category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the cone of a functor is an abstract notion used to define the limit of that

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

. Cones make other appearances in category theory as well.

Definition

Let ''F'' : ''J'' → ''C'' be a

diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...

in ''C''. Formally, a diagram is nothing more than a

from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and

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. Al ...

s in ''C''. The

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

''J'' is thought of as an "index category". One should consider this in analogy with the concept of an

indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, wher ...

of objects in

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a

discrete category In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: :hom''C''(''X'', ''X'') = {id''X''} for all objects ''X'' :hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ '' ...

, 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 :

\psi_X\colon N \to F(X)\,

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 In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

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

\psi_X\colon F(X)\to N\,

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 ''C''^''J'' be the category of diagrams of type ''J'' in ''C'' (this is nothing more than a

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 i ...

). Define the

diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct a ...

Δ : ''C'' → ''C''^''J'' 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 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. Hence, a natur ...

from Δ(''N'') to ''F'' * (''N'', ψ) is an object in the

comma category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another ...

(Δ ↓ ''F'') The dual statements are also equivalent: * ψ is a co-cone from ''F'' to ''N'' * ψ is a

from ''F'' to Δ(''N'') * (''N'', ψ) is an object in the

(''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.

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 ''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 colimits 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 suc ...

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. Equivalently, a universal cone to ''F'' is a

universal morphism In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...

from Δ to ''F'' (thought of as an object in ''C''^''J''), or a

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

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. Equivalently, a universal cone from ''F'' is a universal morphism from ''F'' to Δ, or an

initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...

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'')).

References

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External links