Diagonal functor, the Glossary

Table of Contents

1. 18 relations: Adjoint functors, Category (mathematics), Category theory, Comma category, Complete category, Cone (category theory), Coproduct, Diagonal morphism, Diagram (category theory), Discrete category, Functor, Functor category, Limit (category theory), Mathematics, Morphism, Natural transformation, Product (category theory), Universal property.

Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories.

See Diagonal functor and Adjoint functors

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". Diagonal functor and category (mathematics) are category theory.

See Diagonal functor and Category (mathematics)

Category theory

Category theory is a general theory of mathematical structures and their relations.

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Comma category

In mathematics, a comma category (a special case being a slice category) is a construction in category theory.

See Diagonal functor and Comma category

Complete category

In mathematics, a complete category is a category in which all small limits exist.

See Diagonal functor and Complete category

Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Diagonal functor and cone (category theory) are category theory.

See Diagonal functor and Cone (category theory)

Coproduct

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.

See Diagonal functor and Coproduct

Diagonal morphism

In category theory, a branch of mathematics, for every object a in every category \mathcal where the product a\times a exists, there exists the diagonal morphism satisfying where \pi_k is the canonical projection morphism to the k-th component. Diagonal functor and diagonal morphism are category theory stubs.

See Diagonal functor and Diagonal morphism

Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory.

See Diagonal functor and Diagram (category theory)

Discrete category

In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.

See Diagonal functor and Discrete category

Functor

In mathematics, specifically category theory, a functor is a mapping between categories.

See Diagonal functor and Functor

Functor category

In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in the category).

See Diagonal functor and Functor category

Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

See Diagonal functor and Limit (category theory)

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Diagonal functor and Mathematics

Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.

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Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

See Diagonal functor and Natural transformation

Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

See Diagonal functor and Product (category theory)

Universal property

In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Diagonal functor and universal property are category theory.

See Diagonal functor and Universal property

References

[1] https://en.wikipedia.org/wiki/Diagonal_functor

Also known as Diagonal functors.