In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...

Definition

Let F : J → C be a diagram in C. Formally, a diagram is nothing more functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as a "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, an index set is another name for a function domain. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

Given an object N of C, a cone from N to F is a family of morphisms

for each object X of J such that for every morphism f : X → Y in J, we have F(f)∘ψ_X = ψ_Y. This situation may be depicted by a commutative diagram: In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

The (usually infinite) collection of all these triangles can be (partially) depicted in the shape of a cone with the apex N. The cone ψ is sometimes said to have vertex N and base F. Image File history File links FunctorCone-01. ... This article is about the geometric object, for other uses see Cone. ...

One can also define the dual notion of a cone from F to N (also called a co-cone) by reversing all the arrows above. Explicitly, a cone from F to N is a family of morphisms In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

for each object X of J such that for every morphism f : X → Y in J the following diagram commutes:

Image File history File links FunctorCone-02. ...

Equivalent formulations

At first glance cones seem to be slightly abnormal constructions in category theory. They are maps from an object to a functor (or vice-versa). In keeping with the spirit of category theory we would like to define them as morphisms or objects in some suitable category. In fact, we can do both.

Let J be a small category and let C^J be the category of diagrams of type J in C (this nothing more than a functor category). Define the diagonal functor Δ : C → C^J as follows: Δ(N) : J → C is the constant functor to N for all N in C. In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In category theory, for any object a in any category C where the product a×a exists, there exists the diagonal morphism δa: a → a×a, satisfying Ï€kδa = ida for k=1,2, where Ï€k is the canonical projection morphism to the k-th component. ...

If F is a diagram of type J in C, the following statements are equivalent:

• ψ is a cone from N to F
• ψ is a natural transformation from Δ(N) to F
• (N, ψ) is an object in the comma category (Δ ↓ F)

The dual statements are also equivalent: In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... A comma category is a construction in category theory, a branch of mathematics. ...

• ψ is a cone from F to N
• ψ is a natural transformation from F to Δ(N)
• (N, ψ) is an object in the comma category (F ↓ Δ)

These statements can all be verified by a straightforward application of the definitions. Thinking of cones, as natural transformations we see that they are just morphisms in C^J with source (or target) a constant functor. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... A comma category is a construction in category theory, a branch of mathematics. ...

Category of cones

By the above, we can define the category of cones to F as the comma category (Δ ↓ F). Morphisms of cones are then just morphisms in this category. As one might expect a morphism from a cone (N, ψ) to a cone (L, φ) is just a morphism N → L such that all the "obvious" diagrams commute (see the first diagram in the next section).

Likewise, the category of cones from F is the comma category (F ↓ Δ).

Universal cones

Limits and colimits are defined as universal cones. That is, cones through which all other cones factor. A cone φ from L to F is a universal cone if for any other cone ψ from N to F there is a unique morphism from ψ to φ. In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...

Equivalently, a universal cone to F is a universal morphism from Δ to F (thought of as an object in C^J), or a terminal object in (Δ ↓ F). Image File history File links FunctorCone-03. ... In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...

Dually, a cone φ from F to L is a universal cone if for any other cone ψ from F to N there is a unique morphism from φ to ψ.

Equivalently, a universal cone from F is a universal morphism from F to Δ, or an initial object in (F ↓ Δ). Image File history File links FunctorCone-04. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...

The limit of F is a universal cone to F, and the colimit is a universal cone from F. As with all universal constructions, universal cones are not guaranteed to exist for all diagrams F, but if the do exist they are unique up to a unique isomorphism.

References

• Mac Lane, Saunders (1998). Categories for the Working Mathematician, 2nd ed., New York: Springer. ISBN 0-387-98403-8.