Cone

definition example

Let $D:\mathcal{J}\to \mathcal{C}$ be a Diagram. A cone $(C,c_{\ast })$ over $D$ consists of an object $C\in \mathcal{C}$ (the apex) and, for each $j\in \mathcal{J}$, a morphism $c_j:C\to D(j)$ (a leg), such that for every $f:j\to k$ in $\mathcal{J}$, $c_k=c_j;D(f)$ (all triangles commute). A morphism of cones $(C,c_{\ast })\to (C^{\prime },c_{\ast }^{\prime })$ is $a:C\to C^{\prime }$ with $c_j=a;c_j^{\prime }$ for all $j$. Cones over $D$ form the category $\mathrm{Cone}(D)$, whose Terminal Object is the Limit of $D$.

Equivalently (DaoFP, Kittenlab): a cone with apex $x$ is a Natural Transformation ${\Delta }_x\Rightarrow D$ from the Constant Functor; “the constant functor shrinks all vertices to one, so naturality squares shrink to triangles”. A cocone is the dual: legs $D(j)\to C$, i.e. $D\Rightarrow {\Delta }_x$ (Cocone).

Sources: 7 Sketches Definition 3.92, §3.5.2 (Cone$(X,Y)$, Exercise 3.91); DaoFP §9.4–9.5 (“Cospans as natural transformations”, “Limits and Colimits”); Kittenlab Lecture 9, 15; CTfS §4.5.2–4.5.3 (Definitions 4.5.2.6, 4.5.3.18, Construction 4.5.3.15)

Cones as diagrams of a bigger shape (CTfS Definition 4.5.3.18). Adding a new initial object $-\infty$ to the indexing category gives its left cone ${\mathcal{J}}^{◃}$; a cone over $D$ is then simply a diagram $S:{\mathcal{J}}^{◃}\to \mathcal{C}$ restricting to $D$, and cone morphisms are natural transformations that are the identity on $D$. This is the slice category ${\mathcal{C}}_{/D}$, and the limit is its terminal object: “none shall map to $X$ and $Y$ except through me!“.

Examples. For the discrete diagram $\{X,Y\}$, a cone is a Span $X\leftarrow C\to Y$ (an “object equipped with morphisms to $X$ and $Y$”); for a Cospan $X\to A\leftarrow Y$, a cone is a commuting square with apex $C$ (a Pullback candidate); for a parallel pair $f,g:a\rightrightarrows b$, a cone is $p:e\to a$ with $f\circ p=g\circ p$ (Equalizer). For the empty diagram a cone is just an object. In $\mathbf{Set}$, a cone with apex $1$ over $D$ is an element of $\lim D$ (Finite Limits in Set).

JuliaLeanHaskell

Docs: FinSets · Limits & colimits · Free diagrams — Kittenlab Lecture 9

using Catlab
# a cone over the cospan f : X → A ← Y : g is a Multispan with legs into X and Y agreeing in A
f = FinFunction([1, 1, 2], 2); g = FinFunction([2, 1], 2)
lim = pullback(f, g)              # the limit cone
cone = Multispan(apex(lim), legs(lim))

#check CategoryTheory.Limits.Cone          -- structure Cone F: pt, π : (const J).obj pt ⟶ F
#check CategoryTheory.Limits.ConeMorphism
#check CategoryTheory.Limits.Cocone

-- a cone over a diagram with objects indexed by j: an apex type c and legs c -> d j (commutation unenforced)
newtype Cone c d = Cone (forall j. j -> (c -> d))   -- schematic: legs selected by an index