The Basics of Category Theory (Part 5)

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 G_1,G_2 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 \mathcal{C} be a category. We say that an object I\in Ob(\mathcal{C}) is initial in \mathcal{C} if for every object A, there exists exactly one morphism I\to A. We say that an object F\in Ob(\mathcal{C}) is final in \mathcal{C} if for every object A, there exists exactly one morphism A\to F.

Proposition 1.5.2: If I_1,I_2 are both initial objects in \mathcal{C}, then they are isomorphic. If F_1,F_2 are both final objects in \mathcal{C}, then they are isomorphic.

Proof. Let I_1,I_2 be initial objects. Let f:I_1\to I_2 be the unique morphism in Hom_{\mathcal{C}} (I_1,I_2 ) and let g:I_2\to I_1 be the unique morphism in Hom_{\mathcal{C}} (I_2,I_1). Then gf\in End_{\mathcal{C}} (I_1 ) which contains only one element, namely id_{I_1}. So gf=id_{I_1 } and similarly fg=id_{I_2 }. The proof for final objects holds similarly. \square

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 \mathcal{C} be a category and A,B be objects. The product of A and B is an object A\times B together with a pair of morphisms \pi_A:A\times B\to A, \pi_B:A\times B\to B such that it is a final object of the double slice category (\mathcal{C}\rightrightarrows A,B).

In other words, the product is the object

Fig1-18

Fig1-18

such that for every object

Fig1-19

Fig1-19

there exists a unique morphism \sigma:Z\to A\times B in \mathcal{C} such that the following diagram commutes:

Fig1-20

Fig1-20

In general, given a family of objects X_i where i\in I, we can define their product as the object \Pi_{i\in I} X_i with canonical projection morphisms \pi_j:\Pi_{i\in I} X_i\to X_j 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 (\mathbb{Z},\le) is min
  • the product in the category induced by the preordering on \mathbb{Z}^+ 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 \mathcal{C} be a category and A,B be objects. The coproduct of A and B is an object A\amalg B together with a pair of morphisms \iota_A:A\to A\amalg B, \iota_B:B\to A\amalg B such that it is an initial object of the double coslice category (A,B\rightrightarrows\mathcal{C}). That is, for all objects Z and morphisms f_A:A\to Z, f_B:B\to Z, there exists a unique morphism \sigma:A\amalg B\to Z in \mathcal{C} such that the following diagram commutes:

Fig1-21

Fig1-21

In general, given a family of objects X_i where i\in I, we can define their coproduct as the object \amalg_{i\in I} X_i with canonical “inclusion” morphisms \iota_j:X_j\to\amalg_{i\in I} X_i 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 (\mathbb{Z},\le) is max
  • the coproduct in the category induced by the preordering on \mathbb{Z}^+ 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 \mathcal{C},\mathcal{D},\mathcal{E} be categories and \mathcal{F},\mathcal{G} be functors as follows:

\mathcal{D}\xrightarrow{\mathcal{F}}\mathcal{C}\xleftarrow{\mathcal{G}}\mathcal{E}

We can form the comma category (\mathcal{F}\downarrow\mathcal{G}) as follows:

  • The objects are triples (A,B,f) where A is an object in \mathcal{D}, B is an object in \mathcal{E} and f:\mathcal{F}(A)\to\mathcal{G}(B) is a morphism in \mathcal{C}. Pictorially, it is arrows of the form in figure 1.22.
Fig1-22

Fig1-22

  • Morphisms from (A,B,f) to (A',B',f' ) are all pairs (g,h) where g:A\to A' and h:B\to B' are morphisms in \mathcal{D} and \mathcal{E} respectively with the following diagram (figure 1.23) commuting:
Fig1-23

Fig1-23

  • Composition of morphisms is done by taking (g,h)\circ(g',h') to be (g\circ g',h\circ h' )

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 \mathcal{D}=\mathcal{C} with \mathcal{F} the identity functor, and take \mathcal{E} to be the category with a single object B and denote X:=\mathcal{G}(B) which is an object in \mathcal{C}. Then we obtain the slice category (\mathcal{C}\downarrow X). 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 \mathcal{F}:\mathcal{C}\to\mathcal{D} is a functor, and let X be an object of \mathcal{D}.

  • An initial morphism from X to \mathcal{F} is an initial object in the comma category (X\downarrow\mathcal{F}). In other words, it is a pair (A,f) where A is an object of \mathcal{C} and f:X\to\mathcal{F}(A) is a morphism in \mathcal{D}, such that whenever A' is an object of \mathcal{C} and f':X\to\mathcal{F}(A') is a morphism in \mathcal{D}, there exists a unique morphism \phi:A\to A' that makes the below diagram (fig 1.24) commute.
Fig1-24

Fig1-24

  • A terminal morphism from \mathcal{F} to X is a terminal object in the comma category (\mathcal{F}\downarrow X). In other words, it is a pair (A,f) where A is an object of \mathcal{C} and f:\mathcal{F}(A)\to X is a morphism in \mathcal{D}, such that whenever A' is an object of \mathcal{C} and f':\mathcal{F}(A')\to X is a morphism in \mathcal{D}, there exists a unique morphism \phi:A\to A' that makes the above diagram (fig 1.25) commute.
Fig1-25

Fig1-25

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 \mathcal{C} be a category and A,B be objects. We let X:=(A,B), which is an object of the product category \mathcal{C}\times\mathcal{C}. The relevant functor \mathcal{F}:\mathcal{C}\to\mathcal{C}\times\mathcal{C} here is the diagonal functor x\mapsto(x,x). The reader can check that the universal property of product given earlier is equivalent to saying that (A\times B,(\pi_A,\pi_B ) ) is a terminal object in the comma category (\mathcal{F}\downarrow X).

Example 1.5.9: What is the universal property of quotient constructions, such as the quotient group? Let N be a normal subgroup of G. Consider the subcategory of the coslice category (G\downarrow \text{Grp}) where the objects are pairs (f,H) with H a group and f satisfies N\subseteq \text{ker}f. Then the quotient group G/N 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 \mathcal{D} whose objects are pairs (K,H) where K is a normal subgroup of H, and morphisms from (K,H) to (K',H') are group homomorphisms f:H\to H' satisfying K\subseteq \text{ker}f (also include the identity morphisms for the special case of H=H',K=K' so that the identity axiom is satisfied). Consider the functor \mathcal{F}: \text{Grp} \to \mathcal{D} given by H\mapsto(\{e\},H) (and \phi\mapsto\phi). Then we claim that (G/N,\pi) is an initial object in the comma category ((N,G)\downarrow\mathcal{F}).

Fig1-26

Fig1-26

Indeed, let us be given any group H and f':(N,G)\to(\{e\},H) any morphism of \mathcal{D} (i.e. f':G\to H is a group homomorphism and N\subseteq\text{ker}f'). Then we know (e.g. from the ‘cheating’ version of universal property above) that there exists a unique group homomorphism \phi:G/N\to H such that f'=\phi\circ\pi. 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!

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