Cone (category theory)

Last updated

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.

Contents

Definition

Let F : JC be a diagram 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 morphisms as well. Thus, for example, when J is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes J to be a span. J can also be taken to be the empty category, leading to the simplest cones.

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 : XY in J the following diagram commutes:

Part of a cone from N to F Functor cone.svg
Part of a cone from N to F

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 vertexN and baseF.

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 co-cone from F to N is a family of morphisms

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

Part of a cone from F to N Functor co-cone.svg
Part of a cone from F to N

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). Define the diagonal functor Δ : CCJ as follows: Δ(N) : JC 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.

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. This equivalence is rooted in the observation that a natural map between constant functors Δ(N), Δ(M) corresponds to a morphism between N and M. In this sense, the diagonal functor acts trivially on arrows. In similar vein, writing down the definition of a natural map from a constant functor Δ(N) to F yields the same diagram as the above. As one might expect, a morphism from a cone (N, ψ) to a cone (L, φ) is just a morphism NL 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 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.

Functor cone (extended).svg

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.

Functor co-cone (extended).svg

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 (in the comma category (Δ  F)).

See also

Related Research Articles

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

Universal property Central object of study in category theory

In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism. Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category. Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

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. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

In 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 can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In category theory, a 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 IX.

In mathematics, a direct limit is a way to construct a object from many objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by .

In category theory, the product of two 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. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

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. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.

In 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. Lawvere, although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category".

In mathematics, the derived categoryD(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

This is a glossary of properties and concepts in category theory in mathematics.

Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.

In category theory, a branch of mathematics, the diagonal functor is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the category : a product is a universal arrow from to . The arrow comprises the projection maps.

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 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.

In algebraic geometry and algebraic topology, branches of mathematics, A1homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A1, which is. The theory has seen spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.

In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we have a diagram

In mathematics, specifically category theory, an overcategory is a distinguished class of categories used in multiple contexts, such as with covering spaces. They were introduced as a mechanism for keeping track of data surrounding a fixed object in some category . There is a dual notion of undercategory, which is defined similarly.

References