In the previous post, we discussed natural transformations.
(A pdf notes version of the same material can be found under the mathematics page.)
1.5 Universal Properties
In this section, we give a definition of what the oft-quoted phrase “universal property” means. Before we begin, let us provide some motivation for this somewhat abstract concept. When we study a mathematical object, there are frequently two ways we can understand it. The first is opening the blackbox and examining its concrete construction. For example, we may understand the free product of two groups through the conventional process of explaining what “words” and “reduced words” mean, how to reduce words, etc. The second way is just treat it as a black box and examine instead the relationships between the object-of-interest and other objects. If there is a certain aspect of the relationship web that is specific to the object-of-interest, then it is as valid a defining feature as the concrete construction.
In fact, take a moment to contemplate about the broader philosophy of mathematics: when we make constructions, we care more about the utilities and functionalities of the constructions than the exact recipes that go into them. In category theory, things happen only when objects “talk” to each other. It is for this reason that the second way of understanding the purpose of objects via universal properties outshines the first way of looking at concrete constructions.
The word “relationship” suggests that universal properties will involve studying morphisms surrounding the object-of-interest, rather than the object itself. This should not be surprising given our previous discussion surrounding the equivalence of categories, where we focused on morphisms because we cared more about matching isomorphism classes rather than individual objects. In fact, when we use universal properties to describe objects-of-interest, we do not care about the variety of possible concrete constructions that lead to the same universal property – all these possibilities will turn out to be isomorphic.
Now, we shall begin the formal definitions.
Definition 1.5.1: Let be a category. We say that an object is initial in if for every object , there exists exactly one morphism . We say that an object is final in if for every object , there exists exactly one morphism .
Proposition 1.5.2: If are both initial objects in , then they are isomorphic. If are both final objects in , then they are isomorphic.
Proof. Let be initial objects. Let be the unique morphism in and let be the unique morphism in . Then which contains only one element, namely . So and similarly . The proof for final objects holds similarly.
Definition 1.5.3: We say that a construction satisfies a universal property if it can be viewed as an initial or final object of a category.
As our first example, we shall examine how universal properties can be used to define what is meant by products and coproducts in an arbitrary category. The reader may have encountered notions like the direct product, direct sum, free product, etc. and wondered about the greater purpose behind these constructions. For example, why do we care about direct sum of abelian groups but not so much direct sum of groups in general? It turns out that direct sum is a coproduct in the category Ab but for the category Grp, the corresponding coproduct is the free product.
Example 1.5.4: (Products) Let be a category and be objects. The product of and is an object together with a pair of morphisms , such that it is a final object of the double slice category .
In other words, the product is the object
such that for every object
there exists a unique morphism in such that the following diagram commutes:
In general, given a family of objects where , we can define their product as the object with canonical projection morphisms via a universal property similar to above.
Here are some examples:
- the product in Set is the cartesian product
- the product in the category induced by the preordered set is
- the product in the category induced by the preordering on via the divisibility relation is the greatest common divisor
- the direct product along with the natural projection maps is a product in both Grp and Ab
- the direct product along with the natural projection maps is a product in both Ring and CRing
- the direct product along with the natural projection maps is a product in both R-Mod and Mod-R
Example 1.5.5: (Coproducts) Let be a category and be objects. The coproduct of and is an object together with a pair of morphisms , such that it is an initial object of the double coslice category . That is, for all objects and morphisms , , there exists a unique morphism in such that the following diagram commutes:
In general, given a family of objects where , we can define their coproduct as the object with canonical “inclusion” morphisms via a universal property similar to above.
Again, some examples:
- the coproduct in Set is the disjoint union
- the coproduct in the category induced by the preordered set is
- the coproduct in the category induced by the preordering on via the divisibility relation is the lowest common multiple
- the free product along with the natural inclusion maps is a coproduct in Grp, while the direct sum along with the natural inclusion maps is a coproduct in Ab
- the direct sum along with the natural inclusion maps is a coproduct in R-Mod and Mod-R
Actually, the definition of universal property that we gave earlier (definition 1.5.3) is somewhat of a cheat-version. Before giving the actual definition of universal property, we need to generalize our definitions of slice and coslice categories (definitions 1.1.7 and 1.1.8).
Definition 1.5.6: Let be categories and be functors as follows:
We can form the comma category as follows:
- The objects are triples where is an object in , is an object in and is a morphism in . Pictorially, it is arrows of the form in figure 1.22.
- Morphisms from to are all pairs where and are morphisms in and respectively with the following diagram (figure 1.23) commuting:
- Composition of morphisms is done by taking to be
We observe that the slice categories and coslice categories are special cases of comma categories. For example, using the notation in the definition of comma categories, take with the identity functor, and take to be the category with a single object and denote which is an object in . Then we obtain the slice category . A similar consideration produces the coslice category.
The definition of universal property deals with a comma category of generality in between definition 1.5.6 and slice/coslice categories.
Definition 1.5.7: Suppose is a functor, and let be an object of .
- An initial morphism from to is an initial object in the comma category . In other words, it is a pair where is an object of and is a morphism in , such that whenever is an object of and is a morphism in , there exists a unique morphism that makes the below diagram (fig 1.24) commute.
- A terminal morphism from to is a terminal object in the comma category . In other words, it is a pair where is an object of and is a morphism in , such that whenever is an object of and is a morphism in , there exists a unique morphism that makes the above diagram (fig 1.25) commute.
We say that a construction satisfies a universal property if it can be viewed as an initial or final morphism.
Given enough patience and skills, the right functor can be found so that a universal property presented in the sense of definition 1.5.3 can be turned into an initial or final morphism. However, for most practical purposes, definition 1.5.3 is a good working definition.
Example 1.5.8: Let us take the universal property of product that we gave in example 1.5.4 and turned it into a terminal morphism. Let be a category and be objects. We let , which is an object of the product category . The relevant functor here is the diagonal functor . The reader can check that the universal property of product given earlier is equivalent to saying that is a terminal object in the comma category .
Example 1.5.9: What is the universal property of quotient constructions, such as the quotient group? Let be a normal subgroup of . Consider the subcategory of the coslice category where the objects are pairs with a group and satisfies . Then the quotient group together with the projection map is initial in this category. Here is how one might turn this into a statement about initial morphism:
Consider the category whose objects are pairs where is a normal subgroup of , and morphisms from to are group homomorphisms satisfying (also include the identity morphisms for the special case of so that the identity axiom is satisfied). Consider the functor given by (and ). Then we claim that is an initial object in the comma category .
Indeed, let us be given any group and any morphism of (i.e. is a group homomorphism and ). Then we know (e.g. from the ‘cheating’ version of universal property above) that there exists a unique group homomorphism such that . That is precisely what is required.
As seen from the above example, pursuing the strict definition of universal property can be an exhausting yet unenlightening process. As such, I am usually content with using the ‘cheat’ version of universal property.
This is the last post on category theory for now! Thank you for reading!