Category theory

Limits and colimits



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 .


Let CC and DD be categories; we generally assume that DD is small. Let f:DCf:D\to C be a functor (called a diagram in this situation). Then a cocone (or inductive cone) over ff is a a pair (e,u)(e,u) of an object eCe\in C and a natural transformation u:fΔeu : f\to \Delta e (where Δe\Delta e is the constant diagram Δe:DC\Delta e:D\to C, xex\mapsto e, xDx\in D).

Note that a cocone in CC is precisely a cone in the opposite category C opC^op.

Terminology for natural transformations can also be applied to cocones. For example, a component of a cocone is a component of the natural transformation uu; that is, the component for each object xx of DD is the morphism u(x):f(x)eu(x): f(x) \to e.

A morphism of cocones (e,u)(e,u)(e,u)\to (e',u') is a morphism γ:ee\gamma:e\to e' in CC such that γu 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.

Revised on April 21, 2017 03:55:08 by Urs Schreiber (