## Universal Arrows and Universal Elements (Pt. I)

**Point of Post: **In this post we discuss the notion of universal arrows and universal elements, this is in preparation for our discussion of particular universal problems such as limits and colimits.

*Motivation*

Category theory has the ability to bring together disparate ideas in mathematics under one blanket theory. Category theory accomplishes this in two ways. Firstly, it provides a universal vocabulary which allows us to have a coherent discussion. Secondly, it provides a way of thinking, a way of formulating questions that brings forth the truly important characteristics of an object, separating it from the confounding chaff. In this post we describe a concept that epitomizes both of these categorical methodologies–universal characterizations. The idea behind universal characterizations are simple: in a given category what we really care about are arrows, and thus we should (when possible) define an object entirely in terms of arrows. To be more explicit it would be preferable to define a construction (say) in a way such as “this construction is the unique, up to isomorphism, object such that given maps out of…” opposed to “this construction is defined to be this object, and any object isomorphic to it”. It is clear that defining a construction in the first way, gives a more substantive description about what the construction is “all about”. For explicit example, consider that in the category of abelian groups we could define the “product” of two abelian groups to be either “an abelian group along with group homomorphisms such that given an abelian group and group homomorphism and there exists a unique group homomorphism such that and ” or we could define it as “any group isomorphic to (defined in the usual way)”. A little experimentation shows that these two description are describing the same isomorphism class of objects in , but it’s clear that the first definition clarifies the “use” of the product of groups–what qualities “make it a product”. To drill it in, a definition by mappings tells us how an object is defined, by telling us how it relates (by arrows) to other objects in the category. The idea of defining objects in a way involving only mappings exemplifies the second method of category theory, the rest of this post shall be devoted to the first–the development of the proper language that will allow us to precisely phrase these “universal mapping problems”.

*Universal Arrows and Universal Elements*

So, as stated we’d like to define the general scenario in which we can define objects as solutions to certain “universal mapping problems”. The first step in doing this is to define what we mean by “universal”. To this end, suppose that we have two categories and and a functor . We then define for an object a *universal arrow from to *to be an ordered pair where is an object in and is an arrow satisfying the following property: given any other object in and an arrow there exists a unique arrow such that .

There’s a nice pictoral way to think about universal arrows, but unfortunately my only means of drawing it is by a crude online sketch pad, regardless here’s what the picture looks like

Thus, we can see that a universal arrow is basically a -object and an arrow between the image under of this object and such that any time there is another -object and an arrow from to ‘s image under then we can find a unique arrow whose image under commutatively closes up the triangle started by .

There is another way to interpret universal arrows which shows their uniqueness. Indeed, we have the following theorem:

**Theorem: ***Let and be two categories and a functor . Then, if is an object of an ordered pair where is an object in and an arrow , is a universal arrow from to , if and only if is an initial object in the comma category .*

**Proof: **Suppose first that is a universal arrow from to . We note then that for any other object in the fact that is an arrow guarantees us a unique arrow such that . We then note that is an actual arrow in from to , and moreover if there were another arrow then would need to be an arrow satisfying , contradicting the uniqueness of the mapping . Thus, we see that every object in has a unique arrow emanating from and so is initial in .

Conversely, suppose that is initial in . Then, if is an object in and is an arrow then is an object in . Since is initial we know there exists an arrow in from to . Thus, we see that is an arrow such that . Moreover, it’s clear that such an arrow is unique, otherwise it would create two arrows from to in contradicting the definition of an initial object.

From first principles we can conclude from this that if and are both universal arrows from to then there exists a unique isomorphism such that .

Ok, so now that we have a conceptual understanding of what universal arrows are, let’s take a look at some typifying examples.

Let be some unital ring, consider the forgetful functor . Choose some object (set) in . We claim then that the free left -module and the set map is a universal arrow from to . Indeed, given any left -module and any set map we know there exists a unique -map such that , which is precisely the definition of a universal arrow from to .

For another example of a universal arrow to a forgetful functor, consider the forgetful functor . Choose some set , we claim that an example of a universal arrow from to is the discrete space (i.e. just the discrete topological space, with underlying set –we only used to differentiate between and ) and the identity set map . Indeed, let be some other topological space and suppose that we have a set map . Of course, since the map is also continuous since is discrete (all maps out of it are continuous!) and satisfies . Moreover, it’s fairly evident that this is the only such map.

*Remark: *In general, and it’s not hard to see this if one writes down the mapping properties of such an object, universal arrows to forgetful functors are “free” constructions in some sense.

**References:
**

[1] Mac, Lane Saunders. *Categories for the Working Mathematician*. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. *Abstract and Concrete Categories: the Joy of Cats*. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. *Categories and Modules with K-theory in View*. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. *Abelian Categories.* New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. *Introduction to Homological Algebra*. Springer-Verlag. Print.

[6] Herrlich, Horst, and George E. Strecker. *Category Theory: An Introduction*. Lemgo: Heldermann, 2007. Print.

[…] Universal Arrows and Universal Elements (Pt. II) Point of Post: This is a continuation of this post. […]

Pingback by Universal Arrows and Universal Elements (Pt. II) « Abstract Nonsense | January 12, 2012 |

[…] have indicated in our description of universal arrows that one of their key features was the ability to take constructions found in various parts of […]

Pingback by Categorical Products « Abstract Nonsense | January 24, 2012 |

[…] and . Thus, we see that is an -biadditive map . Therefore if we’d like to consider a “universal” way to define an -module structure on it seems that we should be looking for a […]

Pingback by Extension of Scalars and Change of Ring (Pt. I) « Abstract Nonsense | January 24, 2012 |

[…] of products. Indeed, we start, as with products, with the diagonal functor except instead of universal arrows to we define, for objects in , an ordered pair to be a coproduct of and if it is a universal […]

Pingback by Categorical Coproducts « Abstract Nonsense | February 7, 2012 |

[…] this post we discuss another type of universal arrow which shall be very important for our soon-to-come endeavors–equalizers and coequalizers. To […]

Pingback by Equalizers and Coequalizers (Pt. I) « Abstract Nonsense | February 22, 2012 |

[…] obvious that since this is a universal mapping property that we could have defined to be any module satisfying this property, since any two such […]

Pingback by The Tensor Algebra and Exterior Algebra (Pt. III) « Abstract Nonsense | May 10, 2012 |