Cone (category theory)
This article includes a improve this article by introducing more precise citations. (April 2022) ) |
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.
Definition
Let F : J → C be a
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
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 : 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
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 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 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
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
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
- Inverse limit#Cones – Construction in category theory
References
- ISBN 0-387-98403-8.
- Borceux, Francis (1994). "Limits". Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Vol. 1. Cambridge University Press. ISBN 0-521-44178-1.