.

Usually one works in the following particular case: $Q$ is the poset $[0,\infty]^\text{op}$ and the quantale operation is the sum. In such case, $u : X \to [0,\infty]$ is called a norm on $X$ and denoted with vertical bars $|x|_X$. So, a nonexpansive map is a function $f : X\to Y$ as above, such that $|fx|_Y\le |x|_X$.

One can prove that the category of weighted sets is (symmetric) monoidal closed: regarded as a category, the quantale is closed, and one can use the internal hom $p\Rightarrow q$ in $Q$ to define an internal hom in ${\bf wSet}[Q]$. In the particular case above, $X\times Y$ becomes a weighted set with norm $(x,y)\mapsto |x|+|y|$; the internal hom between $(X,u)$ and $(Y,v)$ is the set of all functions $Y^X$, with norm given by the sup over $x\in X$ of the set of real numbers $|fx| \dot- |x|$ (the sum is truncated at zero: so, note that for the $f$'s that are morphisms of weighted sets, this is actually 0; the norm of $f$ measures how far is $f$ from being nonexpansive).

So, since it is monoidal closed, we can enrich over ${\bf wSet}[Q]$! We can consider the 2-category ${\bf wSet}\textsf{-Cat}$ and do something with it. My problem is the following: I want to compute interesting weighted limits and colimits in ${\bf wSet}\textsf{-Cat}$, motivated by the observation that tensors are kinda interesting constructions.

(I denote just as ${\bf wSet}$ the category of sets weighted over $[0,\infty]^\text{op}$ or, equivalently up to quantale iso, $[0,1]$).

Let $A$ be a set; a category with arbitrary coproducts $\cal C$ is tensored over $\sf Set$ by the assignment $X\mapsto A\odot X := \coprod_{a\in A} X$. It seems to me that for a $\bf wSet$-category $\cal C$, being tensored over ${\bf wSet}$ means that we are building a tensor $\coprod_a X$ where each repeated summand $X$ "weighs" differently, according to the map $\lambda : A\to [0,1]$ (that can be thought of as assigning a probability $\lambda_a \in [0,1]$ to $a\in A$).

In more concrete terms, let's say we are given a set $A$ of wires in parallel and a switch, let's say in an electrical circuit, that chooses port $a$ with a certain probability $\lambda_a$: this intuition is motivated by the fact that if one tries to cast the universal property of the tensor, in order for the hom-set ${\cal C}(A\odot X, Y)$ to be isomorphic (as a weighted set) to the set ${\bf wSet}(A, {\cal C}(X,Y))$, a morphism $A\odot X\to Y$ is a function from the coproduct $\sum_a X$ to $Y$, i.e. a family of functions $f_a : X \to Y$, but "decorated" with the additional information that $|f_a|$ cannot exceed $\lambda_a$ (so it's an upper bound for the probability that, at the switching point, the summand $f_a$ will be chosen to perform the "operation" $X\to Y$).

fosco (Feb 04 2022 at 18:41):

Motivated by this rather enticing intuition, I have tried to find other, more elaborate shapes of weights $W : {\cal J} \to {\bf wSet}$ for which one can compute the weighted co/limit of a diagram $F : {\cal J} \to {\cal C}$ of ${\bf wSet}$-enriched categories.

Unfortunately, I have no idea what kind of $\cal J$ are interesting: contrary to the case where the base of enrichment is $\sf Cat$, there's little information on how to "generate" weights that carry interesting information, and the few attempts I made used very simple shapes that carried little interest.

fosco (Feb 04 2022 at 18:42):

(this question stemmed from @Paolo Perrone 's talk on "The rise of quantitative category theory")

Morgan Rogers (he/him) (Feb 04 2022 at 19:45):

So is the question whether anyone knows what shapes you should look at?

fosco (Feb 04 2022 at 20:01):

Yes, the question is left open on purpose: what are interesting shapes of weights one might want to compute?

Paolo Perrone (Feb 06 2022 at 10:18):

Not much with a meaning of "probability", but rather, "distance", one may want to compute a sequential colimit where the weights are a sequence that converges to zero, a bit as in my example in the talk (Of course one does not need weighted categories for that, Lawvere metric spaces are enough for example.)