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Concept
Diagram (category theory)
Summary
In , 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.
The universal functor of a diagram is the diagonal functor; its right adjoint is the of the diagram and its left adjoint is the colimit. The natural transformation from the diagonal functor to some arbitrary diagram is called a .
Formally, a diagram of type J in a C is a (covariant) functor
The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram. 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 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 CJ, and a diagram is then an object in this category.
Given any object A in C, one has the constant diagram, which is the diagram that maps all objects in J to A, and all morphisms of J to the identity morphism on A.
Official source
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