In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

of that

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

. Cones make other appearances in category theory as well.

Definition

Let ''F'' : ''J'' → ''C'' be a

diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...

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, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

s in ''C''. The

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

''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 Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concer ...

. 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 Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan es ...

. ''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 with the apex ''N''. The cone ψ is sometimes said to have vertex ''N'' and base ''F''. One can also define the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

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 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 ...

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 Δ : ''C'' → ''C''^''J'' as follows: Δ(''N'') : ''J'' → ''C'' is the

constant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

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 na ...

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 to one another, morphisms become ob ...

(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 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 ...

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

* *

External links