Intro to Category Theory

Refer Aluffi (2009, p. 18).

Categories

Definition 1(Category).

A category $\text{C}$ consists of

• a class $\text{Obj}(\text{C})$ of objects of this category; and
• for every two objects $A,B$ of $\text{C}$, a set ${\text{Hom}}_{\text{C}}(A,B)$ of morphisms, with the properties listed below.

Properties of morphisms:

1. For every object $A$ of $\text{C}$, there exists (at least) one morphism $1_A\in {\text{Hom}}_{\text{C}}(A,A)$ of morphisms, the identity on $A$.
2. One can compose morphisms: two morphisms $f\in {\text{Hom}}_{\text{C}}(A,B)$ and $g\in {\text{Hom}}_{\text{C}}(B,C)$ determine a morphism $gf\in {\text{Hom}}_{\text{C}}(A,C)$. That is, for every triple of objects $A,B,C$ of $\text{C}$, there is a function (of sets) ${\text{Hom}}_{\text{C}}(A,B)\times {\text{Hom}}_{\text{C}}(B,C)\to {\text{Hom}}_{\text{C}}(A,C)$, and the image of the pair $(f,g)$ is denoted by $gf$.
3. This ‘composition law’ is associative: if $f\in {\text{Hom}}_{\text{C}}(A,B)$, $g\in {\text{Hom}}_{\text{C}}(B,C)$, and $h\in {\text{Hom}}_{\text{C}}(C,D)$, then $(hg)f=h(gf)$.
4. The identity morphisms are identities with respect to composition: for all $f\in {\text{Hom}}_{\text{C}}(A,B)$, we have $f{1}_A=f$, $1_{B}f=f$.

Also, note that if two morphisms are the same, then necessarily they have the same source and the same target: source and target are part of the datum of a morphism (and by extension, set-functions too).

Refer (Aluffi, 2009, p. 20) for examples.

Note

In Aluffi (2009, p. 22) Example 3.5, we could drop the requirement for morphisms in ${\text{C}}_A$ to be commutative diagrams. In this new category, call it ${\text{D}}_A$, morphisms $f_1\to f_2$ correspond to all morphisms $\sigma :Z_1\to Z_2$. However, such a category doesn’t yield anything new, since ${\text{Hom}}_{{\text{D}}_A}(f,f^{\prime })$ is the same as long as the domains of $f$ and $f^{\prime }$ are fixed. In other words, morphisms in ${\text{D}}_A$ depend only on the $Z$‘s and ignore the maps to $A$. So, ${\text{D}}_A$ is essentially $\text{C}$ itself, where the objects are the pairs $(Z,f)$, but the $f$‘s are forgotten in defining the morphisms.

Example 2(Example).

Let $\text{C}$ be a category. Let $A,B\in \text{Obj}(\text{C})$. Define a new category ${\text{C}}_{A,B}$ like so:

• $\text{Obj}({\text{C}}_{A,B})$ consists of diagrams

  in $\text{C}$, denoted by $(Z,f,g)$; and

• morphisms

  are commutative diagrams

As in example 3.5, the identities are inherited from the identities in $\text{C}$: for $(Z,f,g)$ in ${\text{C}}_{A,B}$, the identity $1_{(Z,f,g)}$ corresponds to the diagram

The composition $(Z_1,f_1,g_1)\stackrel{\sigma }{\to }(Z_2,f_2,g_2)\stackrel{\tau }{\to }(Z_3,f_3,g_3)$ is achieved by combining the diagrams of $\sigma$ and $\tau$ in the following manner

and since $\text{C}$ is a category, it follows that the diagram obtained by removing $Z_2$, i.e,

commutes. Associativity immediately follows from the fact that composition is associative in $\text{C}$.

Example 3(Example).

(Formalizing Example 3.10)
Let $\text{C}$ be a category. Choose two fixed morphisms $\alpha :A\to C$ and $\beta :B\to C$ in $\text{C}$, with the same target $C$. Define a category ${\text{C}}_{\alpha ,\beta }$ as follows:

• $\text{Obj}({\text{C}}_{\alpha ,\beta })$ contains commutative diagrams

  in $\text{C}$, and

• morphisms correspond to commutative diagrams

Again, the identities are inherited from $\text{C}$.

The composition $(Z_1,f_1,g_1)\stackrel{\sigma }{\to }(Z_2,f_2,g_2)\stackrel{\tau }{\to }(Z_3,f_3,g_3)$ is obtained by combining the diagrams of $\sigma$ and $\tau$ like so:

and as before, it follows that

commutes. Associativity follows from $\text{C}$ being a category.

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Morphisms

Isomorphisms and automorphisms

Definition 4(Isomorphism).

A morphism $f\in {\text{Hom}}_{\text{C}}(A,B)$ is an isomorphism if it has a (two-sided) inverse under composition: that is, if $\exists g\in {\text{Hom}}_{\text{C}}(B,A)$ such that

$$gf=1_A,fg=1_B.$$

Groupoids

A category in which every morphism is an isomorphism is called a groupoid. A category can be constructed form any set endowed with a reflexive and transitive relation (ensuring identities and composition respectively). If the relation also happens to be symmetric, the category constructed is a groupoid (every morphism now has an inverse).

Note

The notion of a “bijective homomorphism” does NOT always correspond to what an isomorphism is in general categories. Consider $\text{Poset}$ as a counter example.

Proposition 5(Proposition).

The inverse of an isomorphism is unique.

Proof.

This is an utterly standard verification.□

To hammer the point home, if $f$ is a morphism with a left-inverse $g_1$ and a right inverse $g_2$, then necessarily $f$ is an isomorphism, $g_1=g_2$, and this morphism is the (unique) inverse of $f$.

More utterly standard stuff:

Proposition 6(Proposition).

• Each identity $1_A$ is an isomorphism and its own inverse.
• If $f$ is an isomorphism, then $f^{-1}$ is an isomorphism and further $(f^{-1}{)}^{-1}=f$.
• If $f\in {\text{Hom}}_{\text{C}}(A,B)$, $g\in {\text{Hom}}_{\text{C}}(B,C)$ are isomorphisms, then $gf$ is an isomorphism and $(gf{)}^{-1}=f^{-1}{g}^{-1}$.

A morphism of an object $A$ of a category $\text{C}$ to itself is called an endomorphism. ${\text{Hom}}_{\text{C}}(A,A)$ is denoted by ${\text{End}}_{\text{C}}(A)$.

Definition 7(Automorphism).

An automorphism of an object $A$ of a category $\text{C}$ is an isomorphism form $A$ to itself. The set of automorphisms of $A$ is denoted by ${\text{Aut}}_{\text{C}}(A)$; it is a subset of ${\text{End}}_{\text{C}}(A)$.

From Proposition 6, we can infer than ${\text{Aut}}_{\text{C}}(A)$ is a group for all objects $A$ of all categories $\text{C}$:

• ${\text{Aut}}_{\text{C}}(A)$ is closed under composition;
• composition is associative
• ${\text{Aut}}_{\text{C}}(A)$ contains $1_A$, the identity for composition;
• every element $f\in {\text{Aut}}_{\text{C}}(A)$ has an inverse $f^{-1}\in {\text{Aut}}_{\text{C}}(A)$.

Note that we have already shown this for the category $\text{Gp}$, the category of groups with morphisms being group homomorphisms.

Monomorphisms and epimorphisms

Note that defining qualities of morphisms by their actions on ‘elements’ (as we did in $\text{Set}$) is not an option here, because objects of an arbitrary category do not (in general) have ‘elements’. However, recall that properties of morphisms in $\text{Set}$ such as injectivity and surjectivity did have alternative formulations which did not reference ‘elements’ at all (left cancellable and right cancellable functions, respectively). It turns out that these formulations of these notions do transfer nicely into the categorical setting.

Definition 8(Monomorphism).

Let $\text{C}$ be a category. A morphism $f\in {\text{Hom}}_{\text{C}}(A,B)$ is a monomorphism if the following holds: for all objects $Z$ of $\text{C}$ and all morphisms $\alpha ,{\alpha }^{\prime }\in {\text{Hom}}_{\text{C}}(Z,A)$,

$$f\circ \alpha =f\circ {\alpha }^{\prime }⟹\alpha ={\alpha }^{\prime }.$$

In other words, $f$ is a monomorphism if it is left cancellable.

Definition 9(Epimorphism).

Let $\text{C}$ be a category. A morphism $f\in {\text{Hom}}_{\text{C}}(A,B)$ is an epimorphism if the following holds: for all objects $Z$ of $\text{C}$ and all morphisms $\alpha ,{\alpha }^{\prime }\in {\text{Hom}}_{\text{C}}(B,Z)$,

$$\alpha \circ f={\alpha }^{\prime }\circ f⟹\alpha ={\alpha }^{\prime }.$$

In other words, $f$ is an epimorphism if it is right cancellable.

In $\text{Set}$, the monomorphisms are precisely the injective functions, and the epimorphisms are precisely the surjective functions.

Important

In $\text{Set}$, a function is an isomorphism iff it is both injective and surjective, hence iff it is both a monomorphism and an epimorphism. But in the category defined by $\le$ on $\mathbb{Z}$, every morphism is both a monomorphism and an epimorphism (since there is at most one morphism between any two objects, the defining conditions become vacuously true), while the only isomorphisms are identities. Thus, this is not a property one should expect to hold in every category.

As another example of $\text{Set}$ being special, notice that while

$$\begin{array}{rl}& f\text{ is right invertible }⟹f\text{ is an epimorphism},\text{and}\\ & f\text{ is left invertible }⟹f\text{ is a monomorphism}\end{array}$$

can be easily proven universally, the converse is not generally true (It is true, of course, in $\text{Set}$). For example ¹, the map $n\mapsto 2n$ defines a left-cancellable group homomorphism $f:\mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$. However, there is no group homomorphism $g:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ such that $gf={\mathrm{id}}_{\mathbb{Z}/2\mathbb{Z}}$. Similarly, the map $n\mapsto n\mathrm{mod}2$ defines a right-cancellable group homomorphism $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$. However, since every homomorphism $g:\mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ must map $[1{]}_2$ to $[2{]}_4$, $f$ is not right invertible.

A group theoretic digression

At this point, the question “When does a surjective homomorphism have a right inverse?” naturally arises. If $\phi :G\to H$ is a surjective homomorphism, I conjectured that the answer is yes when $H$ is isomorphic to a subgroup of $G$ (based on having constructed a right inverse for this homomorphism). However, a much stronger condition is required: The existence of a right inverse for $\phi$ is equivalent to the short exact sequence

$$1\to \ker \phi \to G\to H\to 1$$

splitting (Cf. Conrad (22 C.E.) Theorem 3.3).

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Universal Properties

Definition 10(Definition).

Let $\text{C}$ be a category.

• We say that $I\in \text{Obj}(\text{C})$ is initial in $\text{C}$ if for all $A\in \text{Obj}(C)$, ${\text{Hom}}_{\text{C}}(I,A)$ is a singleton.
• We say that $F\in \text{Obj}(\text{C})$ is final in $\text{C}$ if for all $A\in \text{Obj}(\text{C})$, ${\text{Hom}}_{\text{C}}(A,F)$ is a singleton.

A category need not have initial and final objects, and when they exist, they may not be unique. However, they are unique up to a unique isomorphism. Initial and final objects are collectively referred to as terminal objects.

Proposition 11(Proposition).

Let $\text{C}$ be a category.

• If $I_1,I_2$ are both initial objects in $\text{C}$, then $I_1\cong I_2$.
• If $F_1,F_2$ are both final objects in $\text{C}$, then $F_1\cong F_2$.

Moreover, these isomorphisms are uniquely determined.

Proof.

Since$I_1,I_2$ are initial objects, $|{\text{Hom}}_{\text{C}}(I_1,I_2)|=|{\text{Hom}}_{\text{C}}(I_2,I_1)|=$ $|{\text{Hom}}_{\text{C}}(I_1,I_1)|=|{\text{Hom}}_{\text{C}}(I_2,I_2)|=1$. Let ${\phi }_1\in {\text{Hom}}_{\text{C}}(I_1,I_2)$, ${\phi }_2\in {\text{Hom}}_{\text{C}}(I_2,I_1)$. It follows that ${\phi }_1{\phi }_2={\text{Id}}_{I_2}$ and ${\phi }_2{\phi }_1={\text{Id}}_{{\phi }_1}$, whence ${\phi }_1$ and ${\phi }_2$ are isomorphisms. The same proof works for final objects.□

The same object can be both initial and final, as singletons are in the category of pointed sets.

Definition 12(Definition).

We say that a construction satisfies a universal property when it may be viewed as a terminal object of a category.

Note

Try to see the universal property we stated for free groups in the context of this new definition.

Quotients

Let $\sim$ be an equivalence relation defined on a set $A$. Let $\text{C}$ be a category with objects $A\stackrel{\phi }{\to }Z$, where $Z$ is any set, satisfying the property

$$a\sim a^{\prime }⟹\phi (a)=\phi (a^{\prime }).$$

Let objects be denoted by $(\phi ,Z)$. Morphisms $({\phi }_1,Z_1)\to ({\phi }_2,Z_2)$ are commutative diagrams

Denote by $\pi$ the canonical projection from $A$ to $A/\sim$. Then, $(\pi ,A/\sim )$ is an initial object of $\text{C}$. Indeed, for any arbitrary $(\phi ,Z)$ in $\text{C}$, we can find a unique $\overline{\phi }$ such that

commutes.

This information can be sloppily summarized like so:

The quotient $A/\sim$ is universal with respect to the property of mapping $A$ to a set in such a way that equivalent elements have the same image.

Products

Definition 13(Definition).

The product of objects $A,B$ in a category $\text{C}$ is the isomorphism class of final objects in the category ${\text{C}}_{A,B}$, as defined in Example 2.

Example 14(Products of sets).

Let $\text{C}=\text{Set}$, and let $A,B\in \text{Obj}(\text{C})$. Consider the product $A\times B$ with the two natural projections:

Then for every $(Z,f,g)\in \text{Obj}({\text{C}}_{A,B})$, there exists a unique morphism $\sigma :Z\to A\times B$ such that

commutes. In other words, $\text{Hom}((Z,f,g),(A\times B,{\pi }_A,{\pi }_B))$ is a singleton, whence $(A\times B,{\pi }_A,{\pi }_B)$ is final in ${\text{C}}_{A,B}$.

Coproducts

Definition 15(Definition).

The coproduct of objects $A,B$ in a category $\text{C}$ is the isomorphism class of initial objects in the category ${\text{C}}^{A,B}$.

Example 16(Disjoint union of Sets).

Let $\text{C}=\text{Set}$. Let $A,B\in \text{Obj}(\text{C})$. Consider the disjoint union $A⨿B$ with the inclusion maps $i_A$ and $i_B$:

Then for every $(Z,f_A,f_B)$, there exists a unique morphism $\sigma :A⨿B\to Z$ such that

commutes. So, $(A⨿B,i_A,i_B)$ is initial in ${\text{C}}^{A,B}$.

Footnotes

1. https://math.stackexchange.com/questions/3824875/monomorphism-that-is-not-left-invertible-epimorphism-that-is-not-right-invertib ↩

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References

Aluffi, P. (2009). Algebra: Chapter 0. American Mathematical Society.

Conrad, K. (22 C.E.). SPLITTING OF SHORT EXACT SEQUENCES FOR GROUPS. https://kconrad.math.uconn.edu/blurbs/grouptheory/splittinggp.pdf