Weighted Limit

definition theorem example

Ordinary cones pick single “wires” ${\gamma }_j:1\to \mathcal{C}(x,Dj)$ from hom-sets; the set of cones with apex $x$ is $[\mathcal{J},\mathbf{Set}]({\Delta }_1,\mathcal{C}(x,D-))$. In an enriched setting there is no constant $\mathcal{V}$-functor ${\Delta }_1$ (the unit $I$ need not be terminal), so one “smears the singularity” with a weight $W:\mathcal{J}\to \mathcal{V}$ selecting a thicker “cylinder” in each hom-object. A weighted (indexed) limit ${\lim }^{W}D$ of $D:\mathcal{J}\to \mathcal{C}$ is defined by

$$\mathcal{C}(x,\lim {}^{W}D)\cong [\mathcal{J},\mathcal{V}](W,\mathcal{C}(x,D-)),$$

and dually the weighted colimit by $\mathcal{C}({\mathrm{colim}}^{W}D,x)\cong [{\mathcal{J}}^{\mathrm{op}},\mathcal{V}](W,\mathcal{C}(D-,x))$ with $W:{\mathcal{J}}^{\mathrm{op}}\to \mathcal{V}$. Ordinary (conical) limits are the case $W={\Delta }_1$.

Sources: DaoFP §20.5 (“Weighted Limits”), §20.6 (“Ends as Weighted Limits”), §20.7 (“Kan Extensions”), §20.8 (“Useful Formulas”), Exercises 20.6.1, 20.6.2, 20.7.1; §20.4 (enriched Yoneda).

• Formulas (DaoFP Exercise 20.6.2): ${\lim }^{W}D\cong {\int }_{j}Wj⋔Dj$ and ${\mathrm{colim}}^{W}D\cong {\int }^{j}Wj\cdot Dj$ (power and copower, Kan Extension).
• Ends as weighted limits (§20.6): for $P:{\mathcal{C}}^{\mathrm{op}}\otimes \mathcal{C}\to \mathcal{D}$, ${\int }_{c}P⟨c,c⟩={\lim }^{{\mathrm{Hom}}_{\mathcal{C}}}P$ with the hom-functor as weight; proof by the Yoneda trick, Fubini and ninja Yoneda over $c^{\prime }$. Dually ${\int }^{c}P⟨c,c⟩={\mathrm{colim}}^{{\mathrm{Hom}}_{{\mathcal{C}}^{\mathrm{op}}}}P$ (DaoFP Exercise 20.6.1). This defines ends and coends in the enriched setting.
• Kan extensions as weighted (co)limits (§20.7): $({\mathrm{Ran}}_{P}F)e={\lim }^{\mathcal{B}(e,P-)}F$ and $({\mathrm{Lan}}_{P}F)e={\mathrm{colim}}^{\mathcal{B}(P-,e)}F$ (DaoFP Exercise 20.7.1) — the weights are representables, replacing the functor from the terminal category $1$ used for ordinary (co)limits.
• Enriched Yoneda (§20.4): weak form $\mathcal{V}\text{-nat}(\mathcal{C}(c,-),F)\cong \mathcal{V}(I,Fc)$ (a set of $\mathcal{V}$-natural transformations vs. the global elements of $Fc$); strong form ${\int }_x[\mathcal{C}(c,x),Fx]\cong Fc$ as an object of $\mathcal{V}$, using the internal hom.

JuliaLeanHaskell

Docs: FinSets · Limits & colimits

using Catlab
# a weighted limit in Set with a discrete shape J = {1, 2}: lim^W D = ∏_j (W j ⋔ D j) = ∏_j D j^{W j}
D = [FinSet(2), FinSet(3)]; W = [2, 1]            # weights = sizes of W j
prod(length(D[j])^W[j] for j in 1:2)              # 2^2 · 3^1 = 12 elements
# conical limit (W = Δ1) is the plain product: 2 · 3 = 6
ob(product(D))

import Mathlib
open CategoryTheory
-- Mathlib has ordinary (conical) limits; weighted limits appear through ends / Kan extensions
#check @CategoryTheory.Limits.limit
#check @CategoryTheory.Functor.ran            -- (Ran_P F) e = lim^{B(e, P-)} F

{-# LANGUAGE RankNTypes #-}
-- a weighted cone with apex x: a natural transformation W ⇒ C(x, D-), i.e. for each j
-- a function from the weight W j to arrows x -> D j; in Hask with W j = weights as types:
type WeightedCone w d x = forall j. w j -> (x -> d j)