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Cone (category theory)
Encyclopedia
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 C. Formally, a diagram is nothing more than a 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 an "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 morphism
s as well.
Let N be an object 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 the following diagram commutes
:
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.
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
for each object X of J such that for every morphism f : X → Y in J the following diagram commutes:
Let J be a small category and let CJ be the category of diagrams of type J in C (this is nothing more than a functor category
). Define the diagonal functor Δ : C → CJ as follows: Δ(N) : J → C is the constant functor to N for all N in C.
If F is a diagram of type J in C, the following statements are equivalent:
The dual statements are also equivalent:
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 CJ with source (or target) a constant functor.
Likewise, the category of co-cones from F is the comma category (F ↓ Δ).
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.
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Equivalently, a universal cone to F is a universal morphism from Δ to F (thought of as an object in CJ), or a terminal object in (Δ ↓ F).
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.
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Equivalently, a universal cone from F is a universal morphism from F to Δ, or an initial object
in (F ↓ Δ).
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 they do exist they are unique up to a unique isomorphism.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the cone of a functor is an abstract notion used to define the limit
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 and inverse limits....
of that functor. Cones make other appearances in category theory as well.
Definition
Let F : J → C be a diagramDiagram (category theory)
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. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...
in C. Formally, a diagram is nothing more than a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
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
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family
Indexed family
In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers....
of objects in set theory. The primary difference is that here we have morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s as well.
Let N be an object 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 the following diagram commutes
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
:
be (partially) depicted in the shape of a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
with the apex N. The cone ψ is sometimes said to have vertex N and base F.
One can also define the dual
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
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
for each object X of J such that for every morphism f : X → Y in J the following diagram commutes:
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 CJ be the category of diagrams of type J in C (this is nothing more than a functor category
Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
). Define the diagonal functor Δ : C → CJ as follows: Δ(N) : J → C is the constant functor to N for all N in C.
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 transformationNatural transformationIn category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
from Δ(N) to F - (N, ψ) is an object in the comma categoryComma categoryIn mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
(Δ ↓ F)
The dual statements are also equivalent:
- ψ is a co-cone from F to N
- ψ is a natural transformationNatural transformationIn category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
from F to Δ(N) - (N, ψ) is an object in the comma categoryComma categoryIn mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
(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 CJ with source (or target) a constant functor.
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 co-cones from F is the comma category (F ↓ Δ).
Universal cones
Limits and colimitsLimit (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 and inverse limits....
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.
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.
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
in (F ↓ Δ).
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 they do exist they are unique up to a unique isomorphism.