cocone (changes) in nLab

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Idea

A cocone under a diagram is an object equipped with morphisms from each vertex of the diagram into it, such that all new diagrams arising this way commute.

A cocone which is universal is a colimit.

The dual notion is cone .

Definition

Let CC and DD be categories; we generally assume that DD is small. Let f:D→Cf:D\to C be a functor (called a diagram in this situation). Then a cocone (or inductive cone) over ff is a pair (e,u)(e,u) of an object e∈Ce\in C and a natural transformation u:f→Δeu : f\to \Delta e (where Δe\Delta e is the constant diagram Δe:D→C\Delta e:D\to C, x↦ex\mapsto e, x∈Dx\in D ). In other words, a diagram as follows, together with a natural transformation going south west to north east.

Note \begin{tikzcd} that D a \ar[r] cocone \ar[dr, in f, swap] & 1 \ar[d, e] \ & C \end{tikzcd}CC is precisely a cone in the opposite category C opC^op.

Terminology Note for that a cocone innatural transformationCC s can is also precisely be applied to cocones. For example, acomponentcone of in a cocone is a component of the natural transformationuuopposite category ; that is, the component for each objectxC op x C^op of DD is the morphism u(x):f(x)→eu(x): f(x) \to e.

A Terminology formorphism of coconesnatural transformation s can also be applied to cocones. For example, a(e,u)→(e′,u′)(e,u)\to (e',u')component of a cocone is a morphism component of the natural transformation γ u:e→e′ \gamma:e\to u e' ; in that is, the component for each object C x C x such of that γ D∘u x=u′ x \gamma\circ D u_x=u'_x for is all the objects morphismu(x):f(x)→e x u(x): f(x) \to e in DD (symbolically (Δγ)∘u=u′(\Delta \gamma)\circ u = u'); the composition being the composition of underlying morphisms in CC. Thus cocones form a category whose initial object if it exists is a colimit of ff.

A morphism of cocones (e,u)→(e′,u′)(e,u)\to (e',u') is a morphism γ:e→e′\gamma:e\to e' in CC such that γ∘u x=u′ x\gamma\circ u_x=u'_x for all objects xx in DD (symbolically (Δγ)∘u=u′(\Delta \gamma)\circ u = u'); the composition being the composition of underlying morphisms in CC. Thus cocones form a category whose initial object if it exists is a colimit of ff.