Diagram (category theory)

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In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well. Diagrams are used in the definition of limit and colimits and the related notion of cones.

Contents 1 Definition 2 Examples 3 Cones and limits 4 References

[edit] Definition

Formally, a diagram of type J in a category C is a functor

D : J → C

The category J is called the index category or the scheme of the diagram D. The actual objects and morphisms in J are largely irrelevant, only the way in which they are interrelated matters. The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J.

Although, technically, there is no difference between a diagram and a functor or between a scheme and a category, the change in terminology reflects a change in perspective, just as in the set theoretic case.

One is most often interested in the case where the scheme J is a small or even finite. A diagram is said to be small or finite whenever J is.

A morphism of diagrams of type J in a category C is just a natural transformation between functors. One can then interpret the category of diagrams of type J in C as the functor category C^J.

[edit] Examples

If J is a (small) discrete category, then a diagram of type J is essentially just an indexed family of objects in C.

If J is a poset category then a diagram of type J is a family of objects D_i together with a unique morphism f_ij : D_i → D_j whenever i ≤ j. If J is directed then a diagram of type J is called a direct system of objects and morphisms. If the diagram is contravariant then it is called an inverse system.

[edit] Cones and limits

A cone of a diagram D : J → C is a morphism from the constant diagram Δ(N) to D. The constant diagram is the diagram which sends every object of J to an object N of C (and every morphism to the identity morphism on N).

The limit of a diagram D is a universal cone to D. That is, a cone through which all other cones uniquely factor. If the limit exists in a category C for all diagrams of type J one obtains a functor

lim : C^J → C

which sends each diagram to its limit.

Dually, the colimit of diagram D is a universal cone from D. If the colimit exists for all diagrams of type J one has a functor

colim : C^J → C

which sends each diagram to its colimit.

[edit] References

• Adámek, Jiří; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories. John Wiley & Sons. ISBN 0-471-60922-6.  Now available as free on-line edition (4.2MB PDF).