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+Category:Group actions; +Category:Topological groups using HotCat

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In [[category theory]], a branch of [[mathematics]], the '''cone of a functor''' is an abstract notion used to define the [[limit (category theory)|limit]] of that functor. Cones make other appearances in category theory as well.

==Definition==

Let ''F'' : ''J'' → ''C'' be a [[diagram (category theory)|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 (mathematics)|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
:<math>\psi_X\colon N \to F(X)\,</math>
for each object ''X'' of ''J'' such that for every morphism ''f'' : ''X'' → ''Y'' in ''J'' the following diagram [[commutative diagram|commutes]]:

[[Image:Functor cone.svg|175px|center|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 (geometry)|cone]] with the apex ''N''. The cone ψ is sometimes said to have '''vertex''' ''N'' and '''base''' ''F''.

One can also define the [[dual (category theory)|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
:<math>\psi_X\colon F(X)\to N\,</math>
for each object ''X'' of ''J'' such that for every morphism ''f'' : ''X'' → ''Y'' in ''J'' the following diagram commutes:

[[Image:Functor co-cone.svg|175px|center|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 ''C''''J'' be the [[category of diagrams]] of type ''J'' in ''C'' (this is 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''.

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:
* ψ is a co-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.

== 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 ==

[[Limit (category theory)|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.

[[Image:Functor cone (extended).svg|250px|center]]

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'').

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.

[[Image:Functor co-cone (extended).svg|250px|center]]

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.

== References ==

*{{cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | edition = 2nd ed. | publisher = Springer | location = New York | isbn = 0-387-98403-8}}

[[Category:Category theory]]
[[Category:Limits (category theory)]]

Continuous group action - Revision history

en>Tyrol5: +Category:Group actions; +Category:Topological groups using HotCat