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: 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), while 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). Formally, a diagram is an element of a functor category.
Diagrams are used in the definition of limit and colimits and the related notion of cones.
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[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 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. A diagram is said to be small or finite whenever J is.
One may defined a morphism of diagrams of type J in a category C as 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.
[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 (indexed by J).
- If J is a poset category then a diagram of type J is a family of objects Di together with a unique morphism fij : Di → Dj 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.
- If
, then a diagram of type J (
) is called "two parallel morphisms": its limit is an equalizer, and its colimit is a coequalizer.
[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 : CJ → 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 : CJ → C
which sends each diagram to its colimit.
[edit] 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 as 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 (because infinite) or messy (because many objects or morphisms) to draw; 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.
[edit] See also
[edit] Limits
[edit] Examples
[edit] References
- Adámek, Jiří; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories. John Wiley & Sons. ISBN 0-471-60922-6. http://katmat.math.uni-bremen.de/acc/acc.pdf. Now available as free on-line edition (4.2MB PDF).
