Post 6: Perverse Sheaves Part 2.5

28 Dec 2023

In this post, I will discuss the formal definition of an abelian category. It isn’t necessary to read if you just want to follow the “intuition” parts of my perverse sheaves posts.

The heavy lifting is going to be in the category section. I will define universal properties, so that we can define (co)products, (co)kernels, and (co)final objects.

Intuition

Monic and epic

Last time, I mentioned “injective” and “surjective” morphisms. Let me discuss the actual notions here.

The “correct” categorical abstraction of an “injective” morphism is a monomorphism, or a monic morphism. A morphism $f$ is a monomorphism, or is monic, if it is “left cancellable”; if $f\circ g = f\circ h$, then $g=h$.

The “correct” categorical abstraction of an “surjective” morphism is an epimorphism, or an epic morphism. A morphism $f$ is an epimorphism, or is epic, if it is “right cancellable”; if $g\circ f = h\circ f$, then $g=h$.

A function $f$ is monic in Set if and only if it is injective; it is epic in Set if and only if it is surjective.

However, even in familiar categories, the notions can fail to agree in general. For a good example, the usual injective map from $\Z$ to $\Q$ is an epimorphism in the category of commutative rings!

Universal Properties

Universal properties are perhaps one of the most important categorical concepts to understand; they show up everywhere. For example, the axioms for an abelian category involve some concept that is defined in terms of a universal property. You may have learned about the tensor product in algebra; these are also defined in terms of a universal property.

Here is how Emily Riehl1 describes universal properties “colloquially”: a universal property of an object $X$ is a description of the functor $\Hom(X, −)$ or of the functor $\Hom(−, X)$ associated to that object. Notice that the functors $\Hom(X,-)$ and $\Hom(-,X)$ carry the information of morphisms into or out of $X$, respectively.

Here are some important and hopefully illustrative examples:

The important part about objects defined by universal properties is that they are “unique up to unique isomorphism”. For example, if $X_1$ and $X_2$ are two initial objects, then there is a unique isomorphism $X_1\to X_2$. As an exercise, prove that!

Another important thing to note is that a category may fail to have some object that is defined by a universal product. For instance, axioms 1 and 2 of an abelian category assert the existence of such objects.

However, sometimes universal properties are not defined in terms of morphisms in the category you’re working in. Here is an important example:

“Final with respect to this data” means that if we have some other triple $(Z,f_1:Z\to X,f_2:Z\to Y)$, then there is a unique morphism $f:Z\to P$ that is compatible with the given morphisms: $f_1=pr_1\circ f$ and $f_2=pr_2\circ f$.

Often times, $P$ is denoted by $X\times Y$.

Of course, there is a more formal way to phrase this, but hopefully you get the idea. See if you can figure out the formal meaning of this definition:

Often times, $C$ is denoted by $X\coprod Y$.

You can also define products and coproducts of arbitrary collections of objects. A product or coproduct of one object $X$ is just $(X,1_X)$. A product of an empty collection is usually defined to be a terminal object, and a coproduct of an empty collection is usually defined to be an initial object.

Also, instead of saying “products of finite collections”, we often say “finite products”. Similarly we say “finite coproducts” and “finite biproducts” (see below).

Now assume your category has a zero object $0$, so that it also has zero morphisms $0$, the morphisms defined by composing $A\to 0\to B$.

  1. $pr_k\circ i_k = 1_{X_k}$ for all $k$,

  2. $pr_\ell\circ i_k=0=pr_k\circ i_\ell$ if $k\neq\ell$,

  3. $(B,\{pr_k\})$ is a product of the $\{X_k\}$,

  4. $(B,\{i_k\})$ is a coproduct of the $\{X_k\}$.

We often denote $B$ by $\bigoplus X_k$.

A biproduct of an empty collection is a zero object.

Finally, let me discuss the kernel. Again, assume your category has a zero object.

This means that if you have some $g:G\to A$ with $f\circ g$, there is a unique morphism $h:G\to K$ such that $g=k\circ h$.

As an exercise, work out the formal meaning of the this definition:

You now have the necessary definitions to define an abelian category, at least in the way I’m approaching it. I will discuss an alternative approach in the formal section.

Abelian categories

Here are the axioms for a category to be abelian:

  1. It has finite biproducts. In particular, it has an empty biproduct, i.e. a zero object.

  2. Any morphism $f:A\to B$ has a kernel and a cokernel.

  3. Every monomorphism is the kernel of some morphism, and every epimorphism is the cokernel of some morphism.

As a consequence of the first two axioms, you can “enrich” $\mc C$ so that you can add two morphisms $f,g:A\to B$ to get a new morphism $f+g:A\to B$. In fact, this makes $\Hom(A,B)$ into an abelian group. The $0$ morphism is the group identity, and the morphism $-f$ denotes the group inverse of $-f$. Furthermore, “composition is bilinear”:

Formalities

Universal properties

In order to formalize the notion of universal properties, we need to discuss some more about functors.

Let $F,G:\mc C\to \mc D$ be two functors. A natural transformation $\eta:F\to G$ is a collection of morphisms $\eta(A):F(A)\to G(A)$ for each $A\in\mc C$, with the requirement that for any $f:A\to B$ in $\mc C$, we have $G(f)\circ \eta(A)=\eta(B)\circ F(f)$.

The morphisms $\eta(A)$ are called the components of $A$. A natural isomorphism is a natural transformation in which each component is an isomorphism.

A functor $F:\mc C\to \text{Set}$ is representable if there is an $X\in\mc C$ and a natural isomorphism $F\to\Hom(X,-)$. We then say $F$ represents $X$.

Now we can say the following: representable functors “encode” universal properties for the objects they represent. For example, consider the “constant” functor $*:\mc C\to\Set$ that sends every object to a fixed set with one object and sends every morphism to the identity morphism of that set. Then if $*$ is representable and represents $X$, then $X$ is initial. Conversely, any initial object represents this functor.

In order to get access to terminal objects in terms of representable functors, you either need to talk about contravariant functors, or just talk about the opposite category. I prefer the second approach:

In other words, $\mc C^{op}$ is the result of “reversing the arrows” in $\mc C$.

It follows that $\Hom(-,X)$ is a functor from $\mc C^{op}\to \Set$.

A terminal object in $\mc C$ is an initial object in $\mc C^{op}$ and vice versa. In particular, an object $X$ is terminal if and only if $\Hom(-,X)$ is isomorphic to the constant functor $*:\mc C^{op}\to \Set$ that sends everything to a one element set and its identity morphism.

Now, how do you get things like products and kernels in terms of representable functors? Well, you have to construct an appropriate category. For example, to define the product of objects $X$ and $Y$, consider the category whose objects are triples $(Z,f_1:Z\to X, f_2:Z\to Y)$, and whose morphisms are defined as follows:

Then a product of $X$ and $Y$ is a terminal object in this category, which has the previously mentioned interpretation in terms of representable functors.

Here is a nice result, which is a corollary of the Yoneda lemma, which I won’t talk about:

In particular, two objects that represent the same functor are isomorphic.

Enriched categories (or a special case thereof)

In order to define enriched categories in general, one needs to define a monoidal category, and I feel that going through that definition would lead us too far astray. Instead, I’m going to focus on the definition of being enriched over the category of abelian groups, AbGrp.

Below, $\otimes$ will denote the tensor product of abelian groups, and $\Z$ will denote the usual group of integers under addition.

A category $\mc C$ enriched over AbGrp is the following data:

  1. Objects,

  2. An abelian group $\Hom(A,B)\in$ for all objects $A,B\in\mc C$,

  3. An element $1_A\in\Hom(A,A)$ for all objects $A$,

  4. A morphism $\circ_{ABC}:\Hom(B,C)\otimes\Hom(A,B)\to\Hom(A,C)$ in AbGrp for all objects $A,B,C\in\mc C$.

A morphism $f:A\to B$ in $\mc C$ is defined to be an element of the group $\Hom(A,B)$. The composition of two morphisms $f:A\to B$ and $g:B\to C$, written $g\circ f$, is defined to be $\circ_{ABC}(g\otimes f)$.

Furthermore, we must have the usual conditions on composition:

  1. Composition is associative: $f\circ(g\circ h)=(f\circ g)\circ h$.

  2. $1_A$ is an identity for composition: $1_A\circ f = f$ and $f\circ 1_A=f$.

If you don’t like the usage of tensor product, we can equivalently define $\circ_{ABC}$ to be a $\Z$-bilinear function $\Hom(B,C)\times\Hom(A,B)\to\Hom(A,C)$, and then composition is defined as $\circ_{ABC}(g,f)$.

Abelian categories, again

In most sources, abelian categories are defined in four steps. The second step, additive categories, is actually important, since for instance, the homotopy category is always additive, but not abelian in general. For this reason, I’ve decided to include this definition.

Let AbGrp be the category of abelian groups; also known as $\Z$-mod.

  1. A category enriched over AbGrp is called preadditive.

  2. A preadditive category is additive if it has finite biproducts. In particular, it has an empty biproduct, i.e. a zero object.

  3. An additive category is preabelian if every morphism has a kernel and a cokernel.

  4. A preabelian category is abelian if every monomorphism is a kernel and every epimorphism is a cokernel.

That’s all for now!

Footnotes:

  1. See Emily Riehl’s book, “Category Theory in Context”. ↩