A Category Theory (Mini)Tour

Prologue~
“To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.”
— William Blake, Auguries of Innocence
With this post we delve into the very basics of a category and functors. We understand the adjunction between the $\mathrm{Hom}$ functor and the Tensor product- something which is quite interesting since it reveals to you more than you ask of it. There are several adjunctions which arise naturally, similar to this one and by the uniqueness property they can be shown to be isomorphic to each other, which is a very interesting result (if you ask me)!

Verse 1: Categories~
A category $\mathcal C$ consists of the following:
1. A class of objects denoted as $ \mathrm{Ob}(\mathcal C)$;
2. for $X,Y \in  \mathrm{Ob}(\mathcal C)$, a class of morphisms from $X$ into $Y$ denoted as $\mathrm{Mor}_{\mathcal C}(X,Y)$;
3. for each $X, Y, Z \in  \mathrm{Ob}(\mathcal C),$ a composition map $$ \mathrm{Mor}_{\mathcal C} (X,Y) \times  \mathrm{ Mor}_{\mathcal C} (Y,Z) \to  \mathrm{Mor}_{\mathcal C} (X, Z)  $$ satisfying the associative property for all $ f \in  \mathrm{Mor}_{\mathcal C} (X,Y), g \in  \mathrm{Mor}_{\mathcal C} (Y,Z)$ and $h \in  \mathrm{Mor}_{\mathcal C}(Z,W)$, $(h \circ g) \circ f =  h \circ (g \circ f)$ holds, for every $X,Y,Z,W \in  \mathrm{Ob}(\mathcal C).$
4. Identity morphisms id_X ∈ Mor_C(X,X) for each X, satisfying f ∘ id_X = f = id_Y ∘ f.

Verse 2: Monoids~

For fun's sake, let us recall the definition of a monoid- A monoid is a set $M$ with a binary operation, which is associative and has an identity. We can also state the same definition as- a monoid being a category with one object. Thus for any category $\mathcal C$ and any object $a \in  \mathrm{Ob}(\mathcal C)$, the class $ \mathrm{Mor}_{\mathcal C}(a,a)$ of all morphisms from $a$ to $a$ is a monoid. Essentially, the single object’s endomorphisms form a monoid under composition, with identity being the identity morphism.

Verse 3: Functors~ A functor is a morphism between categories. Consider Categories $\mathcal A$ and $\mathcal B$, then a functor, say $F$ is defined as, $$F: \mathcal A \to \mathcal B$$ with domain $\mathcal A$ and the codomain $\mathcal B$, consisting of two related functions, namely the object function and the arrow function. The object function assigns each object  $a \in  \mathrm{Ob}(\mathcal A)$  with an object  $Fa$  of  $\mathcal B$  and the arrow function assigns to each arrow, say  $f: a \to a'$  of $\mathcal A$,  an arrow  $Ff: Fa \to Fa'$  of  $\mathcal B$.
$F$  hence carries each identity of  $\mathcal A$  to an identity of  $\mathcal B$  and each composable pair  $<g,f>$  in  $\mathcal A$  to a composable pair  $<Fg, Ff>$  in  $\mathcal B$  with  $Fg \circ Ff= F(g \circ f)$.

Verse 4: Hom bifunctor~
Now, consider the set of all morphisms from the definition of a category and let us define it functorially, i.e., we consider Hom to be a functor and define it as
$$ \mathrm{Hom}_{\mathcal C} ( - ,-) :=  \mathrm{Mor}_{\mathcal C} (-,-)$$ Let  $\mathcal C$  be a category. For objects  $A,B \in  \mathrm{Ob}(\mathcal C)$,  the set of morphisms from  $A$  to  $B$  is denoted by  $\mathrm{Mor}_{\mathcal C}(A,B)$.  Then the assignment  $(A,B) \longmapsto \mathrm{Mor}_{\mathcal C}(A,B)$  defines a bifunctor $$\mathrm{Hom}_{\mathcal C}(-,-) : \mathcal C^{\mathrm{op}} \times \mathcal C \to \mathbf{Set}$$ whose variance is described as follows:
1. Contravariance in the first argument.
Let  $f : A' \to A$  be a morphism in  $\mathcal C$,  and let  $B$  be a fixed object. Define a function $$\mathrm{Hom}_{\mathcal C}(A,B) \to \mathrm{Hom}_{\mathcal C}(A',B)$$ by  $h \longmapsto h \circ f .$  This assignment satisfies $$\mathrm{id}_A^\ast = \mathrm{id}_{\mathrm{Hom}(A,B)}, \qquad (g \circ f)^\ast = f^\ast \circ g^\ast, $$ and therefore defines a contravariant functor in the first variable.
2. Covariance in the second argument.
Let  $g : B \to B'$  be a morphism in  $\mathcal C$,  and let  $A$  be a fixed object. Define a function $$\mathrm{Hom}_{\mathcal C}(A,B) \to \mathrm{Hom}_{\mathcal C}(A,B')$$ by  $ h \longmapsto g \circ h .$  This assignment satisfies $$\mathrm{id}_B{}_\ast = \mathrm{id}_{\mathrm{Hom}(A,B)}, \qquad (g' \circ g)_\ast = g'_\ast \circ g_\ast,$$ and therefore defines a covariant functor in the second variable.
This is exactly what the  $\mathrm{Hom}$  functor does, and hence it is contravariant in its first argument and covariant in its second, and consequently defines a bifunctor $$\mathrm{Hom}_{\mathcal C}(-,-) : \mathcal C^{\mathrm{op}} \times \mathcal C \to \mathbf{Set}.$$ Note that in the first argument, $ \mathrm{Hom}$ admits an object from the opposite category ${\mathcal C}^{op}$.  

Verse 5: Tensor Product~
Let  $R$  be a commutative ring, and let  $\mathbf{Mod}_R$  denote the category of  $R$-modules.
For  $R$-modules  $M$  and  $N$ , the tensor product  $M \otimes_R N$  is an  $R$-module equipped with a bilinear map $$\tau : M \times N \to M \otimes_R N$$ satisfying the following universal property.
For every  $R$-module  $P$, composition with  $\tau$  induces a bijection $$\mathrm{Hom}_R(M \otimes_R N, P) \;\cong\; \mathrm{Bil}_R(M \times N, P),$$ where  $\mathrm{Bil}_R(M \times N, P)$  denotes the set of  $R$-bilinear maps from  $M \times N$  to  $P$.
Equivalently, for every bilinear map  $\beta : M \times N \to P$  there exists a unique  $R$-linear map $\widetilde{\beta} : M \otimes_R N \to P,$   such that  $\beta = \widetilde{\beta} \circ \tau .$
The tensor product defines a bifunctor $$-\otimes_R- : \mathbf{Mod}_R \times \mathbf{Mod}_R \to \mathbf{Mod}_R $$ which is right exact in each variable.
Fixing an  $R$-module $N$, the assignment  $M \longmapsto M \otimes_R N$  defines a covariant functor $$-\otimes_R N : \mathbf{Mod}_R \to \mathbf{Mod}_R.$$ Tensors products and Hom functors are bifunctors in a monoidal category.

Verse 6: Adjunctions~

Let $\mathcal C$ and $\mathcal D$ be categories, and let  $F : \mathcal C \to \mathcal D$  and  $G : \mathcal D \to \mathcal C $  be functors.

The functor  $F$  is said to be left adjoint to  $G$ , and  $G$ is said to be right adjoint to  $F$ , if for every object $C \in \mathcal C$  and every object  $D \in \mathcal D$  there exists a bijection $$\mathrm{Hom}_{\mathcal D}(F(C), D)  \;\cong\; \mathrm{Hom}_{\mathcal C}(C, G(D)).$$

This bijection is required to be natural in both variables, that is, natural in  $C$  and natural in  $D$. Equivalently, the adjunction is specified by a pair of natural transformations
$$\eta : \mathrm{id}_{\mathcal C} \to G \circ F \quad \text{and} \quad \varepsilon : F \circ G \to \mathrm{id}_{\mathcal D},$$ called the unit and counit of the adjunction, satisfying the identities $$(\varepsilon \circ F) \circ (F \circ \eta) = \mathrm{id}_F, \qquad (G \circ \varepsilon) \circ (\eta \circ G) = \mathrm{id}_G.$$ For an adjunction  $F \dashv G$  between categories  $\mathcal{C}$ and $\mathcal{D}$ , the unit and counit are (said to be) natural transformations $$\eta : \mathrm{id}_{\mathcal{C}} \longrightarrow G \circ F, \qquad \varepsilon : F \circ G \longrightarrow \mathrm{id}_{\mathcal{D}}.$$ Explicitly, for each object  $C \in \mathcal{C}$  and  $D \in \mathcal{D}$  there are morphisms $$ \eta_C : C \longrightarrow G(F(C)), \qquad \varepsilon_D : F(G(D)) \longrightarrow D, $$ which are natural in  $C$  and  $D$ , respectively. These satisfy the triangle identities. For every object  $C \in \mathcal{C}$ , $$ F(C) \xrightarrow{F(\eta_C)} F(G(F(C))) \xrightarrow{\varepsilon_{F(C)}} F(C)$$ is equal to  $\mathrm{id}_{F(C)}$;  and for every object  $D \in \mathcal{D}$,  $$ G(D) \xrightarrow{\eta_{G(D)}} G(F(G(D))) \xrightarrow{G(\varepsilon_D)} G(D)$$ is equal to  $\mathrm{id}_{G(D)}$.

$(\varepsilon_{F(-)} \circ F(\eta_-)) = \mathrm{id}_F$ and $(G(\varepsilon_-) \circ \eta_{G(-)}) = \mathrm{id}_G$

These identities express the coherence of the adjunction: going from  $F(C)$  to  $G(F(C))$  via the unit and back via the counit does nothing overall, and similarly for  $G(D)$.

Verse 7: Hom-⊗ Adjunction~

Let $R$ be a commutative ring. For $R$-modules $M,N,P$, there is a natural isomorphism $$\mathrm{Hom}_R(M \otimes_R N, P) \;\cong\; \mathrm{Hom}_R\bigl(M,\mathrm{Hom}_R(N,P)\bigr). $$ One should also note the symmetric role of $M$ and $N$, that is we finally obtain the following, $$\mathrm{Hom}_R(M \otimes_R N, P) \;\cong\; \mathrm{Hom}_R\bigl(M,\mathrm{Hom}_R(N,P)\bigr) \;\cong\; \mathrm{Hom}_R(N \otimes_R M, P)$$ For fixed $N, \ − ⊗_R N$ is left adjoint to $\mathrm{Hom}_R(N, −)$, with  $Φ$  as the adjunction isomorphism.

Epilogue~
“For last year's words belong to last year's language
And next year's words await another voice.”
— T. S. Eliot

One can refer to Categories for the Working Mathematician by Saunders Maclane, for a detailed study.

Of Algebra and Diagrams...

Prologue~
“Algebra is generous; she often gives more than is asked of her.”
— Jean le Rond d'Alembert

The aim of this post is to discuss a few theorems based of ring homomorphisms, while giving an intuition of diagram chasing, since our magic tool here would be commutative diagrams. We also encounter what a natural map would be in such a setting.

Verse I: The Epimorphism Theorem~

Let $f : R \to S$ be a ring homomorphism then there exists an isomorphism from $R/ Ker\ f$ to $S$, iff $f$ is an epimorphism.

Fig. 1

Now if you consider the adjacent figure, then clearly $f$ has been rightfully depicted as a map from $R$ to $S$.

We were supposed to find the map between the quotient ring and $S$, which here is represented as $\tilde{f}$.

Notice that the map denoted by $\eta$ here is a natural (or canonical) surjection, and it is always a surjective ring homomorphism. 

The meaning of surjective here is simply onto, i.e., the map $\eta$ takes an element of $R$ and lands it in $R/Ker \ f$, such that every element of $R/Ker\ f$ has a pre-image in $R$.

Now we must define $\tilde{f}$ to complete the proof. So what our $\tilde{f}$ does here is takes an element from $R/Ker \ f$ and maps it to the image of $R$ under $f.$

Hence, the above paragraph(s), for all elements $x \in R$ can be concluded as, 

$f(x)= \tilde{f} \circ \eta (x)$ 

and we say that the diagram in Fig. 1 commutes!

Here $\circ$ implies pre-right composition.

Verse II: Quotient of a Quotient Theorem~

If $I \subseteq J$ are both 2-sided ideals in $R$, then $(R/I)/(J/I)$ is naturally isomorphic to $R/J.$

Fig. 2

Let us dissect the adjacent figure which, one can claim that gives a precise and swift proof of the theorem stated above.

Here again, we have two quotient rings $R/I$ and $R/J$, where our $I$ and $J$ are ideals. And we have been given that $I \subseteq J$. (In)formally, our $I$ sits in $J$ and hence from what we did in the previous proof, we can say that the map $\eta_{IJ}$ forms a natural (or canonical) surjection, which again means that for every element of $R/I$, there exists a pre-image in $R$. 

Similar applies for $\eta_J$ and the map here is (again) a canonical surjection.

Interestingly, $\eta_{R/J}$ and $\eta_{IJ}$ are also canonical surjections from their respective domains to their codomains.

Fig. 2.1

Now, talking about the map which actually concerns us, because everything so far arose naturally from the ideal and quotient ring relations. 

We are interested in $\tilde{\eta}$, which is a map from $(R/I)/(J/I)$ to $R/J$, and since $I \subseteq J$, this again turns out to be another natural surjection, the only difference being that it's slightly more demanding than the others.

What this map does is take an element of the form $x+I+J/I$, where $x \in R$ and maps it to an element in $R/J$ of the form $x+J$, which is obvious, since we started with $x+I+J/I= x+I + j+I , \  \forall j \in J$

and since $I \subseteq J$, $j+I, \ \forall j \in J,$ belongs to $J$. Therefore, we can concisely write the $  J+I = J$.

Hence, $x+I+J/I = x+I + j+I= x+I+J= x+J $.

And our Fig. 2 commutes, which implies that $  \eta_I \circ \eta_{IJ}= \eta_J $ and $\eta_{IJ}= \tilde{\eta} \circ \eta_{R/ J}$

One may verify these.

Also see that we can combine the other two relations and write 

$ \eta_I \circ (\tilde{\eta} \circ \eta_{R/ J} )= \eta_J $

Verse III: Extension and Ideal Theorem~

If $I$ is a 2-sided ideal in $R$, then $I[X]$ is a 2-sided ideal of $R[X]$ and the quotient ring $R[X]/I[X]$ is naturally isomorphic to $(R/I)[X]$.

Fig. 3

The first part of the theorem can be proved by considering $I$ to be the 2-sided ideal and then we can proceed by writing $I[X]$ as $$ I[X]=\{  a_0 +a_1 X + a_2 X^2 + \cdots + a_r X^r , \ \forall \ a_i \in I \} $$

And, $I[X]$ is a 2-sided ideal in $R[X]$.

Now, consider the adjacent figure. 

Here, the maps $\gamma_{R/I}$ and $\gamma_{I[X]}$ are both natural surjections.

And our $\gamma_{R/I}$ is defined by 

$a_0 +a_1 X + a_2 X^2 + \cdots + a_r X^r  = \overline{a}_0 + \overline{a}_1 X + \overline{a}_2 X^2 + \cdots + \overline{a}_r X^r  $

where $\overline{a}= a+I, \ \forall \  a \in R$.

So, we are now concerned with our map $\tilde{\gamma}$ which takes objects from $R[X]/I[X] $ to an object in $(R/I)[X]$. Clearly, the objects of the domain are of the form 

$\tilde{\gamma}(a(X) + I[X]) = (a_0 + I) + (a_1 + I)X + \cdots + (a_n + I)X^n$

 and clearly, $(a_0 + I) + (a_1 + I)X + \cdots + (a_n + I)X^n \ \in (R/I)[X]$ and it's a surjection. every element of $(R/I)[X]$ has a pre-image in $R[X]/I[X] $.

Verse IV: Quotient maps~

Fig. 4

In all the above commutative diagrams, we obtain a figure of the adjacent form, and if all the maps in the figure, i.e., $\alpha, \beta$ and $\delta$ are onto, then the map $\alpha$ is said to factor through the object $C$ such that $ \alpha = \delta \circ \beta$ and hence in all the above figures, we obtain this universal property.

In general, a universal property describes how a particular object (such as a quotient ring, product, coproduct, or free object) is characterized by its relationships to other objects via maps that uniquely factor through it. 

Note that the diagram in Fig. 4 commutes.

Epilogue~

Algebra's real generosity is in its ability to unify results.

Many fundamental theorems, such as the isomorphism theorems for ring homomorphisms, become clear and accessible when approached through the lens of commutative diagrams and natural maps. These tools reveal underlying connections and provide a systematic framework for understanding why these results hold and how they fit together.

Reference- C. Musili, Rings and Modules.

On Siken and Being~

Richard Siken’s writing has become a touchstone, especially the book I recently devoured- The War of the Foxes. Because of the way he blurs certainty and leaves the reader suspended between clarity and disarray. His poems often weave images of intimacy with undercurrents of danger, collapsing the distinct between love and violence, desire and loss.
Inspired by him, I tried to write something, titled- Being and the Blur.

I took a deep breath and let it out.

What was exhaled was no more a breath,
instead an air that mixes with the atmosphere.
We call it life.

Something you give out is life.
Something you have to hold in is death.
And if you hold everything in,
maybe you are dead.

I too took my anger and created a demon out of it.
I call that demon an angel.
You look too long and the lines begin to blur.
What is a 'land' and what is a 'sea'?
Do you see?

The see glass is mint green.
Green- the color of jealousy they say.
I see trees of green and not an ounce of hatred.

We won't lead anywhere in a world like this.

~Purnima

In search of a nicer $G$...

Prologue~
“What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.”
—From Science et Méthode (1908), Livre Premier, Chapitre 2, p. 25; Translated in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method (1913), p. 372. by  Henri Poincaré

Verse I: What is a 'nicer' G?
So I have a group $G$- (say, a non-trivial one) which can either have a commutative or non-commutative structure, that is, be abelian or non-abelian. Now, abelian groups are very 'nice' to deal with- they make things easier for us, but non-abelian groups, not so much...
So how can one make things easier in a non-abelian group, or rather- how can we work in or deal with a 'nice' part of this non-abelian group which can make things (again, say-) 'nicer'? We explore this question in this post.

Verse II: The canvas~
Let $G$ be any non-trivial group, then if $a,b \in G$, then the commutator of $a$ and $b$ is the element $ab a^{-1} b^{-1}$.  
A good observation would be to see that if our $G$ is abelian, then our commutator is simply the identity element $e \in G$. Now, let us define $C$ to be the set\[ C =\{ x_1 x_2 x_3 \cdots x_n \ | \ n \geq 1, \text{each $x_i$ is a commutator in $G$} \} \] Basically, our $C$ here is the collection of all finite products of commutators in $G$. One can intuitively see that $C$ then will be a normal subgroup of $G$. \[ C \lhd G\] If not, maybe we can try thinking about it- as mentioned above, at least the identity element is a commutator and hence our $C$ is non-empty. And given any two elements in $C$, they are of the form $x_1 x_2 \cdots x_n$ and $y_1 y_2 \cdots y_n$ and since this is just a finite product of commutators, we also have their inverses existing, which then again belong to $C$. Now we can take an element $g \in G$ and apply the left conjugation action by $g$ on an element of $C$. As depicted below, we can play around!\[ g c g^{-1} = g x_1 x_2 \cdots x_n g^{-1} = (g x_1 g^{-1}) (g x_2 g^{-1}) \cdots g(x_n g^{-1}) \] Also notice that given any arbitrary element $c \in C$, we have, \[ (g c g^{-1})^{-1} = g c^{-1} g^{-1}\] and we obtain our equation back. Moreover, $g c g^{-1}$ was a product of commutators and hence it belongs in $C$.
$\therefore$ $\forall \ g \in G$ and $c \in C$, $gcg^{-1} \in C$, and hence, $C \lhd G$.
The normal subgroup $C$ of $G$ is  called the commutator subgroup or the derived subgroup of $G$ and is usually denoted by $C= G' = [G,G]$.
Also, if our $G$ is abelian (to begin with,) we then obtain $C=\{e\}$ and hence one can see the commutator subgroup as one measure of how far away a group is from being abelian.
To answer the question we began with, what does it mean to abelianize a group?

Verse III: Towards Abelianization~
What it intuitively means is that whenever $g,h \in G$, we must have $hg=gh$ with respect to the group operation (which has been suppressed here). But from the sketch of the proof done above, we realise that an abelian group has $C=\{e\}$ and therefore, in particular, we want every element of the form $ghg^{-1} h^{-1}$ to be the identity. And indeed, this is all that we need to do!
That is to say, if $gh g^{-1} h^{-1} = e, \ \forall g,h \in G$, then our $G$ is abelian!
We can see this as, \[ ghg^{-1}h^{-1}= e \Rightarrow ghg^{-1}= h  \Rightarrow gh =hg \]
What does this tell us though?
If we make every commutator of our group $G$ trivial, then we will abelianize $G$, and conversely, to make $G$ abelian, we have to make every commutator of our group $G$ trivial.
But, how do we do that?
We have a commutator subgroup $C$ which is normal in $G$. We can take the quotient of $G$ by $C$ (since it will give us a quotient group) and it makes good sense as the first step to play around!
Now, what we obtain will be a quotient group consisting of all the cosets of $C$ in $G$ and one can see that this very well could be (rather will be) the 'nicer', that is, the 'abelian version' of our group $G$ which we have been looking for. Now let's see if this works!
Consider the abelianization of $G$: \(G^{\mathrm{ab}} = G/[G,G]\)
Consider the quotient \[G^{\mathrm{ab}} \;:=\; G/[G,G]\] It is abelian because for any \(a,b \in G\), which is to say that, \[(a[G,G])\,(b[G,G]) \;=\; ab[G,G]\] while \[(b[G,G])\,(a[G,G]) \;=\; ba[G,G]\] But \(ab\) and \(ba\) differ by the commutator- what does that mean?
Well, since $ab= [a,b] (ba)$, the two cosets coincide. 
We have $[a,b] = aba^{-1} b^{-1}$ and if we multiply on the right  by $(ba)$, we obtain \[ [a,b] (ba)= aba^{-1}b^{-1} (ba) = ab a^{-1} (b^{-1} b) a = ab a^{-1} a = ab\] \[\therefore ab = [a,b](ba),\] so \[ab[G,G] = [G,G] (ba)\] since \([a,b]\in [G,G]\). Hence the cosets commute and \(G^{\mathrm{ab}}\) is abelian. Conversely, if every commutator is trivial in a quotient \(G/N\), then \([G,G]\subseteq N\). This gives the minimality of \([G,G]\): \([G,G]\) is the smallest normal subgroup of \(G\) with abelian quotient, i.e. for any normal \(N\trianglelefteq G\), \(G/N\) is abelian if and only if \([G,G]\subseteq N\).

Verse IV: Universality in Action~
Now, the abelianization of our group $G$ is not just an abelian quotient; it is the universal one.
If \(A\) is abelian and \(\varphi:G\to A\) is any homomorphism, then there exists a unique homomorphism \(\overline{\varphi}:G^{\mathrm{ab}}\to A\) such that \(\varphi = \overline{\varphi}\circ \pi\), where \(\pi:G\to G^{\mathrm{ab}}\) is the natural projection.
Why? Since \(A\) is abelian, \(\varphi([a,b])=e\) for all \(a,b\in G\), so \([G,G]\subseteq \ker\varphi\).
And by the First Isomorphism Theorem, \(\varphi\) factors uniquely through \(G/[G,G]\).
And guess what? This universal property is often the most efficient definition of abelianization.

Epilogue~
When we begin with a non-abelian group, we constantly worry about the turbulence of non-commutativity and what abelianization does for us is- it takes away that turbulence and offers us a 'nicer' environment to work with. 
The process gives a precise answer: we take the quotient by the commutator subgroup, leaving the 'largest' abelian image of the original group. This is not just convenient—it is universal. Any map from our group to an abelian group must factor uniquely through this construction. That is why abelianization matters: it formalizes the idea of extracting commutativity in the most general and economical way, without discarding more structure than necessary.

References- (1), (2).

In search of $G$...

Prologue~
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
— From A Mathematician’s Apology by G. H. Hardy

Verse I: The Problem~
An extension problem in group theory typically deals with construction of new groups with the already existing ones.
What that asks of you is to imagine two groups- $N$ and $Q$, and then search the group(s) (namely $G$) that would consist of $N$ at its core (as a normal subgroup) and $Q$ as a visible skeleton (as a quotient).
This problem can look something like:$$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1 $$
So, can we construct $G$ knowing only $Q$ and $N$?
Well, there is more to a sequence like that!

Verse II: The Rise~
Hurewicz introduced the idea of exact sequences in 1941 (without naming it) to describe boundary maps in cohomology. Kelley and Pitcher coined the term "exact sequence" in 1947, acknowledging its use by Eilenberg and Steenrod in earlier courses.
The  variations of the story surrounding the naming of the term, 'exact sequences' can be accessed here.
It seems rather funny to me that the term 'right' was actually 'exact'!  
Another variation can be accessed here.

Verse III: The sequence~
A split extension is an extension  $1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1$ with a homomorphism  $\phi : H \rightarrow G$, if the short exact sequence induces the identity map on $H$ i.e., $p \circ \phi = id_H$ where $p: G \rightarrow H$ is the projection . In this situation, it is usually said that $\phi$ splits the above exact sequence.

Verse IV: The Way~
So to tackle the problem, one approach could be (will be) to construct the group using direct products of $N$ and $Q$ (a trivial case, perhaps) and every element of $G$ is then simply represented as an ordered pair, where one element comes from $N$ and the other from $Q$. In such a case, we find $G$ to contain a copy of $N$ as well as $Q$!
$$G \cong N \times Q$$ And, nothing mystical really happens— instead the two groups coexist peacefully without entanglements (in a trivial sense, that is).
But are all cases trivial? Certainly no!
Consider a semidirect product, $$G \cong N \rtimes_\phi Q$$ Here, $Q$ no longer simply co-inhabits $G$ with $N$, instead it acts upon the automorphism group of $N$, and the fabric of our beloved group $G$ here, becomes richer in tapestry.
Is that where our thinking stops? Certainly no (again)!
There exist cases where $G$ cannot be expressed as a direct or a semidirect product of $N$ and $Q$ and those give rise to the non-split extensions. Which implies that even though $N$ and $Q$ are present in the groups, one cannot unseam the stitches of the fabric and obtain copies of them.
A small example would be to consider $N= \mathbb{Z}/ 2 \mathbb{Z}$ and $Q= \mathbb{Z}/ 2 \mathbb{Z}$. Now, constructing it's direct product, we obtain $\mathbb{Z}/ 2 \mathbb{Z} \times \mathbb{Z}/ 2 \mathbb{Z}$ which is the split extension and surprisingly enough, the semi direct product also leads us to the same group, which (voilà!) is isomorphic to the Klein 4 group. 

Verse V: The other Way~
But, there also exists a possible non-split extension- $G \cong \mathbb{Z} / 4 \mathbb{Z}$.
Then how do we know which one is our $G$? Because $\mathbb{Z} / 4 \mathbb{Z}$ is a non-split extension and hence cannot be written as a semidirect product.
Well, this ambiguity, with a small example underlines our extension problem which arises from this multiplicity of possibilities.

Verse VI: The 'exact'~
An exact sequence is a sequence of morphisms between objects (groups, rings, modules, objects of abelian categories, etc.) such that the image of a morphism equals the kernel of the next.   
And a sequence is short exact if in- $$ 0 \rightarrow A\xrightarrow{f} B \xrightarrow{g} C \rightarrow 0$$ our $f$ is injective and $g$ is surjective and $im (f)= ker (g)$.
Actually, short exact of complexes leads to a long exact sequence in homology (let's delve into them at a later time). 

Epilogue~
Mathematics often (but not always) begins with the concrete—numbers, shapes, groups—and then dissolves into patterns of relationships (our beloved categories).
The extension problem is a quiet metaphor for this: knowing the parts does not guarantee knowing the whole. We see $N$ and $Q$, yet $G$ resists complete revelation, for structure is more than just the sum. This is the quiet truth mathematics shares with art: form matters as much as substance.
Maybe structure is not an afterthought; but simply the essence.
It is indeed the pattern that breathes life into pieces, and perhaps this is why the language of exactness feels profound—it teaches us that understanding lies not in what we have, but in how what we have holds together.

The Mirror Law of Algebra

Prologue~
“What I have done... will not be lost. It is a study which must lie in wait for its time, perhaps for many years. The mathematical language I have created will someday be common property.”
— From Galois' final letter before his death

Verse I: The Ground~
Even before a formal proof was established for a very fundamental result that we so easily take for granted, algebraists observed empirically that polynomials with real coefficients had non-real roots occurring in pairs. This was a much under-appreciated founding listed in the works of Cardano, Euler and Lagrange.

The rigorous and structural explanation — that conjugate roots arise because of the action of a Galois automorphism — was made possible through the revolutionary framework introduced by Évariste Galois, whose ideas provided a structural approach to the roots of polynomials. These insights were later expanded and rigorously developed by modern algebraists like Dedekind, Kronecker, Artin, and Zariski.
What we encounter here is a proof of the statement.

Verse II: The statement and the proof~

Theorema
Let $f(x) \in \mathbb{R[x]}$. If $ \alpha \in \mathbb{C}$ is a root of $f$, then $\overline{\alpha}$  is also a root of $f$.

Probationem
To prove the above statement, let us rely a little on our morphism skills and try defining an automorphism from $\mathbb{C}$ to $\mathbb{C}$ such that: $$\phi : \mathbb{C} \rightarrow \mathbb{C}$$ $$ \phi(z)= \overline{z}$$ A few more properties of this map would then be:
1) $\phi(z+ w)= \phi(z) + \phi(w)$
2) $\phi(zw)=\phi(z) \phi(w)$
3) $\phi(1/z)= \frac{1}{ \phi(z)}$, for $z \neq 0$
4) $\phi(a)= a, \ \forall \ a \in \mathbb{R}$

Hence, for every $a$ in $\mathbb{R}$, the automorphism map $\phi$ fixes $a$.
Therefore, complex conjugation is a field automorphism of the field extension $\mathbb{C} / \mathbb{R}$ that fixes every element of $\mathbb{R}$.
We now take into consideration $\mathbb{R} \ \subset \mathbb{C}$ and a polynomial $P(z) \in \mathbb{R[x]}$ such that the coefficients are real.

$\Rightarrow P(z)= a_n z^n + a_{n-1} z^{n-1}+ \cdots + a_1 z + a_0$ , where $a_i \in \mathbb{R}$.

Now, let us consider some $\alpha \in \mathbb{C}$ as a root of $P(z)$, such that $P(\alpha)=0$.

If we attempt to apply the field automorphim to the polynomial $P(z)$ at $\alpha$, we obtain:
$$ \phi(P(\alpha))= \overline{P(\alpha)}= \overline{0}= 0$$

This implies that for any $\alpha \in \mathbb{C}, \ \overline{\alpha} \in \mathbb{C}$ is also a root of the polynomial $P(z)$.

Epilogue~
Upon deeper reflection, we realise that the Galois group of the field extension $\mathbb{C}/ \mathbb{R}$ is remarkably simple — it contains only two automorphisms: the identity map, which leaves every complex number unchanged, and the complex conjugation map $\phi$, which reflects each number across the real axis. These are the only field automorphisms of $\mathbb{C}$ that fix every element of $\mathbb{R}$. $$ Gal (\mathbb{C} / \mathbb{R}) = \{ id, \phi \}$$ This Galois group, though minimal in size, captures a powerful symmetry: the necessity that every non-real root of a real polynomial must be accompanied by its mirror image — its complex conjugate. The structure of the field dictates the symmetry of the solutions.
Thus, the pairing of complex roots in real polynomials is no coincidence — it is instead a reflection of deep algebraic symmetry. A symmetry so deeply etched into the soul of mathematics that it whispers through every structure it touches.

A Life so little...

"What he knew, he knew from books, and books lied, they made things prettier."

A recent book I completed was A Little Life—a novel that came out in 2015, gained immense critical and popular attention, both praise and criticism, and was later adapted into a play. I have since then wanted to put my thoughts into words. Here is a humble attempt at that.

"You don’t visit the lost; you visit the people who search for the lost."

The book begins with four characters—four friends in a shared apartment on Lispenard Street, whose lives have been intertwined since college. What unfurls is not just a story, but a life—so fragile and bruised that at some point, you may find yourself wanting to fight the author, to protect the characters not from an external villain, but from themselves.

The book has a non-linear narrative with jumps in timelines happening all around but in no way does it hamper the storyline, it instead makes it extremely interesting and a very grappling read. Hana Yanagihara (the author) paints scenes without initially naming characters, compelling the reader to piece together identities and the reader- almost in a frenzy tries to read as much as possible in order to connect the dots and uncover the truth. This task, rather than made irritating for the reader has been very carefully spaced throughout the novel, so instead of feeling frustrated at some point, the reader feels responsible for figuring out the character mentioned, in a veiled manner with utmost sincereity and urgency. 

Yanagihara handles time playfully- the flashbacks, fractured memories, and layered pasts flow seamlessly into the present. The resulting structure is not just unique, but haunting. The intentional disorientation of timelines and intersecting stories makes the reader more involved in the book. What results of such an unorthodox writing style is a gripping plot- well, not even a plot that one might see coming. And what it unearths is profoundly human.
I don't think that the writer overextends the portrayal of suffering, instead she creates whitespaces for the reader to fill in- things that cannot be spoken of. 

Yanagihara has said that A Little Life was the book she had been trying not to write her whole life. She described it as already existing, waiting to be transcribed. She envisioned it as an ombré cloth—beginning light blue and ending a deep indigo. When constructing Jude’s story, she had in mind this picture of a very light blue that shaded to a very dark indigo. which indeed reflects in Jude's descriptions throughout the book. 

The book starts with portraying Jude almost as a background character- somebody who would rather seep into the darkness and blur himself out rather than emerge at the surface as a prominent figure. But the novel takes a turn and starts revolving around Jude, but that does not mean that things start to look good for him, the readers instead find Jude questioning himself and life in general. He is of the opinion that maybe, happiness is simply not meant for him. With each passing year of his life, the events that have occured led him to believe that hummanness is something that he does not deserves.

Survival quietly stretches into prolonged suffering in certain lives- is one of the most subtle and devastating nuances Yanagihara offers us. In Jude’s case, survival is not a triumph—it becomes a slow, relentless state of endurance. His life stops being his own in many ways; it begins to orbit around the people who love him, around their desperate hope that their presence might redeem his past. This, too, is a kind of tragedy—the way a person, in surviving, begins to live not for themselves but for the sake of others’ love, others’ belief, others’ grief. There is something profoundly human in this—the tension between wanting to spare others pain and the unbearable weight of continuing for their sake. Jude does not seek attention or pity. Instead, his suffering becomes a silent, permanent background hum to his existence—so familiar that it feels inevitable. Yanagihara does not dramatize this; she lets it unfold with a quiet brutality that makes it all the more poignant.

For me, the protagonist is not just Jude, but in equal parts- Willem, JB, Malcom (the other three friends) and most importantly, Harold and Julia (Jude's adoptive parents)- the people who quietly shape the novel- appear when they must and as unnoticingly receed into the background but with a carfeul gaze, always on the outlook to support whenever needed, but not smother. They embody the kind of steadfast, unobtrusive support we all hope to have.

Personally, I suspected early on that Jude would die. That intuition was confirmed when the first mention of the excruciating pain exhalted by Jude's own body on him was mentioned, I knew that the book would end in him dying- of his own accord. The book spirals around this knowledge, but it never simplifies it.  However, as I progressed with the book, there were so many instances where love and presence come so close to eradicating the loneliness that Jude feels- in almost an overwhelming manner, but then the world is again, very quietly destroyed that it almost feels like coming home- as in the events were supposed to occur that way and they indeed would and there is nothing one can do about it. And yet, just as the reader begins to hope, the world collapses again—inevitably, almost naturally. The novel teaches you to expect this, to brace for impact.

There have been several instances throughout the book when my heart went out specifically to Jude, Willem, or Harold and I found myself sighing over what I have read- from helplessnes and from a sense of conviction that it was bound to happen- the world in the book (as it in reality does too) is created and destroyed over and over again. Rebuilt everytime with a new and fresh hope, however the destruction does not seem to break one either- it instead offers a quiet shoulder to lean on.
The cycles of destruction do not necessarily devastate; rather, they offer a strange kind of comfort. As Adrienne Rich once wrote: "There must be those among whom we can sit down and weep and still be counted as warriors." This is what Yanagihara offers her reader: a space to sit, weep, and survive.

"Every year, his right to humanness diminished."

The central theme, I believe, is not just trauma or survival, but the quiet, liminal space between life and death—what it means to live without ever truly feeling alive. For Jude, survival was rarely living. The living between life and death was often- if not always, a haze for him. The moments of kindness and compassion offered to him made him overwhelmingly grateful for his life- appearing as undeserved gifts, as though life were apologizing for all it had done to him. His life, which had been so unkind to him throughout that it is presenting him with a gift of compassion and love and camaraderie as a way of making up for its deeds in the past. But here is where the problem lies- the past is never the past for him. He always lives in it- almost constantly.
In the latter half of the novel his past and the three characters whom he remembers as life's protrayal of angelic, yet diabolical characteristsics- both personified, left an indelible mark on his life- Brother Luke, Dr. Traylor and Caleb. He remembers them not because they were the only ones who had been unkind to him but because they were the ones who had been kind but had used him- to an extent which made him believe that humanity is not something he deserves. 

The novel also explores love, friendship, shame, identity, and the terrifying fragility of self-worth. Love is not enough—Yanagihara makes this painfully clear. And healing is not always possible. Jude's death becomes something the reader, heartbreakingly, begins to wish for—not because he deserves to die, but because he does not deserve to suffer any longer.

At times, Jude’s self-harm becomes a habit more than an act of acute pain. But he cannot stop—because he was taught early on that harming himself was how to process feeling. The novel never excuses this, but it does explain it, and the reader watches helplessly as Jude spirals again and again.

The novel breaks convention by constructing a character who never "gets better." A Little Life from the get go was suppsoed to break the norms- to follow a reverse path, where Jude and the plot itself begins healthy (or at least appears to) and ends sick.

It is a portrait of loneliness—not isolation in the absence of people, but the deeper solitude of being surrounded and still untouched. Yanagihara describes the invisible line we draw between the desire not to be alone and the fear of letting others in. Jude walks that line his entire life.

Towards the end, Harold becomes the reader’s stand-in. He tries to keep Jude alive—through logic, guilt, love, even pleading. And yet, he must live with the unbearable knowledge that he cannot rescue someone who does not want to stay.

"You pounce upon the happy moments... you hold them up as proof... See? This is why it's worth living."

After Willem's death, Jude never made promises for the future- he did not wish to disappoint anyone as he felt his last breaths approaching- his patience with life breaking apart.

Harold eventually understands that keeping Jude alive was his own need, his own selfish desire. And Jude dies believing he is alone, surrounded by love but unable to feel it. That grief does not end with Jude’s death. It lives on in Harold, in Julia, in everyone who tried to hold him.

                 "It isn’t only that he died, or how he died; it is what he died believing.

And so I try to be kind to everything I see, and in everything I see, I see him."

In its final moments, the novel leaves us with questions not only about Jude but about ourselves:

"Why wasn’t friendship as good as a relationship? Why wasn’t it even better?"

In the play adaptation of the book, the directors made a striking choice: the same actor played the three characters who returned as hyenas in Jude’s memories. It is a chilling personification of how trauma collapses identities and blurs past and present.

A Little Life is not only a chronicle of Jude’s pain, but also of Harold’s and Willem’s—the pain of loving someone you cannot save. The book captures the quiet grief of walking beside someone you love, knowing they are slipping away.

A Little Life does not leave you. It lingers like an ache, like a bruise you can’t locate but feel every time you move. It does not offer healing in the traditional sense. It simply asks: What does it mean to keep living, even when life no longer makes sense?

Every quote included here is drawn from the pages of Hanya Yanagihara’s A Little Life itself.

Almost There!

Prologue~
“The heart asks pleasure first,
And then, excuse from pain”
~Emily Dickinson

In life some things are always close to being perfect. An almost lingers there- half a syllable away from being the perfect thing that can be. A heartbreak then seeps in...
Similarly, in mathematics we have these structures and objects which suffer a(n almost) heartbreak! Overqualified for certain things but underqualified for the others- lingering between a here and a there... an almost there!

Verse I~ A king with no field.
We have a Ring and a king who speaks 
But the subjects are civilians with unique voices, indeed. 
The king demands to bow before him,
Once under 1, these civilians will have kins
but the adamant civilians refuse to bow~
And hence remain kin-less in a field of doubt.
Not a field but more than a ring;
With no inverses in sight
The king then rules a Ring so kin-less...
What an autumnal plight!

The ring here is  $\mathbb{Z}$ which does not qualify for a field, simply because the multiplicative inverses do not exist. When one comes to ponder, one sympathises with the set of integers due to the lack of their kins. No hand in sight, and none to be called a friend.
There exists an identity, and two operations indeed, where one stands complete but the other- suffers defeat.

Verse II~ Exempla
We can also consider $2 \mathbb{Z}$ which is certainly a ring but lacks identity. Another almost there, another heartbreak...
For instance, the symmetric group $S_n$ which consists of all the possible permutations of $n$ elements of the underlying set. Essentially, this group is a set of maps. For $n \geq 3$, the group is non-abelian, but given any element in $S_n$, the element commutes with itself!
A set of introverts, communicating within themselves, while the world around doesn't reciprocate?
Also, consider the General Linear Group $GL_n$ which consists of all the invertible matrices of order $n \times n$ but fails to fulfil another essential condition and hence the group (again), suffers an unfortunate fate, but worse- it does not even form a ring! Since the sum of $n \times n$ invertible matrices might not be invertible.
Or perhaps, Cantor Set which is uncountable but has a zero (Lebesgue) measure! Or a Möbius strip which is locally orientable but globally not! 

Verse III~ Wabi-Sabi (Imperfect, Impermanent and Incomplete)
Mathematics, much like life, reveals to us the imperfections- a mirror in the face of adversity and self retrospection. A mirror that tells you of your imperfections, which might even mock you but to look at these in the light of optimisim, where the deeper truths reveal, is an art, and an essential one–very much so.
The set of integers is not a field but that flaw reveals to us the elegnace of irreducible wholes! The rationals- incomplete but adamant, force us to look into the expansive continuum of the reals.
A very fun example of the Möbius strip- an extremely fascinating surface, in refusal of being orientable, becomes something stranger and richer- alluring people towards its charms.

Epilogue~
Perhaps the cracks and the imperfections in the structures are simply a way to let the light in. 

"There is a crack in everything, that’s how the light gets in."
~ Leonard Cohen

Mathematics speaks to us of the ‘almost’s while teaching one to hover at the edge of revelation. The integers are not broken—they are unfinished. The Möbius strip is not flawed—it is unfathomable. And we, like Cauchy sequences in $\mathbb{Q}$, are always converging towards something, just beyond our grasp (for now).

Through three minds: The Permutation Representation Theorem

Verse I: The act, the play, the vision~

Cayley's Act:
Cayley's theorem- a profound result in the theory of groups was proven in his 1854 paper titled On The Theory of Groups, which states that 'every finite group is a subgroup of a symmetric group', however he considered only faithful actions on the group.

Jordan's Play:

Camille Jordan in his work Traité des Substitutions, formalized permutation groups and extended Cayley's ideas by closely observing non-faithful actions and thereby determining that homomorphisms need not be injective and hence Cayley's idea was then extended to actions on arbitrary sets.

Noether's Vision:
Emmy Noether- a name not unknown, worked futher on this idea and remoulded it in terms of abstract thinking. The theorem was then reframed as a universal property of group actions, much influenced by her axiomatic reasoning. 

Verse II: The statement and the proof~
Theorema
The actions of $G$ on a set $X$ are in fact, the group homomorphisms from $G$ to $Sym(X)$.

Probationem
Consider a group $G$ acting on a non-empty set $X$, and the group action $\phi$ is defined using the left action as, $\phi : G \times X \rightarrow X$  such that  $\phi (g,x)= g \cdot x$

Now, what this map does is ­­­­take an element from the set $X$ and map it to some element in this set itself .
This is simply a permutation on the set $X$ and hence, this map would then lie in $Sym(X).$
That is, $\phi \in Sym(X)$
Now, consider elements $g_1, g_2\in G$ then, $$\phi(g_1, \phi (g_2, x))= \phi(g_1, g_2 \cdot x)= g_1 \cdot (g_2 \cdot (x))= (g_1 \cdot g_2) \cdot (x)= \phi(g_1 g_2) \cdot (x)$$ and, $\phi (e, x)= e \cdot x= x$ (the identity map).
Now,  $\phi (g^{-1})$ exists, and the composition of $\phi(g)$ and $\phi(g^{-1})$ will be $\phi(e) .$
Hence, $g \mapsto \phi(g)$ is the homomorphism $G \rightarrow Sym(X)$.

Conversely, consider the homomorphism $f_g :G \rightarrow Sym(X)$ defined as $f_g (x)= g \cdot x$.

Here, $f_g$ is the permutation associated to $g$ under the homomorphism.
The stated map satisfies the axioms for the group actions, i.e., the existence of an identity and fulfillment of the associative property.

For the identity element $e \in G, \  e \cdot x = f_e (x) =e \cdot x= x$
(since $f$ is a homomorphism and must map the identity of $G$ to the identity permutation)

Now, for $g_1, g_2 \in G$ and $x \in X,$ 

$(g_1 g_2) \cdot x = f_{g_1 g_2}(x) = (f_{g_1} \circ f_{g_2}(x) )= f_{g_1}(f_{g_2}(x))
= g_1 \cdot (g_2 \cdot x)$

Here we have used the homomorphism property $\psi(g_1 g_2) = \psi(g_1) \circ \psi(g_2)$.

Thus, every homomorphism $f_g: G \to \text{Sym}(X)$ induces a valid group action.

According to this map, for each $g \in G$, we have a permutation $f_g \in Sym(X).$

We therefore conclude that the actions of a group on a set are essentially the group homomorphisms from the group to the group of permutations of the set.

Much obliged to K. Conrad's papers, which always come to the rescue.

On Dickinson and Glory~

This poem is inspired and birthed from Emily Dickinson's Fame is a Fickle Food. When reading her writings, one begins to explore the world of words and not simply words but the enjambment that characterises her writings. This was my humble take on her writing style which also does not shy away from using her analogy of a crow. So here we go with Glory.

Glory is an erratic dune-
Set upon the bank of endeavor And the beds of patience, Upon the grave of dread In my eyes~

Like fame, it's a fickle food

The crows inspect and fly.

But men eat of it and some savour;

While some meet fate and die.

~Purnima