Diagram (category theory)

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In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.

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The universal functor of a diagram is the diagonal functor; its right adjoint is the limit of the diagram and its left adjoint is the colimit. [1] The natural transformation from the diagonal functor to some arbitrary diagram is called a cone.

Definition

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

D : JC.

The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram. [2] 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 an individual 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 fixes the index category, and allows the functor (and, secondarily, the target category) to vary.

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

A morphism of diagrams of type J in a category C is a natural transformation between functors. One can then interpret the category of diagrams of type J in C as the functor category CJ, and a diagram is then an object in this category.

Examples

Cones and limits

A cone with vertex N of a diagram D : JC 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 : CJC

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 : CJC

which sends each diagram to its colimit.

Commutative diagrams

Diagrams and functor categories are often visualized by commutative diagrams, particularly if the index category is a finite poset category with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way.

Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism (), or with two parallel arrows (; ) need not commute. Further, diagrams may be impossible to draw (because they are infinite) or simply messy (because there are too many objects or morphisms); however, schematic commutative diagrams (for subcategories of the index category, or with ellipses, such as for a directed system) are used to clarify such complex diagrams.

See also

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References

  1. Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in geometry and logic a first introduction to topos theory . New York: Springer-Verlag. pp.  20–23. ISBN   9780387977102.
  2. May, J. P. (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. p. 16. ISBN   0-226-51183-9.